Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 395628 14 pageshttpdxdoiorg1011552013395628

Research ArticleExact Traveling Wave Solutions fora Nonlinear Evolution Equation of GeneralizedTzitzeacuteica-Dodd-Bullough-Mikhailov Type

Weiguo Rui

College of Mathematics of Honghe University Mengzi Yunnan 661100 China

Correspondence should be addressed to Weiguo Rui weiguorhhuyahoocomcn

Received 18 March 2013 Accepted 9 May 2013

Academic Editor Shiping Lu

Copyright copy 2013 Weiguo Rui This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By using the integral bifurcation method a generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM) equation is studied Underdifferent parameters we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation Manysingular traveling wave solutions with blow-up form and broken form such as periodic blow-up wave solutions solitary wavesolutions of blow-up form broken solitary wave solutions broken kink wave solutions and some unboundary wave solutions areobtained In order to visually show dynamical behaviors of these exact solutions we plot graphs of profiles for some exact solutionsand discuss their dynamical properties

1 Introduction

In this paper we consider the following nonlinear evolutionequation

119906119909119905= 120572119890119898119906

+ 120573119890119899119906

(1)

where 120572 120573 are two non-zero real numbers and 119898 119899 are twointegers We call (1) generalized Tzitzeica-Dodd-Bullough-Mikhailov equation because it contains Tzitzeica equationDodd-Bullough-Mikhailov equation and Tzitzeica-Dodd-Bullough equation When 120572 = 1 120573 = minus1 119898 = 1 119899 = minus2

or 120572 = minus1 120573 = 1 119898 = minus2 119899 = 1 especially (1) becomesclassical Tzitzeica equation [1ndash3] as follows

119906119909119905= 119890119906

minus 119890minus2119906

(2)

which was originally found in the field of geometry in 1907by G Tzitzeica and appeared in the fields of mathematics andphysics alike Equation (2) usually called the ldquoDodd-Bulloughequationrdquo which was initiated by Bullough and Dodd [4] andZiber and Sabat [5] Indeed (2) has another form

119906119909119909minus 119906119905119905= 119890119906

minus 119890minus2119906

(3)

see [6 7] and the references cited therein

When 120572 = minus1 120573 = minus1 and 119898 = 1 119899 = minus2 (1) becomesDodd-Bullough-Mikhailov equation

119906119909119905+ 119890119906

+ 119890minus2119906

= 0 (4)

When 120572 = 1 120573 = 1 and119898 = 1 119899 = minus2 (1) becomes Tzitzeica-Dodd-Bullough equation

119906119909119905minus 119890119906

minus 119890minus2119906

= 0 (5)

The Dodd-Bullough-Mikhailov equation and Tzitzeica-Dodd-Bullough equation appeared in many problemsvarying from fluid flow to quantum field theory

Moreover when 119898 = 1 119899 = minus1 120572 = 12 120573 = minus12 or119898 = minus1 119899 = 1 120572 = minus12 120573 = 12 (1) becomes the sinh-Gordon equation

119906119909119905= sinh 119906 (6)

which was shown in [7ndash10] and the references cited thereinWhen 119898 = 1 119899 = minus1 120572 = 120573 = 12 or 119898 = minus1 119899 = 1120572 = 120573 = 12 (1) becomes the cosh-Gordon equation

119906119909119905= cosh 119906 (7)

2 Journal of Applied Mathematics

which was shown in [11 12] and the references cited thereinWhen 119898 = 1 119899 = 0 especially (1) becomes the Liouvilleequation

119906119909119905= 120572119890119906

(8)

All the equations mentioned above play very significant rolesin many scientific applications And some of them werestudied by many authors in recent years see the followingbrief statements

In [13] by using the tanh method Wazwaz consid-ered some solitary wave and periodic wave solutions forthe Dodd-Bullough-Mikhailov and Dodd-Bullough equa-tions In [14] Andreev studied the Backlund transformationfor Bullough-Dodd-Jiber-Shabat equation In [15] Cher-dantzev and Sharipov obtained finite-gap solutions of theBullough-Dodd-Jiber-Shabat equation In [16] Cherdantzevand Sharipov investigated solitons on the finite-gap back-ground in the Bullough-Dodd-Jiber-Shabat model In addi-tion the Darboux transformation self-dual Einstein spacesand consistency and general solution of Tzitzeica equationwere studied in [17ndash19]

In this paper by using integral bifurcation method [20]we will study (1) Indeed the integral bifurcation methodhas successfully been combined with the computer method[21] and some transformations [22 23] for investigating exacttraveling wave solutions of some nonlinear PDEs Thereforeusing this method we will obtain some new results which aredifferent from those in the above references

The rest of this paper is organized as follows in Section 2wewill derive two-dimensional planar systemwhich is equiv-alent to (1) and give its first integral equation In Sections 3and 4 by using the integral bifurcationmethodwewill obtainsome new traveling wave solutions of (1) and discuss theirdynamic properties

2 Two-Dimensional Planar Dynamical Systemof (1) and Its First Integral

Applying the transformations 119906(119909 119905) = ln |V(119909 119905)| we change(1) into the following form

VV119909119905minus V119909V119905= 120572V119898+2

+ 120573V119899+2

(9)

Let V(119909 119905) = 120601(119909 minus 119888119905) = 120601(120585) substituting 120601(119909 minus 119888119905) into (9)respectively we obtain

119888(1206011015840

)

2

minus 11988812060112060110158401015840

= 120572120601119898+2

+ 120573120601119899+2

(10)

where ldquo1015840rdquo is the derivative with respect to 120585 and 119888 is wavespeed

Clearly (10) is equivalent to the following two-dimensional systems

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 120572120601119898+2

minus 1205731206012+119899

119888120601

(11)

However when 120601 = 0 (11) is not equivalent to (10) and the119889119910119889120585 cannot be defined so we call the 120601 = 0 singular line

and call (11) singular system In order to obtain an equivalentsystem of (10) making a scalar transformation

119889120585 = 119888120601119889120591 (12)

equation (11) can be changed into a regular two-dimensionalsystem as follows

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601119898+2

minus 120573120601119899+2

(13)

Systems (11) and (13) are two integrable systems and theyhave the same first integral as follows

1199102

= ℎ1206012

minus

2120572

119888119898

120601119898+2

minus

2120573

119888119899

120601119899+2

(14)

where ℎ is integral constantSystems (11) and (13) are planar dynamical systems

defined by the 5-parameter (120572 120573 119888 119898 119899) Usually their firstintegral (14) can be rewritten as

119867(120601 119910) equiv

2120572

119888119898

120601119898

+

2120573

119888119899

120601119899

+

1199102

1206012= ℎ (15)

When 119899 minus 119898 = 2119896 minus 1 (119896 isin 119873) system (13) has twoequilibrium points 119874(0 0) and 119860((minus120572120573)1(119899minus119898) 0) in the 120601-axes When 119899 minus 119898 = 2119896 (119896 isin 119873) and 120572120573 lt 0 system (13) hasthree equilibrium point119874(0 0) and119861

12(plusmn(minus120572120573)

1(119899minus119898)

0) inthe 120601-axesWhen 119899minus119898 = 2119896 (119896 isin 119873) and 120572120573 gt 0 system (13)has only one equilibrium points 119874(0 0) When 119899 = 119898 = 119896(119896 isin 119873) system (13) has only one equilibrium point 119874(0 0)

Respectively substituting every equilibrium point (thepoint 119874(0 0) is exception) into (15) we have

ℎ119860= 119867((minus

120572

120573

)

1(119899minus119898)

0) =

2120572 (119899 minus 119898)

119888119898119899

(minus

120572

120573

)

119898(119899minus119898)

ℎ1198611= 119867 (119861

1) = 119867((minus

120572

120573

)

1(119899minus119898)

0)

=

2120572 (119899 minus 119898)

119888119898119899

(minus

120572

120573

)

119898(119899minus119898)

ℎ1198612= 119867 (119861

2) = 119867(minus(minus

120572

120573

)

1(119899minus119898)

0) = (minus1)119898

ℎ1198611

(16)

From the transformation 119906(119909 119905) = ln |120601(119909 minus 119888119905)| weknow that ln |120601(119909 minus 119888119905)| approaches to minusinfin if a solution120601(120585) approaches to 0 in (14) In other words this solutiondetermines an unboundary wave solution or blow-up wavesolution of (1) It is easy to see that the second equation ofsystem (11) is not continuous when 120601 = 0 In other wordson such straight line 120601 = 0 in the phase plane (120601 119910) thefunction 12060110158401015840 is not defined and this implies that the smoothwave solutions of (1) sometimes become nonsmooth wavesolutions

Journal of Applied Mathematics 3

3 Exact Solutions of (1) and Their Propertiesunder the Conditions 119899 = plusmn 119898 plusmn2119898 119898 isin Z+and 119898 = 2 119899 = minus1

In this section we investigate exact solutions of (1) underdifferent kinds of parametric conditions and their properties

31 Different Kinds of Solitary Wave Solutions and Unbound-ary Wave Solutions in the Special Cases 119899 = 2119898 isin Z+ or119899 = 119898 isin Z+ (i) When ℎ gt 0 120573119888 lt 0 119899 = 2119898 and 119898 iseven then (14) can be reduced to

119910 = plusmn120601radicℎ minus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(17)

Taking 1205850= 0 and 120601(0) = 120601

1as initial values substituting (17)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(18)

where 1206011= (minus120572120573 + radic(120572

2+ 120573119888119898ℎ)120573

2)1119898 Ω

1= 120601119898

1(1205722

+

120573119888119898ℎ) 1205961= minus119898radicℎ 120574

1= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

1) 1205751= 120572(119888119898ℎ minus

120572120601119898

1) By using program of Maple it is easy to validate that

(18) is the solution equation (9) when119898 is given Substituting(18) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtain asolitary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(19)

(ii) When 119899 = 119898 ℎ gt 0 119888(120572 + 120573) gt 0 and119898 is even (14)can be reduced to

119910 = plusmn120601radicℎ minus

2 (120572 + 120573)

119888119898

120601119898

(20)

Taking 1205850= 0 and 120601(0) = (119898ℎ1198882(120572 + 120573))

1119898 as initialvalues substituting (20) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = plusmn(119898ℎ119888 sech2120596

2(119909 minus 119888119905)

2 (120572 + 120573)

)

1119898

(21)

where 1205962= (12)119898radicℎ Substituting (21) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain a solution withoutboundary of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119898ℎ119888 sech21205962(119909 minus 119888119905)

2 (120572 + 120573)

100381610038161003816100381610038161003816100381610038161003816

(22)

Equation (21) is smooth solitary wave solution of (9) but (22)is an exact solution without boundary of (1) because 119906 rarr

minusinfin as V rarr 0 In order to visually show dynamical behaviorsof solutions V and 119906 of (21) and (22) we plot graphs of theirprofiles which are shown in Figures 1(a) and 1(b) respectively

(iii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 120572119888 lt 0 and 119898 is even(14) can be reduced to

119910 = plusmn120601radicminus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(23)

Taking 1205850= 0 and 120601(0) = 0 as initial values substituting (23)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(2 (minus120572119888)

120573119888 + 1198981205722(119909 minus 119888119905)

2)

1119898

(24)

Substituting (24) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain an exact solutions of rational function type of (1)as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

2120572119888

120573119888 + 1198981205722(119909 minus 119888119905)

2

100381610038161003816100381610038161003816100381610038161003816

(25)

(iv) When 119899 = 2119898 ℎ gt 0 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(26)

Respectively taking 1205850= 0 120601(0) = 120601

1 and 120585

0= 0 120601(0) = 120601

2

as initial values substituting (26) into the 119889120601119889120585 = 119910 of (11)and then integrating them yield

V (119909 119905) = (1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(27)

V (119909 119905) = (1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

)

1119898

(28)

where 1206011

is given above and 1206012

=

(minus120572120573 minus radic(1205722+ 120573119888119898ℎ)120573

2)

1119898

Ω2

= 120601119898

2(1205722

+ 120573119888119898ℎ)1205962= 119898radicℎ 120574

2= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

2) 1205752= 120572(119888119898ℎ minus 120572120601

119898

2)

Respectively substituting (27) and (28) into thetransformation 119906(119909 119905) = ln |V(119909 119905)| we obtain two smoothunboundary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(29)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

100381610038161003816100381610038161003816100381610038161003816

(30)

(v) When 119899 = 2119898 0 lt ℎ lt ℎ119860 120572120573 lt 0 120573119888 lt 0 (or 120572120573 gt 0

120573119888 lt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradicminus

120573

119888119898

120601radicminus

119888119898ℎ

120573

+

2120572

120573

120601119898+ (120601119898)2

(31)

Similarly by using (31) we obtain two smooth solitary wavesolutions of (1) which are the same as the solutions (29) and(30)

4 Journal of Applied Mathematics

minus2 minus1 1 2 3x

0

05

1

15

2

(a) Smooth solitary wave

02

04

06

minus02 minus01 01 02 03 04 05 06x

u

(b) Unboundary wave

Figure 1 The phase portraits of solutions (21) and (22) for119898 = 4 120572 = 3 120573 = minus1 119888 = 2 ℎ = 9 119905 = 01

(vi)When 119899 = 119898 ℎ gt 0 and119898 is odd (14) can be reducedto (20) so the obtained solution is as the same as the solution(22)

(vii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 and 119898 is odd (14) canbe reduced to

119910 = plusmnradic120573

119888119898

120601radicminus

2120572

120573

120601119898minus (120601119898)2

(32)

Taking 1205850= 0 and 120601(0) = (minus2120572120573)

1119898 as initial valuessubstituting (32) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = (minus2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

)

1119898

(33)

Substituting (33) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain a smooth unboundary wave solution of rationalfunction type of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816100381610038161003816

2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

(34)

32 Periodic Blow-UpWave Solutions Broken KinkWave andAntikink Wave Solutions in the Special Case 119899 = 2119898 isin Z+ (1)When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ

1198611lt ℎ lt 0 (or ℎ

1198612lt ℎ lt 0)

and119898 is even (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(35)

Taking 1205850= 0 120601(0) = (minus120572120573 + radic120572

21205732+ 119898ℎ119888120573)

1119898

and

1205850= 0 120601(0) = (minus120572120573 minus radic120572

21205732+ 119898ℎ119888120573)

1119898

as the initial

values substituting (35) into the 119889120601119889120585 = 119910 of (11) and thenintegrating them yield

V (119909 119905) = (119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

)

1119898

(36)

V (119909 119905) = minus(119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

)

1119898

(37)

where Ω3

= radic(1205722+ 119888119898ℎ120573)120573

2 1205963

= 119898radicminusℎ 1198631

=

(1205722

+ 120572120573Ω3+ 120573119888119898ℎ)120573Ω

3(120572 + 120573Ω

3) 1198632= (1205722

minus 120572120573Ω3+

120573119888119898ℎ)120573Ω3(minus120572 + 120573Ω

3) Respectively substituting (36) and

(37) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtaintwo periodic blow-up wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(38)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(39)

(2) When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ1198612lt ℎ lt 0 and 119898 is

odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(40)

which is the same as (35) so the obtained solution is the sameas solution (39) Similarly when 119899 = 2119898 120572120573 lt 0 120573119888 gt 0

Journal of Applied Mathematics 5

ℎ1198611lt ℎ lt 0 and 119898 is odd the obtained solution is also the

same as solution (38)(3) when 119899 = 2119898 120572120573 lt 0 120573119888 lt 0 ℎ = ℎ

1198611= ℎ1198612=

minus1205722

120573119888119898 and119898 is even (14) can be reduced to

119910 = plusmnradicminus

1

119888119898120573

120601 (120573120601119898

+ 120572) (41)

Substituting (41) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it and setting the integral constants as zero yield

V (119909 119905) = (minus120572

2120573

(1 plusmn tanh Ω42

(119909 minus 119888119905)))

1119898

(42)

where Ω4= 120572radicminus119898119888120573 Substituting (42) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain an unboundary wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816

120572

2120573

[1 plusmn tanh Ω42

(119909 minus 119888119905)]

10038161003816100381610038161003816100381610038161003816

(43)

Equation (42) is smooth kink wave solution of (9) but (43) isa broken kink wave solution without boundary of (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (42) and (43) we plot graphsof their profiles which are shown in Figures 2(a) and 2(b)respectively

(4) When 119899 = 2119898 120572120573 gt 0 120573119888 lt 0 ℎ = ℎ119860or 120572120573 lt 0

120573119888 lt 0 ℎ = ℎ119860and 119898 is odd (14) can be reduced to (41) So

the obtained solution is also the same as the solution (43)

33 Periodic Blow-Up Wave Solutions and Solitary WaveSolutions of Blow-Up Form in the Special Case119898 = 2 119899 = minus1When 119899 = minus1 (14) can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

2120573

119888

120601 (44)

(a) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (120601 minus 0) (120601 minus 120574

1) (120601 minus 120575

1) 120601 isin (0 120599

1]

(45)

where 1205991 1205741 1205751are three roots of equation 2120573120572 + (119888ℎ120572)120601 minus

1206013 and 120599

1gt 0 gt 120574

1gt 1205751 These three roots 120599

1 1205741 1205751can

be obtained by Cardano formula as long as the parameters120572 120573 119888 ℎ are fixed (or given) concretely For example taking120572 = 40 120573 = 20 119888 = 40 ℎ = 90 the 120599

1≐ 3054084215

1205741≐ minus01112641576 120575

1≐ minus2942820058 Taking (0 120599

1) as

the initial values of the variables (120585 120601) substituting (45) intothe 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =1205991(minus1205751) cn2 (Ω

1(120585) 1198961)

(minus1205751) + 1205991sn2 (Ω

1120585 1198961)

120585 = 1198791

(46)

where Ω1

= (12)radicminus1205721205751(1205991minus 1205751)119888 119896

1=

radic1205991(1205741minus 1205751) minus 120575

1(1205991minus 1205741) 1198791

= (1Ω1)snminus1(1 119896

1) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (46) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205991(minus1205751) cn2 (Ω

1(119909minus119888119905) 119896

1)

(minus1205751)+1205991sn2 (Ω

1(119909minus119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 =1198791

(47)

(b) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (0 minus 120601) (120574

1minus 120601) (120601 minus 120575

1) 120601isin[120575

1 1205741]

(48)

Similarly taking (0 1205751) as the initial values of the variables

(120585 120601) substituting (48) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1120585 1198961)

(49)

By using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (49) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1(119909 minus 119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

(50)

(c) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120578

2minus 120601) (0 minus 120601) (120601 minus 120575

2) 120601 isin [120575

2 0)

(51)

where 1205992 1205782 0 1205752are roots of equation 120601(2120573120572 + (119888ℎ120572)120601 minus

1206013

) = 0 and 1205992gt 1205782gt 0 gt 120575

2 As in the above cases these

three roots can be obtained by Cardano formula as long asthe parameters 120572 120573 119888 ℎ are given concretely so the similarcases will be not commentated anymore in the followingdiscussions Similarly taking (0 120575

2) as the initial values of the

variables (120585 120601) substituting (51) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) =12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 120575

2sn2 (Ω

2120585 1198962)

120585 = 1198792

(52)

where Ω2

= (12)radic1205721205992(1205782minus 1205752)119888 119896

2=

radicminus1205752(1205992minus 1205782)1205992(1205782minus 1205752) 1198792

= (1Ω2)snminus1(0 119896

2) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (52) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 1205752sn2 (Ω

2120585 1198962)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198792

(53)

(d) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120601 minus 120578

2) (120601 minus 0) (120601 minus 120575

2) (54)

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

which was shown in [11 12] and the references cited thereinWhen 119898 = 1 119899 = 0 especially (1) becomes the Liouvilleequation

119906119909119905= 120572119890119906

(8)

All the equations mentioned above play very significant rolesin many scientific applications And some of them werestudied by many authors in recent years see the followingbrief statements

In [13] by using the tanh method Wazwaz consid-ered some solitary wave and periodic wave solutions forthe Dodd-Bullough-Mikhailov and Dodd-Bullough equa-tions In [14] Andreev studied the Backlund transformationfor Bullough-Dodd-Jiber-Shabat equation In [15] Cher-dantzev and Sharipov obtained finite-gap solutions of theBullough-Dodd-Jiber-Shabat equation In [16] Cherdantzevand Sharipov investigated solitons on the finite-gap back-ground in the Bullough-Dodd-Jiber-Shabat model In addi-tion the Darboux transformation self-dual Einstein spacesand consistency and general solution of Tzitzeica equationwere studied in [17ndash19]

In this paper by using integral bifurcation method [20]we will study (1) Indeed the integral bifurcation methodhas successfully been combined with the computer method[21] and some transformations [22 23] for investigating exacttraveling wave solutions of some nonlinear PDEs Thereforeusing this method we will obtain some new results which aredifferent from those in the above references

The rest of this paper is organized as follows in Section 2wewill derive two-dimensional planar systemwhich is equiv-alent to (1) and give its first integral equation In Sections 3and 4 by using the integral bifurcationmethodwewill obtainsome new traveling wave solutions of (1) and discuss theirdynamic properties

2 Two-Dimensional Planar Dynamical Systemof (1) and Its First Integral

Applying the transformations 119906(119909 119905) = ln |V(119909 119905)| we change(1) into the following form

VV119909119905minus V119909V119905= 120572V119898+2

+ 120573V119899+2

(9)

Let V(119909 119905) = 120601(119909 minus 119888119905) = 120601(120585) substituting 120601(119909 minus 119888119905) into (9)respectively we obtain

119888(1206011015840

)

2

minus 11988812060112060110158401015840

= 120572120601119898+2

+ 120573120601119899+2

(10)

where ldquo1015840rdquo is the derivative with respect to 120585 and 119888 is wavespeed

Clearly (10) is equivalent to the following two-dimensional systems

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 120572120601119898+2

minus 1205731206012+119899

119888120601

(11)

However when 120601 = 0 (11) is not equivalent to (10) and the119889119910119889120585 cannot be defined so we call the 120601 = 0 singular line

and call (11) singular system In order to obtain an equivalentsystem of (10) making a scalar transformation

119889120585 = 119888120601119889120591 (12)

equation (11) can be changed into a regular two-dimensionalsystem as follows

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601119898+2

minus 120573120601119899+2

(13)

Systems (11) and (13) are two integrable systems and theyhave the same first integral as follows

1199102

= ℎ1206012

minus

2120572

119888119898

120601119898+2

minus

2120573

119888119899

120601119899+2

(14)

where ℎ is integral constantSystems (11) and (13) are planar dynamical systems

defined by the 5-parameter (120572 120573 119888 119898 119899) Usually their firstintegral (14) can be rewritten as

119867(120601 119910) equiv

2120572

119888119898

120601119898

+

2120573

119888119899

120601119899

+

1199102

1206012= ℎ (15)

When 119899 minus 119898 = 2119896 minus 1 (119896 isin 119873) system (13) has twoequilibrium points 119874(0 0) and 119860((minus120572120573)1(119899minus119898) 0) in the 120601-axes When 119899 minus 119898 = 2119896 (119896 isin 119873) and 120572120573 lt 0 system (13) hasthree equilibrium point119874(0 0) and119861

12(plusmn(minus120572120573)

1(119899minus119898)

0) inthe 120601-axesWhen 119899minus119898 = 2119896 (119896 isin 119873) and 120572120573 gt 0 system (13)has only one equilibrium points 119874(0 0) When 119899 = 119898 = 119896(119896 isin 119873) system (13) has only one equilibrium point 119874(0 0)

Respectively substituting every equilibrium point (thepoint 119874(0 0) is exception) into (15) we have

ℎ119860= 119867((minus

120572

120573

)

1(119899minus119898)

0) =

2120572 (119899 minus 119898)

119888119898119899

(minus

120572

120573

)

119898(119899minus119898)

ℎ1198611= 119867 (119861

1) = 119867((minus

120572

120573

)

1(119899minus119898)

0)

=

2120572 (119899 minus 119898)

119888119898119899

(minus

120572

120573

)

119898(119899minus119898)

ℎ1198612= 119867 (119861

2) = 119867(minus(minus

120572

120573

)

1(119899minus119898)

0) = (minus1)119898

ℎ1198611

(16)

From the transformation 119906(119909 119905) = ln |120601(119909 minus 119888119905)| weknow that ln |120601(119909 minus 119888119905)| approaches to minusinfin if a solution120601(120585) approaches to 0 in (14) In other words this solutiondetermines an unboundary wave solution or blow-up wavesolution of (1) It is easy to see that the second equation ofsystem (11) is not continuous when 120601 = 0 In other wordson such straight line 120601 = 0 in the phase plane (120601 119910) thefunction 12060110158401015840 is not defined and this implies that the smoothwave solutions of (1) sometimes become nonsmooth wavesolutions

Journal of Applied Mathematics 3

3 Exact Solutions of (1) and Their Propertiesunder the Conditions 119899 = plusmn 119898 plusmn2119898 119898 isin Z+and 119898 = 2 119899 = minus1

In this section we investigate exact solutions of (1) underdifferent kinds of parametric conditions and their properties

31 Different Kinds of Solitary Wave Solutions and Unbound-ary Wave Solutions in the Special Cases 119899 = 2119898 isin Z+ or119899 = 119898 isin Z+ (i) When ℎ gt 0 120573119888 lt 0 119899 = 2119898 and 119898 iseven then (14) can be reduced to

119910 = plusmn120601radicℎ minus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(17)

Taking 1205850= 0 and 120601(0) = 120601

1as initial values substituting (17)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(18)

where 1206011= (minus120572120573 + radic(120572

2+ 120573119888119898ℎ)120573

2)1119898 Ω

1= 120601119898

1(1205722

+

120573119888119898ℎ) 1205961= minus119898radicℎ 120574

1= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

1) 1205751= 120572(119888119898ℎ minus

120572120601119898

1) By using program of Maple it is easy to validate that

(18) is the solution equation (9) when119898 is given Substituting(18) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtain asolitary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(19)

(ii) When 119899 = 119898 ℎ gt 0 119888(120572 + 120573) gt 0 and119898 is even (14)can be reduced to

119910 = plusmn120601radicℎ minus

2 (120572 + 120573)

119888119898

120601119898

(20)

Taking 1205850= 0 and 120601(0) = (119898ℎ1198882(120572 + 120573))

1119898 as initialvalues substituting (20) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = plusmn(119898ℎ119888 sech2120596

2(119909 minus 119888119905)

2 (120572 + 120573)

)

1119898

(21)

where 1205962= (12)119898radicℎ Substituting (21) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain a solution withoutboundary of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119898ℎ119888 sech21205962(119909 minus 119888119905)

2 (120572 + 120573)

100381610038161003816100381610038161003816100381610038161003816

(22)

Equation (21) is smooth solitary wave solution of (9) but (22)is an exact solution without boundary of (1) because 119906 rarr

minusinfin as V rarr 0 In order to visually show dynamical behaviorsof solutions V and 119906 of (21) and (22) we plot graphs of theirprofiles which are shown in Figures 1(a) and 1(b) respectively

(iii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 120572119888 lt 0 and 119898 is even(14) can be reduced to

119910 = plusmn120601radicminus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(23)

Taking 1205850= 0 and 120601(0) = 0 as initial values substituting (23)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(2 (minus120572119888)

120573119888 + 1198981205722(119909 minus 119888119905)

2)

1119898

(24)

Substituting (24) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain an exact solutions of rational function type of (1)as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

2120572119888

120573119888 + 1198981205722(119909 minus 119888119905)

2

100381610038161003816100381610038161003816100381610038161003816

(25)

(iv) When 119899 = 2119898 ℎ gt 0 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(26)

Respectively taking 1205850= 0 120601(0) = 120601

1 and 120585

0= 0 120601(0) = 120601

2

as initial values substituting (26) into the 119889120601119889120585 = 119910 of (11)and then integrating them yield

V (119909 119905) = (1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(27)

V (119909 119905) = (1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

)

1119898

(28)

where 1206011

is given above and 1206012

=

(minus120572120573 minus radic(1205722+ 120573119888119898ℎ)120573

2)

1119898

Ω2

= 120601119898

2(1205722

+ 120573119888119898ℎ)1205962= 119898radicℎ 120574

2= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

2) 1205752= 120572(119888119898ℎ minus 120572120601

119898

2)

Respectively substituting (27) and (28) into thetransformation 119906(119909 119905) = ln |V(119909 119905)| we obtain two smoothunboundary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(29)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

100381610038161003816100381610038161003816100381610038161003816

(30)

(v) When 119899 = 2119898 0 lt ℎ lt ℎ119860 120572120573 lt 0 120573119888 lt 0 (or 120572120573 gt 0

120573119888 lt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradicminus

120573

119888119898

120601radicminus

119888119898ℎ

120573

+

2120572

120573

120601119898+ (120601119898)2

(31)

Similarly by using (31) we obtain two smooth solitary wavesolutions of (1) which are the same as the solutions (29) and(30)

4 Journal of Applied Mathematics

minus2 minus1 1 2 3x

0

05

1

15

2

(a) Smooth solitary wave

02

04

06

minus02 minus01 01 02 03 04 05 06x

u

(b) Unboundary wave

Figure 1 The phase portraits of solutions (21) and (22) for119898 = 4 120572 = 3 120573 = minus1 119888 = 2 ℎ = 9 119905 = 01

(vi)When 119899 = 119898 ℎ gt 0 and119898 is odd (14) can be reducedto (20) so the obtained solution is as the same as the solution(22)

(vii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 and 119898 is odd (14) canbe reduced to

119910 = plusmnradic120573

119888119898

120601radicminus

2120572

120573

120601119898minus (120601119898)2

(32)

Taking 1205850= 0 and 120601(0) = (minus2120572120573)

1119898 as initial valuessubstituting (32) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = (minus2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

)

1119898

(33)

Substituting (33) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain a smooth unboundary wave solution of rationalfunction type of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816100381610038161003816

2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

(34)

32 Periodic Blow-UpWave Solutions Broken KinkWave andAntikink Wave Solutions in the Special Case 119899 = 2119898 isin Z+ (1)When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ

1198611lt ℎ lt 0 (or ℎ

1198612lt ℎ lt 0)

and119898 is even (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(35)

Taking 1205850= 0 120601(0) = (minus120572120573 + radic120572

21205732+ 119898ℎ119888120573)

1119898

and

1205850= 0 120601(0) = (minus120572120573 minus radic120572

21205732+ 119898ℎ119888120573)

1119898

as the initial

values substituting (35) into the 119889120601119889120585 = 119910 of (11) and thenintegrating them yield

V (119909 119905) = (119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

)

1119898

(36)

V (119909 119905) = minus(119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

)

1119898

(37)

where Ω3

= radic(1205722+ 119888119898ℎ120573)120573

2 1205963

= 119898radicminusℎ 1198631

=

(1205722

+ 120572120573Ω3+ 120573119888119898ℎ)120573Ω

3(120572 + 120573Ω

3) 1198632= (1205722

minus 120572120573Ω3+

120573119888119898ℎ)120573Ω3(minus120572 + 120573Ω

3) Respectively substituting (36) and

(37) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtaintwo periodic blow-up wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(38)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(39)

(2) When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ1198612lt ℎ lt 0 and 119898 is

odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(40)

which is the same as (35) so the obtained solution is the sameas solution (39) Similarly when 119899 = 2119898 120572120573 lt 0 120573119888 gt 0

Journal of Applied Mathematics 5

ℎ1198611lt ℎ lt 0 and 119898 is odd the obtained solution is also the

same as solution (38)(3) when 119899 = 2119898 120572120573 lt 0 120573119888 lt 0 ℎ = ℎ

1198611= ℎ1198612=

minus1205722

120573119888119898 and119898 is even (14) can be reduced to

119910 = plusmnradicminus

1

119888119898120573

120601 (120573120601119898

+ 120572) (41)

Substituting (41) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it and setting the integral constants as zero yield

V (119909 119905) = (minus120572

2120573

(1 plusmn tanh Ω42

(119909 minus 119888119905)))

1119898

(42)

where Ω4= 120572radicminus119898119888120573 Substituting (42) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain an unboundary wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816

120572

2120573

[1 plusmn tanh Ω42

(119909 minus 119888119905)]

10038161003816100381610038161003816100381610038161003816

(43)

Equation (42) is smooth kink wave solution of (9) but (43) isa broken kink wave solution without boundary of (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (42) and (43) we plot graphsof their profiles which are shown in Figures 2(a) and 2(b)respectively

(4) When 119899 = 2119898 120572120573 gt 0 120573119888 lt 0 ℎ = ℎ119860or 120572120573 lt 0

120573119888 lt 0 ℎ = ℎ119860and 119898 is odd (14) can be reduced to (41) So

the obtained solution is also the same as the solution (43)

33 Periodic Blow-Up Wave Solutions and Solitary WaveSolutions of Blow-Up Form in the Special Case119898 = 2 119899 = minus1When 119899 = minus1 (14) can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

2120573

119888

120601 (44)

(a) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (120601 minus 0) (120601 minus 120574

1) (120601 minus 120575

1) 120601 isin (0 120599

1]

(45)

where 1205991 1205741 1205751are three roots of equation 2120573120572 + (119888ℎ120572)120601 minus

1206013 and 120599

1gt 0 gt 120574

1gt 1205751 These three roots 120599

1 1205741 1205751can

be obtained by Cardano formula as long as the parameters120572 120573 119888 ℎ are fixed (or given) concretely For example taking120572 = 40 120573 = 20 119888 = 40 ℎ = 90 the 120599

1≐ 3054084215

1205741≐ minus01112641576 120575

1≐ minus2942820058 Taking (0 120599

1) as

the initial values of the variables (120585 120601) substituting (45) intothe 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =1205991(minus1205751) cn2 (Ω

1(120585) 1198961)

(minus1205751) + 1205991sn2 (Ω

1120585 1198961)

120585 = 1198791

(46)

where Ω1

= (12)radicminus1205721205751(1205991minus 1205751)119888 119896

1=

radic1205991(1205741minus 1205751) minus 120575

1(1205991minus 1205741) 1198791

= (1Ω1)snminus1(1 119896

1) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (46) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205991(minus1205751) cn2 (Ω

1(119909minus119888119905) 119896

1)

(minus1205751)+1205991sn2 (Ω

1(119909minus119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 =1198791

(47)

(b) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (0 minus 120601) (120574

1minus 120601) (120601 minus 120575

1) 120601isin[120575

1 1205741]

(48)

Similarly taking (0 1205751) as the initial values of the variables

(120585 120601) substituting (48) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1120585 1198961)

(49)

By using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (49) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1(119909 minus 119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

(50)

(c) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120578

2minus 120601) (0 minus 120601) (120601 minus 120575

2) 120601 isin [120575

2 0)

(51)

where 1205992 1205782 0 1205752are roots of equation 120601(2120573120572 + (119888ℎ120572)120601 minus

1206013

) = 0 and 1205992gt 1205782gt 0 gt 120575

2 As in the above cases these

three roots can be obtained by Cardano formula as long asthe parameters 120572 120573 119888 ℎ are given concretely so the similarcases will be not commentated anymore in the followingdiscussions Similarly taking (0 120575

2) as the initial values of the

variables (120585 120601) substituting (51) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) =12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 120575

2sn2 (Ω

2120585 1198962)

120585 = 1198792

(52)

where Ω2

= (12)radic1205721205992(1205782minus 1205752)119888 119896

2=

radicminus1205752(1205992minus 1205782)1205992(1205782minus 1205752) 1198792

= (1Ω2)snminus1(0 119896

2) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (52) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 1205752sn2 (Ω

2120585 1198962)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198792

(53)

(d) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120601 minus 120578

2) (120601 minus 0) (120601 minus 120575

2) (54)

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

3 Exact Solutions of (1) and Their Propertiesunder the Conditions 119899 = plusmn 119898 plusmn2119898 119898 isin Z+and 119898 = 2 119899 = minus1

In this section we investigate exact solutions of (1) underdifferent kinds of parametric conditions and their properties

31 Different Kinds of Solitary Wave Solutions and Unbound-ary Wave Solutions in the Special Cases 119899 = 2119898 isin Z+ or119899 = 119898 isin Z+ (i) When ℎ gt 0 120573119888 lt 0 119899 = 2119898 and 119898 iseven then (14) can be reduced to

119910 = plusmn120601radicℎ minus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(17)

Taking 1205850= 0 and 120601(0) = 120601

1as initial values substituting (17)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(18)

where 1206011= (minus120572120573 + radic(120572

2+ 120573119888119898ℎ)120573

2)1119898 Ω

1= 120601119898

1(1205722

+

120573119888119898ℎ) 1205961= minus119898radicℎ 120574

1= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

1) 1205751= 120572(119888119898ℎ minus

120572120601119898

1) By using program of Maple it is easy to validate that

(18) is the solution equation (9) when119898 is given Substituting(18) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtain asolitary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(19)

(ii) When 119899 = 119898 ℎ gt 0 119888(120572 + 120573) gt 0 and119898 is even (14)can be reduced to

119910 = plusmn120601radicℎ minus

2 (120572 + 120573)

119888119898

120601119898

(20)

Taking 1205850= 0 and 120601(0) = (119898ℎ1198882(120572 + 120573))

1119898 as initialvalues substituting (20) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = plusmn(119898ℎ119888 sech2120596

2(119909 minus 119888119905)

2 (120572 + 120573)

)

1119898

(21)

where 1205962= (12)119898radicℎ Substituting (21) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain a solution withoutboundary of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119898ℎ119888 sech21205962(119909 minus 119888119905)

2 (120572 + 120573)

100381610038161003816100381610038161003816100381610038161003816

(22)

Equation (21) is smooth solitary wave solution of (9) but (22)is an exact solution without boundary of (1) because 119906 rarr

minusinfin as V rarr 0 In order to visually show dynamical behaviorsof solutions V and 119906 of (21) and (22) we plot graphs of theirprofiles which are shown in Figures 1(a) and 1(b) respectively

(iii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 120572119888 lt 0 and 119898 is even(14) can be reduced to

119910 = plusmn120601radicminus

2120572

119888119898

120601119898minus

120573

119888119898

(120601119898)2

(23)

Taking 1205850= 0 and 120601(0) = 0 as initial values substituting (23)

into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = plusmn(2 (minus120572119888)

120573119888 + 1198981205722(119909 minus 119888119905)

2)

1119898

(24)

Substituting (24) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain an exact solutions of rational function type of (1)as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

2120572119888

120573119888 + 1198981205722(119909 minus 119888119905)

2

100381610038161003816100381610038161003816100381610038161003816

(25)

(iv) When 119899 = 2119898 ℎ gt 0 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(26)

Respectively taking 1205850= 0 120601(0) = 120601

1 and 120585

0= 0 120601(0) = 120601

2

as initial values substituting (26) into the 119889120601119889120585 = 119910 of (11)and then integrating them yield

V (119909 119905) = (1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

)

1119898

(27)

V (119909 119905) = (1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

)

1119898

(28)

where 1206011

is given above and 1206012

=

(minus120572120573 minus radic(1205722+ 120573119888119898ℎ)120573

2)

1119898

Ω2

= 120601119898

2(1205722

+ 120573119888119898ℎ)1205962= 119898radicℎ 120574

2= 119888119898ℎ(119888119898ℎ minus 120572120601

119898

2) 1205752= 120572(119888119898ℎ minus 120572120601

119898

2)

Respectively substituting (27) and (28) into thetransformation 119906(119909 119905) = ln |V(119909 119905)| we obtain two smoothunboundary wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205741

Ω1cosh (120596

1(119909 minus 119888119905)) + 120575

1

100381610038161003816100381610038161003816100381610038161003816

(29)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

1205742

Ω2cosh (120596

2(119909 minus 119888119905)) + 120575

2

100381610038161003816100381610038161003816100381610038161003816

(30)

(v) When 119899 = 2119898 0 lt ℎ lt ℎ119860 120572120573 lt 0 120573119888 lt 0 (or 120572120573 gt 0

120573119888 lt 0) and119898 is odd (14) can be reduced to

119910 = plusmnradicminus

120573

119888119898

120601radicminus

119888119898ℎ

120573

+

2120572

120573

120601119898+ (120601119898)2

(31)

Similarly by using (31) we obtain two smooth solitary wavesolutions of (1) which are the same as the solutions (29) and(30)

4 Journal of Applied Mathematics

minus2 minus1 1 2 3x

0

05

1

15

2

(a) Smooth solitary wave

02

04

06

minus02 minus01 01 02 03 04 05 06x

u

(b) Unboundary wave

Figure 1 The phase portraits of solutions (21) and (22) for119898 = 4 120572 = 3 120573 = minus1 119888 = 2 ℎ = 9 119905 = 01

(vi)When 119899 = 119898 ℎ gt 0 and119898 is odd (14) can be reducedto (20) so the obtained solution is as the same as the solution(22)

(vii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 and 119898 is odd (14) canbe reduced to

119910 = plusmnradic120573

119888119898

120601radicminus

2120572

120573

120601119898minus (120601119898)2

(32)

Taking 1205850= 0 and 120601(0) = (minus2120572120573)

1119898 as initial valuessubstituting (32) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = (minus2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

)

1119898

(33)

Substituting (33) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain a smooth unboundary wave solution of rationalfunction type of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816100381610038161003816

2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

(34)

32 Periodic Blow-UpWave Solutions Broken KinkWave andAntikink Wave Solutions in the Special Case 119899 = 2119898 isin Z+ (1)When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ

1198611lt ℎ lt 0 (or ℎ

1198612lt ℎ lt 0)

and119898 is even (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(35)

Taking 1205850= 0 120601(0) = (minus120572120573 + radic120572

21205732+ 119898ℎ119888120573)

1119898

and

1205850= 0 120601(0) = (minus120572120573 minus radic120572

21205732+ 119898ℎ119888120573)

1119898

as the initial

values substituting (35) into the 119889120601119889120585 = 119910 of (11) and thenintegrating them yield

V (119909 119905) = (119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

)

1119898

(36)

V (119909 119905) = minus(119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

)

1119898

(37)

where Ω3

= radic(1205722+ 119888119898ℎ120573)120573

2 1205963

= 119898radicminusℎ 1198631

=

(1205722

+ 120572120573Ω3+ 120573119888119898ℎ)120573Ω

3(120572 + 120573Ω

3) 1198632= (1205722

minus 120572120573Ω3+

120573119888119898ℎ)120573Ω3(minus120572 + 120573Ω

3) Respectively substituting (36) and

(37) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtaintwo periodic blow-up wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(38)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(39)

(2) When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ1198612lt ℎ lt 0 and 119898 is

odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(40)

which is the same as (35) so the obtained solution is the sameas solution (39) Similarly when 119899 = 2119898 120572120573 lt 0 120573119888 gt 0

Journal of Applied Mathematics 5

ℎ1198611lt ℎ lt 0 and 119898 is odd the obtained solution is also the

same as solution (38)(3) when 119899 = 2119898 120572120573 lt 0 120573119888 lt 0 ℎ = ℎ

1198611= ℎ1198612=

minus1205722

120573119888119898 and119898 is even (14) can be reduced to

119910 = plusmnradicminus

1

119888119898120573

120601 (120573120601119898

+ 120572) (41)

Substituting (41) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it and setting the integral constants as zero yield

V (119909 119905) = (minus120572

2120573

(1 plusmn tanh Ω42

(119909 minus 119888119905)))

1119898

(42)

where Ω4= 120572radicminus119898119888120573 Substituting (42) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain an unboundary wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816

120572

2120573

[1 plusmn tanh Ω42

(119909 minus 119888119905)]

10038161003816100381610038161003816100381610038161003816

(43)

Equation (42) is smooth kink wave solution of (9) but (43) isa broken kink wave solution without boundary of (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (42) and (43) we plot graphsof their profiles which are shown in Figures 2(a) and 2(b)respectively

(4) When 119899 = 2119898 120572120573 gt 0 120573119888 lt 0 ℎ = ℎ119860or 120572120573 lt 0

120573119888 lt 0 ℎ = ℎ119860and 119898 is odd (14) can be reduced to (41) So

the obtained solution is also the same as the solution (43)

33 Periodic Blow-Up Wave Solutions and Solitary WaveSolutions of Blow-Up Form in the Special Case119898 = 2 119899 = minus1When 119899 = minus1 (14) can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

2120573

119888

120601 (44)

(a) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (120601 minus 0) (120601 minus 120574

1) (120601 minus 120575

1) 120601 isin (0 120599

1]

(45)

where 1205991 1205741 1205751are three roots of equation 2120573120572 + (119888ℎ120572)120601 minus

1206013 and 120599

1gt 0 gt 120574

1gt 1205751 These three roots 120599

1 1205741 1205751can

be obtained by Cardano formula as long as the parameters120572 120573 119888 ℎ are fixed (or given) concretely For example taking120572 = 40 120573 = 20 119888 = 40 ℎ = 90 the 120599

1≐ 3054084215

1205741≐ minus01112641576 120575

1≐ minus2942820058 Taking (0 120599

1) as

the initial values of the variables (120585 120601) substituting (45) intothe 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =1205991(minus1205751) cn2 (Ω

1(120585) 1198961)

(minus1205751) + 1205991sn2 (Ω

1120585 1198961)

120585 = 1198791

(46)

where Ω1

= (12)radicminus1205721205751(1205991minus 1205751)119888 119896

1=

radic1205991(1205741minus 1205751) minus 120575

1(1205991minus 1205741) 1198791

= (1Ω1)snminus1(1 119896

1) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (46) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205991(minus1205751) cn2 (Ω

1(119909minus119888119905) 119896

1)

(minus1205751)+1205991sn2 (Ω

1(119909minus119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 =1198791

(47)

(b) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (0 minus 120601) (120574

1minus 120601) (120601 minus 120575

1) 120601isin[120575

1 1205741]

(48)

Similarly taking (0 1205751) as the initial values of the variables

(120585 120601) substituting (48) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1120585 1198961)

(49)

By using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (49) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1(119909 minus 119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

(50)

(c) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120578

2minus 120601) (0 minus 120601) (120601 minus 120575

2) 120601 isin [120575

2 0)

(51)

where 1205992 1205782 0 1205752are roots of equation 120601(2120573120572 + (119888ℎ120572)120601 minus

1206013

) = 0 and 1205992gt 1205782gt 0 gt 120575

2 As in the above cases these

three roots can be obtained by Cardano formula as long asthe parameters 120572 120573 119888 ℎ are given concretely so the similarcases will be not commentated anymore in the followingdiscussions Similarly taking (0 120575

2) as the initial values of the

variables (120585 120601) substituting (51) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) =12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 120575

2sn2 (Ω

2120585 1198962)

120585 = 1198792

(52)

where Ω2

= (12)radic1205721205992(1205782minus 1205752)119888 119896

2=

radicminus1205752(1205992minus 1205782)1205992(1205782minus 1205752) 1198792

= (1Ω2)snminus1(0 119896

2) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (52) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 1205752sn2 (Ω

2120585 1198962)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198792

(53)

(d) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120601 minus 120578

2) (120601 minus 0) (120601 minus 120575

2) (54)

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

minus2 minus1 1 2 3x

0

05

1

15

2

(a) Smooth solitary wave

02

04

06

minus02 minus01 01 02 03 04 05 06x

u

(b) Unboundary wave

Figure 1 The phase portraits of solutions (21) and (22) for119898 = 4 120572 = 3 120573 = minus1 119888 = 2 ℎ = 9 119905 = 01

(vi)When 119899 = 119898 ℎ gt 0 and119898 is odd (14) can be reducedto (20) so the obtained solution is as the same as the solution(22)

(vii) When 119899 = 2119898 ℎ = 0 120573119888 gt 0 and 119898 is odd (14) canbe reduced to

119910 = plusmnradic120573

119888119898

120601radicminus

2120572

120573

120601119898minus (120601119898)2

(32)

Taking 1205850= 0 and 120601(0) = (minus2120572120573)

1119898 as initial valuessubstituting (32) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = (minus2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

)

1119898

(33)

Substituting (33) into the transformation 119906(119909 119905) = ln |V(119909 119905)|we obtain a smooth unboundary wave solution of rationalfunction type of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816100381610038161003816

2120572

120573 [1 + (1198981205722120573119888) (119909 minus 119888119905)

2

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

(34)

32 Periodic Blow-UpWave Solutions Broken KinkWave andAntikink Wave Solutions in the Special Case 119899 = 2119898 isin Z+ (1)When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ

1198611lt ℎ lt 0 (or ℎ

1198612lt ℎ lt 0)

and119898 is even (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(35)

Taking 1205850= 0 120601(0) = (minus120572120573 + radic120572

21205732+ 119898ℎ119888120573)

1119898

and

1205850= 0 120601(0) = (minus120572120573 minus radic120572

21205732+ 119898ℎ119888120573)

1119898

as the initial

values substituting (35) into the 119889120601119889120585 = 119910 of (11) and thenintegrating them yield

V (119909 119905) = (119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

)

1119898

(36)

V (119909 119905) = minus(119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

)

1119898

(37)

where Ω3

= radic(1205722+ 119888119898ℎ120573)120573

2 1205963

= 119898radicminusℎ 1198631

=

(1205722

+ 120572120573Ω3+ 120573119888119898ℎ)120573Ω

3(120572 + 120573Ω

3) 1198632= (1205722

minus 120572120573Ω3+

120573119888119898ℎ)120573Ω3(minus120572 + 120573Ω

3) Respectively substituting (36) and

(37) into the transformation 119906(119909 119905) = ln |V(119909 119905)| we obtaintwo periodic blow-up wave solutions of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 minus 120573Ω3sin (arcsin119863

1minus 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(38)

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

119888119898ℎ

120572 + 120573Ω3sin (arcsin119863

2+ 1205963(119909 minus 119888119905))

100381610038161003816100381610038161003816100381610038161003816

(39)

(2) When 119899 = 2119898 120572120573 lt 0 120573119888 gt 0 ℎ1198612lt ℎ lt 0 and 119898 is

odd (14) can be reduced to

119910 = plusmnradic120573

119888119898

120601radic

119888119898ℎ

120573

minus

2120572

120573

120601119898minus (120601119898)2

(40)

which is the same as (35) so the obtained solution is the sameas solution (39) Similarly when 119899 = 2119898 120572120573 lt 0 120573119888 gt 0

Journal of Applied Mathematics 5

ℎ1198611lt ℎ lt 0 and 119898 is odd the obtained solution is also the

same as solution (38)(3) when 119899 = 2119898 120572120573 lt 0 120573119888 lt 0 ℎ = ℎ

1198611= ℎ1198612=

minus1205722

120573119888119898 and119898 is even (14) can be reduced to

119910 = plusmnradicminus

1

119888119898120573

120601 (120573120601119898

+ 120572) (41)

Substituting (41) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it and setting the integral constants as zero yield

V (119909 119905) = (minus120572

2120573

(1 plusmn tanh Ω42

(119909 minus 119888119905)))

1119898

(42)

where Ω4= 120572radicminus119898119888120573 Substituting (42) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain an unboundary wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816

120572

2120573

[1 plusmn tanh Ω42

(119909 minus 119888119905)]

10038161003816100381610038161003816100381610038161003816

(43)

Equation (42) is smooth kink wave solution of (9) but (43) isa broken kink wave solution without boundary of (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (42) and (43) we plot graphsof their profiles which are shown in Figures 2(a) and 2(b)respectively

(4) When 119899 = 2119898 120572120573 gt 0 120573119888 lt 0 ℎ = ℎ119860or 120572120573 lt 0

120573119888 lt 0 ℎ = ℎ119860and 119898 is odd (14) can be reduced to (41) So

the obtained solution is also the same as the solution (43)

33 Periodic Blow-Up Wave Solutions and Solitary WaveSolutions of Blow-Up Form in the Special Case119898 = 2 119899 = minus1When 119899 = minus1 (14) can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

2120573

119888

120601 (44)

(a) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (120601 minus 0) (120601 minus 120574

1) (120601 minus 120575

1) 120601 isin (0 120599

1]

(45)

where 1205991 1205741 1205751are three roots of equation 2120573120572 + (119888ℎ120572)120601 minus

1206013 and 120599

1gt 0 gt 120574

1gt 1205751 These three roots 120599

1 1205741 1205751can

be obtained by Cardano formula as long as the parameters120572 120573 119888 ℎ are fixed (or given) concretely For example taking120572 = 40 120573 = 20 119888 = 40 ℎ = 90 the 120599

1≐ 3054084215

1205741≐ minus01112641576 120575

1≐ minus2942820058 Taking (0 120599

1) as

the initial values of the variables (120585 120601) substituting (45) intothe 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =1205991(minus1205751) cn2 (Ω

1(120585) 1198961)

(minus1205751) + 1205991sn2 (Ω

1120585 1198961)

120585 = 1198791

(46)

where Ω1

= (12)radicminus1205721205751(1205991minus 1205751)119888 119896

1=

radic1205991(1205741minus 1205751) minus 120575

1(1205991minus 1205741) 1198791

= (1Ω1)snminus1(1 119896

1) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (46) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205991(minus1205751) cn2 (Ω

1(119909minus119888119905) 119896

1)

(minus1205751)+1205991sn2 (Ω

1(119909minus119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 =1198791

(47)

(b) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (0 minus 120601) (120574

1minus 120601) (120601 minus 120575

1) 120601isin[120575

1 1205741]

(48)

Similarly taking (0 1205751) as the initial values of the variables

(120585 120601) substituting (48) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1120585 1198961)

(49)

By using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (49) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1(119909 minus 119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

(50)

(c) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120578

2minus 120601) (0 minus 120601) (120601 minus 120575

2) 120601 isin [120575

2 0)

(51)

where 1205992 1205782 0 1205752are roots of equation 120601(2120573120572 + (119888ℎ120572)120601 minus

1206013

) = 0 and 1205992gt 1205782gt 0 gt 120575

2 As in the above cases these

three roots can be obtained by Cardano formula as long asthe parameters 120572 120573 119888 ℎ are given concretely so the similarcases will be not commentated anymore in the followingdiscussions Similarly taking (0 120575

2) as the initial values of the

variables (120585 120601) substituting (51) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) =12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 120575

2sn2 (Ω

2120585 1198962)

120585 = 1198792

(52)

where Ω2

= (12)radic1205721205992(1205782minus 1205752)119888 119896

2=

radicminus1205752(1205992minus 1205782)1205992(1205782minus 1205752) 1198792

= (1Ω2)snminus1(0 119896

2) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (52) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 1205752sn2 (Ω

2120585 1198962)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198792

(53)

(d) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120601 minus 120578

2) (120601 minus 0) (120601 minus 120575

2) (54)

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

ℎ1198611lt ℎ lt 0 and 119898 is odd the obtained solution is also the

same as solution (38)(3) when 119899 = 2119898 120572120573 lt 0 120573119888 lt 0 ℎ = ℎ

1198611= ℎ1198612=

minus1205722

120573119888119898 and119898 is even (14) can be reduced to

119910 = plusmnradicminus

1

119888119898120573

120601 (120573120601119898

+ 120572) (41)

Substituting (41) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it and setting the integral constants as zero yield

V (119909 119905) = (minus120572

2120573

(1 plusmn tanh Ω42

(119909 minus 119888119905)))

1119898

(42)

where Ω4= 120572radicminus119898119888120573 Substituting (42) into the transfor-

mation 119906(119909 119905) = ln |V(119909 119905)| we obtain an unboundary wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln10038161003816100381610038161003816100381610038161003816

120572

2120573

[1 plusmn tanh Ω42

(119909 minus 119888119905)]

10038161003816100381610038161003816100381610038161003816

(43)

Equation (42) is smooth kink wave solution of (9) but (43) isa broken kink wave solution without boundary of (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (42) and (43) we plot graphsof their profiles which are shown in Figures 2(a) and 2(b)respectively

(4) When 119899 = 2119898 120572120573 gt 0 120573119888 lt 0 ℎ = ℎ119860or 120572120573 lt 0

120573119888 lt 0 ℎ = ℎ119860and 119898 is odd (14) can be reduced to (41) So

the obtained solution is also the same as the solution (43)

33 Periodic Blow-Up Wave Solutions and Solitary WaveSolutions of Blow-Up Form in the Special Case119898 = 2 119899 = minus1When 119899 = minus1 (14) can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

2120573

119888

120601 (44)

(a) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (120601 minus 0) (120601 minus 120574

1) (120601 minus 120575

1) 120601 isin (0 120599

1]

(45)

where 1205991 1205741 1205751are three roots of equation 2120573120572 + (119888ℎ120572)120601 minus

1206013 and 120599

1gt 0 gt 120574

1gt 1205751 These three roots 120599

1 1205741 1205751can

be obtained by Cardano formula as long as the parameters120572 120573 119888 ℎ are fixed (or given) concretely For example taking120572 = 40 120573 = 20 119888 = 40 ℎ = 90 the 120599

1≐ 3054084215

1205741≐ minus01112641576 120575

1≐ minus2942820058 Taking (0 120599

1) as

the initial values of the variables (120585 120601) substituting (45) intothe 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =1205991(minus1205751) cn2 (Ω

1(120585) 1198961)

(minus1205751) + 1205991sn2 (Ω

1120585 1198961)

120585 = 1198791

(46)

where Ω1

= (12)radicminus1205721205751(1205991minus 1205751)119888 119896

1=

radic1205991(1205741minus 1205751) minus 120575

1(1205991minus 1205741) 1198791

= (1Ω1)snminus1(1 119896

1) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (46) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205991(minus1205751) cn2 (Ω

1(119909minus119888119905) 119896

1)

(minus1205751)+1205991sn2 (Ω

1(119909minus119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 =1198791

(47)

(b) Under the conditions 120572120573 gt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205991minus 120601) (0 minus 120601) (120574

1minus 120601) (120601 minus 120575

1) 120601isin[120575

1 1205741]

(48)

Similarly taking (0 1205751) as the initial values of the variables

(120585 120601) substituting (48) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1120585 1198961)

(49)

By using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (49) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057411205751

1205751+ (1205741minus 1205751) sn2 (Ω

1(119909 minus 119888119905) 119896

1)

100381610038161003816100381610038161003816100381610038161003816

(50)

(c) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (minusinfin +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120578

2minus 120601) (0 minus 120601) (120601 minus 120575

2) 120601 isin [120575

2 0)

(51)

where 1205992 1205782 0 1205752are roots of equation 120601(2120573120572 + (119888ℎ120572)120601 minus

1206013

) = 0 and 1205992gt 1205782gt 0 gt 120575

2 As in the above cases these

three roots can be obtained by Cardano formula as long asthe parameters 120572 120573 119888 ℎ are given concretely so the similarcases will be not commentated anymore in the followingdiscussions Similarly taking (0 120575

2) as the initial values of the

variables (120585 120601) substituting (51) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) =12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 120575

2sn2 (Ω

2120585 1198962)

120585 = 1198792

(52)

where Ω2

= (12)radic1205721205992(1205782minus 1205752)119888 119896

2=

radicminus1205752(1205992minus 1205782)1205992(1205782minus 1205752) 1198792

= (1Ω2)snminus1(0 119896

2) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (52) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057821205752sn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + 1205752sn2 (Ω

2120585 1198962)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198792

(53)

(d) Under the conditions 120572120573 lt 0 120572119888 gt 0 ℎ isin (ℎ119860 +infin)

and119898 = 2 (44) becomes

119910 = plusmnradic

120572

119888

radic(1205992minus 120601) (120601 minus 120578

2) (120601 minus 0) (120601 minus 120575

2) (54)

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

minus30 minus20 minus10 10x

minus06

minus05

minus04

minus03

minus02

minus01

0

(a) Smooth kink wave

u

minus4 minus2 2 4 6 8x

minus08

minus06

minus04

minus02

0

(b) Unboundary wave

Figure 2 The phase portraits of solutions (42) and (43) for119898 = 4 120572 = 03 120573 = minus03 119888 = 2 119905 = 01

Taking the (0 1205992) as the initial values of the variables (120585 120601)

substituting (54) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) =1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2120585 1198962)

(55)

By using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (55) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1205992(1205782minus 1205752) dn2 (Ω

2(119909 minus 119888119905) 119896

2)

(1205782minus 1205752) + (120599

2minus 1205782) sn2 (Ω

2(119909 minus 119888119905) 119896

2)

100381610038161003816100381610038161003816100381610038161003816

(56)

(e) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205993minus 120601) (120578

3minus 120601) (120601 minus 0) (120601 minus 120575

3) 120601isin (0 120578

3]

(57)

where 1205993 1205783 0 1205753are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205993gt 1205783gt 0 gt 120575

3 Taking the (0 120578

3) as the

initial values of the variables (120585 120601) substituting (57) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12059931205783cn2 (Ω

3120585 1198963)

1205753minus 1205783sn2 (Ω

3120585 1198963)

120585 = 1198793 (58)

where Ω3

= (12)radicminus1205721205993(1205783minus 1205753)119888 119896

3=

radic1205783(1205993minus 1205753)1205993(1205783minus 1205753) 1198793

= (1Ω3)snminus1(1 119896

3) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (58) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12059931205783cn2 (Ω

3(119909 minus 119888119905) 119896

3)

1205753minus 1205783sn2 (Ω

3(119909 minus 119888119905) 119896

3)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198793

(59)

(f) Under the conditions 120572120573 lt 0 120572119888 lt 0 ℎ isin (minusinfin ℎ119860)

and119898 = 2 (44) becomes

119910 = plusmnradicminus

120572

119888

radic(1205994minus 120601) (0 minus 120601) (120601 minus 120574

4) (120601 minus 120575

4) 120601 isin [120574

4 0)

(60)

where 1205994 0 1205744 1205754are roots of equation 120601(minus2120573120572 minus (119888ℎ120572)120601 +

1206013

) = 0 and 1205994gt 0 gt 120574

4gt 1205754 Taking the (0 120574

4) as the

initial values of the variables (120585 120601) substituting (60) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =12057541205744cn2 (Ω

4120585 1198964)

1205754minus 1205744cn2 (Ω

4120585 1198964)

120585 = 1198794 (61)

where Ω4

= (12)radic1205721205754(1205994minus 1205744)119888 119896

4=

radic1205744(1205994minus 1205754)1205994(1205994minus 1205744) 1198794

= (1Ω4)snminus1(1 119896

4) By

using the transformation 119906(119909 119905) = ln |V(119909 119905)| and (61) weobtain a periodic below-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

12057541205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

1205754minus 1205744cn2 (Ω

4(119909 minus 119888119905) 119896

4)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198794

(62)

(g) Under the conditions 119898 = 2 120572119888 lt 0 ℎ = ℎ119860=

minus1205722

2119888120573 (44) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206011minus 120601

1003816100381610038161003816radic120601 (120601 + 2120601

1) 120601 = 0 (63)

where 1206011= (minus120573120572)

13 Taking (0 (12)1206011) as the initial values

of the variables (120585 120601) substituting (63) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011[1 minus

3

cosh (Ω5120585) + 2

] 120585 = 1198795

(64)

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

where Ω5= 1206011radicminus3120572119888 119879

5= (1Ω

5)coshminus1(1) Clearly we

have 120601(1198795) = 0 120601(plusmninfin) = 120601

1 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (64) we obtain a solitary wavesolution of blow-up form of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206011[1 minus

3

cosh (Ω5(119909 minus 119888119905)) + 2

]

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198795

(65)

Equation (64) is smooth solitarywave solution of (9) but (65)is a solitary wave solution of blow-up form for (1) because119906 rarr minusinfin as V rarr 0 In order to visually show dynamicalbehaviors of solutions V and 119906 of (64) and (65) we plot graphsof their profiles which are shown in Figures 3(a) and 3(b)respectively

34 Different Kinds of Periodic Wave Broken Solitary WaveSolutions in the Special Case 119898 = 1 119899 = minus1 (1) Under theconditions ℎ isin (ℎ

119860 +infin) 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0 120573119888 gt 0)

120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 120601

119898) (120601 minus 0) 120601 isin [120601

119898 120601119872]

(66)

where 120601119872119898

= (ℎ119888 plusmn radicℎ21198882+ 16120572120573)4120572 and 0 are three

roots of equation ℎ1206012 minus (2120572119888)1206013 + (2120573119888)120601 = 0 and 120601119872gt

120601119898gt 0 Taking (0 120601

119898) as the initial values of the variables

(120585 120601) substituting (66) into the 119889120601119889120585 = 119910 of (11) and thenintegrating it yield

V (119909 119905) = 120601 (120585) = 120601119898minus (120601119872minus 120601119898) sn2 (Ω

6120585 1198966)

(67)

where Ω6= radic120572120601

1198722119888 119896

6= radic(120601

119872minus 120601119898)120601119872 By using the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (67) we obtain adouble periodic wave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119898minus (120601119872minus 120601119898) sn2 (Ω

6(119909 minus 119888119905) 119896

6)

10038161003816100381610038161003816

(68)

Equation (67) is smooth periodic wave solution of (9) but(68) is a double periodic wave solution of blow-up form of(1) because 119906 rarr minusinfin as V rarr 0 In order to visually showdynamical behaviors of solutions V and 119906 of (67) and (68) weplot graphs of their profiles which are shown in Figures 4(a)and 4(b) respectively

(2) Under the condition ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0 (or

120572120573 lt 0 120573119888 lt 0)120572119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(0 minus 120601) (120601 minus 120601119872) (120601 minus 120601

119898) 120601 isin [120601

119872 0)

(69)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119898lt 120601119872

lt 0 Taking (0 120601119872) as the

initial values of the variables (120585 120601) substituting (69) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 120601119872sn2 (Ω

7120585 1198967) 120585 = 119879

7 (70)

where Ω7

= radicminus1205721206011198722119888 119896

7= radic120601

119872120601119898 1198797

=

(1Ω7)snminus1(0 119896

7) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (70) we obtain a periodic blow-up wavesolution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872sn2 (Ω

7(119909 minus 119888119905) 119896

7)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

7

(71)

(3) Under the conditions ℎ isin (ℎ119860 +infin) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 gt 0 120573119888 gt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(0 minus 120601) (120601119872minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 120601119872]

(72)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 0 gt 120601119872

gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (72) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119872

dn2 (Ω8120585 1198968)

(73)

where Ω8= radic120572120601

1198982119888 119896

8= radic(120601

119872minus 120601119898) minus 120601

119898 By using

the transformation 119906(119909 119905) = ln |V(119909 119905)| and (73) we obtain aperiodic blow-up wave solutions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119872

dn2 (Ω8(119909 minus 119888119905) 119896

8)

100381610038161003816100381610038161003816100381610038161003816

(74)

(4) Under the conditions ℎ isin (minusinfin ℎ119860) 120572120573 lt 0 120573119888 gt 0

(or 120572120573 lt 0 120573119888 lt 0) and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (120601

119898minus 120601) (120601 minus 0) 120601 isin (0 120601

119898]

(75)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 120601119898gt 0 Taking (0 120601

119898) as the

initial values of the variables (120585 120601) substituting (75) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =120601119898cn2 (Ω

9120585 1198969)

dn2 (Ω9120585 1198969)

120585 = 1198799 (76)

where Ω9

= radicminus1205721206011198722119888 119896

9= radic120601

119898120601119872 1198799

=

(1Ω9)snminus1(0 119896

9) By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (76) we obtain periodic blow-up wave solu-tions of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

120601119898cn2 (Ω

9(119909 minus 119888119905) 119896

9)

dn2 (Ω9(119909 minus 119888119905) 119896

9)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 1198799

(77)

(5) Under the conditions ℎ = ℎ119860= minus4120572120601

1119888 120572120573 lt 0

120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

10038161003816100381610038161206011+ 120601

1003816100381610038161003816radicminus120601 120601 isin [minus120601

1 0) (78)

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

minus20 minus15 minus10 minus5 5 10 15x

minus1

minus05

05

1

(a) Smooth solitary wave

minus25

minus2

minus15

minus1

minus05

0

u

minus15 minus10 minus5 5 10x

(b) Solitary wave of blow-up form

Figure 3 The phase portraits of solutions (64) and (65) for119898 = 2 120572 = 03 120573 = minus03 119888 = minus2 119905 = 01

minus2

minus15

minus1

minus05

0

minus8 minus6 minus4 minus2 2 4x

(a) Smooth periodic wave

minus2

minus15

minus1

minus05

05

1

u

minus6 minus4 minus2 2 4 6 8

x

(b) Double periodic wave of blow-up type

Figure 4 The phase portraits of solutions (67) and (68) for ℎ = 8 120572 = 25 120573 = minus25 119888 = 2 119905 = 01

where 1206011= (minus120573120572)

12 Taking (0 minus1206011) as the initial values of

the variables (120585 120601) substituting (78) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206011tanh2 (05Ω

10120585) 120585 = 0 (79)

whereΩ10= radic2120572120601

1119888 By using the transformation 119906(119909 119905) =

ln |V(119909 119905)| and (79) we obtain a solitary wave solution ofblow-up form of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

10(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0

(80)

(6) Under the conditions ℎ = ℎ119860= 4120572120601

1119888 120572120573 lt 0 120573119888 gt

0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

1003816100381610038161003816120601 minus 1206011

1003816100381610038161003816radic120601 120601 isin (0 120601

1] (81)

where 1206011= (minus120573120572)

12 Taking (0 1206011) as the initial values of

the variables (120585 120601) substituting (81) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206011tanh2 (05Ω

11120585) 120585 = 0 (82)

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

where Ω11

= radicminus21205721206011119888 By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (82) we obtain a broken solitarywavesolution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161206011tanh2 (05Ω

11(119909 minus 119888119905))

10038161003816100381610038161003816 119909 minus 119888119905 = 0 (83)

(7) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 lt 0 and119898 = 1 (44) becomes

119910 = plusmnradicminus

2120572

119888

radic(120601119872minus 120601) (0 minus 120601) (120601 minus 120601

119898) 120601 isin [120601

119898 0)

(84)

where 120601119872 120601119898 0 are roots of equation 1206012ℎ minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (84) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) =minus1206011198721198962

12sn2 (Ω

12120585 11989612)

dn2 (Ω12120585 11989612)

120585 = 11987912 (85)

where Ω12= radicminus120572(120601

119872minus 120601119898)2119888 119896

12= radicminus120601

119898(120601119872minus 120601119898)

11987912

= (1Ω12)snminus1(0 119896

12) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (85) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus1206011198721198962

12sn2 (Ω

12(119909 minus 119888119905) 119896

12)

dn2 (Ω12(119909 minus 119888119905) 119896

12)

100381610038161003816100381610038161003816100381610038161003816

119909 minus 119888119905 = 11987912

(86)

(8) Under the conditions ℎ isin (minusinfin 0] 120572120573 gt 0 120573119888 gt 0 and119898 = 1 (44) becomes

119910 = plusmnradic2120572

119888

radic(120601119872minus 120601) (120601 minus 0) (120601 minus 120601

119898) 120601 isin (0 120601

119872]

(87)

where 120601119872 120601119898 0 are roots of equation ℎ1206012 minus (2120572119888)1206013 +

(2120573119888)120601 = 0 and 120601119872

gt 0 gt 120601119898 Taking (0 120601

119872) as the

initial values of the variables (120585 120601) substituting (87) into the119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601119872cn2 (Ω

13120585 11989613) 120585 = 119879

13 (88)

where Ω13

= radic120572(120601119872minus 120601119898)2119888 119896

13= radic120601

119872(120601119872minus 120601119898)

11987913

= (1Ω13)snminus1(0 119896

13) By using the transformation

119906(119909 119905) = ln |V(119909 119905)| and (88) we obtain a periodic blow-upwave solution of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161003816120601119872cn2 (Ω

13(119909 minus 119888119905) 119896

13)

10038161003816100381610038161003816 119909 minus 119888119905 = 119879

13

(89)

35 Periodic Blow-Up Wave Broken Solitary Wave BrokenKink and AntikinkWave Solutions in the Special Cases119898 = 2

or 119898 = 1 and 119899 = minus2 When 119898 = 2 is even and 119899 = minus2 (14)can be reduced to

119910 = plusmnradicℎ1206012minus

2120572

119888119898

120601119898+2

+

120573

119888

(90)

Because the singular straight line 120601 = 0 this implies that(1) has broken kink and antikink wave solutions see thefollowing discussions

(i) Under the conditions 119898 = 2 119899 = minus2 ℎ = radicminus41205721205731198882120572120573 lt 0 120572119888 lt 0 (90) becomes

119910 = plusmnradicminus

120572

119888

100381610038161003816100381610038161206012

minus 1206012

1

10038161003816100381610038161003816 (120601 = 0) (91)

where 1206011= radicminus120573120572 Taking (0 120601

1) as the initial values of the

variables (120585 120601) substituting (91) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = plusmn1206011tanh [Ω

14120585] (92)

where Ω14

= 1206011radicminus120572119888 By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (92) we obtain a broken kink wavesolutions of (1) as follows

119906 (119909 119905) = ln 10038161003816100381610038161206011tanh [Ω

14(119909 minus 119888119905)]

1003816100381610038161003816 (93)

where (119909 minus 119888119905) isin (minusinfin 0) and (119909 minus 119888119905) isin (0 +infin)(ii) Under the conditions ℎ isin (ℎ

1198611 +infin) 120572120573 gt 0 120572119888 gt 0

and119898 = 2 119899 = minus2 (90) becomes

119910 = plusmnradic

120572

119888

radic(1198862minus 1206012) (1206012minus 1198872) (94)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Taking (0 119886) as the initial values of the vari-

ables (120585 120601) substituting (94) into the 119889120601119889120585 = 119910 of (11) andthen integrating it yield

V (119909 119905) = 120601 (120585) = plusmnradic1198862 minus (1198862 minus 1198872) sn2 (Ω15120585 11989615) (95)

where 1198862 = (12120572)(119888ℎ + radic1198882ℎ2+ 4120572120573) 1198872 = (12120572)(119888ℎ minus

radic1198882ℎ2+ 4120572120573) Ω

15= 119886radic120572119888 119896

15= radic(119886

2minus 1198872)1198862 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (95) we obtaina periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = 05 ln 100381610038161003816100381610038161198862

minus (1198862

minus 1198872

) sn2 (Ω15120585 11989615)

10038161003816100381610038161003816 (96)

(iii) Under the conditions 119898 = 1 119899 = minus2 ℎ = ℎ1198611=

31205721206012119888 120572120573 gt 0 120572119888 gt 0 (90) becomes

119910 = plusmnradic

120572

119888

1003816100381610038161003816minus1206012+ 120601

1003816100381610038161003816radicminus1206012minus 2120601 (120601 = 0) (97)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (97) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = minus1206012[

3

cosh (Ω16120585) + 1

minus 1] 120585 = 11987916

(98)

where Ω16= radicminus3120572120601

2119888 11987916= (1Ω

16)coshminus1(2) By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (98) when

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

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Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

(119909 minus 119888119905) isin (minusinfin11987916) ⋃ (119879

16 +infin) we obtain a broken

solitary wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

31206012

cosh (Ω16(119909 minus 119888119905)) + 1

minus 1206012

100381610038161003816100381610038161003816100381610038161003816

(99)

(iv) Under the conditions ℎ = ℎ1198611= 3120572120601

2119888 120572120573 lt 0

120572119888 lt 0 and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradicminus

120572

119888

10038161003816100381610038161206012minus 120601

1003816100381610038161003816radic1206012+ 2120601 (120601 = 0) (100)

where 1206012= (minus120573120572)

13 Taking (0 minus051206012) as the initial values

of the variables (120585 120601) substituting (100) into the 119889120601119889120585 = 119910of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1206012[1 minus

3

cosh (Ω17120585) + 1

] 120585 = 11987917

(101)

where Ω17= radic3120572120601

1119888 11987917= (1Ω

17)coshminus1(2) By using the

transformations 119906(119909 119905) = ln |V(119909 119905)| and (101) when (119909minus119888119905) isin(minusinfin119879

17) ⋃ (119879

17 +infin) we obtain a broken solitary wave

solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

1206012minus

31206012

cosh (Ω17(119909 minus 119888119905)) + 1

100381610038161003816100381610038161003816100381610038161003816

(102)

(v) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 lt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120601 minus 120578

5) (120601 minus 120574

5) 120601 isin [120578

5 1205995]

(103)

where 1205995 1205785 1205745are roots of equation1205732120572 + (119888ℎ2120572)1206012 minus 1206013

and 1205745lt 0 lt 120578

5lt 1205995 Taking (0 120599

5) as the initial values of the

variables (120585 120601) substituting (103) into the 119889120601119889120585 = 119910 of (11)and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205995minus (1205995minus 1205785) sn2 (Ω

18120585 11989618) (104)

where Ω18= radic120572(120599

5minus 1205745)2119888 119896

18= radic(120599

5minus 1205785)(1205995minus 1205745) By

using the transformations 119906(119909 119905) = ln |V(119909 119905)| and (104) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln 100381610038161003816100381610038161205995minus (1205995minus 1205785) sn2 (Ω

18(119909 minus 119888119905) 119896

18)

10038161003816100381610038161003816 (105)

(vi) Under the conditions ℎ isin (ℎ1198611 +infin) 120572120573 gt 0 120573119888 lt 0

and119898 = 1 119899 = minus2 (90) becomes

119910 = plusmnradic2120572

119888

radic(1205995minus 120601) (120578

5minus 120601) (120601 minus 120574

5) 120601 isin [120574

5 1205785]

(106)where 120574

5lt 1205785lt 0 lt 120599

5 Taking (0 120599

5) as the initial values of

the variables (120585 120601) substituting (106) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = 120601 (120585) = 1205745+ (1205785minus 1205745) sn2 (Ω

19120585 11989619) (107)

where 1205745lt 1205785lt 0 lt 120599

5 Ω19

= radicminus120572(1205995minus 1205745)2119888

11989619

= radic(1205995minus 1205785)(1205995minus 1205745) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain a periodic blow-upwave solution of (1) as follows119906 (119909 119905) = ln 1003816100381610038161003816

10038161205745+ (1205785minus 1205745) sn2 (Ω

19(119909 minus 119888119905) 119896

19)

10038161003816100381610038161003816 (108)

36 Periodic Wave Broken Solitary Wave Broken Kink andAntikink Wave Solutions in the Special Cases 119899 =minus119898 or119899 =minus2119898 and119898 isin Z+ (a)When 119899 = minus119898119898 isin Z+ and120572120573 lt 0120573119888 lt 0 120572119888 gt 0 ℎ gt 4radicminus120572120573119888119898 (14) can be reduced to

119910 = plusmn

radic(2120573119888119898) 120601119898+ ℎ(120601

119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120601119898

119892minus 120601119898) (120601119898minus 120601119898

119897) (120601119898minus 0)

120601119898minus1

(109)

where 120601119892119897

= [(14120572)(119888119898ℎ plusmn radic11988821198982ℎ2+ 16120572120573)]

1119898 Taking(0 120601119892) as the initial values of the variables (120585 120601) substituting

(109) into the 119889120601119889120585 = 119910 of (11) and then integrating it yields

V (119909 119905) = 120601 (120585) = plusmn[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19120585 11989619)]

1119898

(110)

where Ω19= 119898radic120572120601

1198922119888119898 119896

19= radic(120601

119892minus 120601119897)120601119892 By using

the transformations 119906(119909 119905) = ln |V(119909 119905)| and (110) we obtaina periodic wave solution of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816[120601119892minus (120601119892minus 120601119897) sn2 (Ω

19(119909 minus 119888119905) 119896

19)]

10038161003816100381610038161003816

(111)

(b) When 119899 = minus2119898 119898 isin Z+and 120572120573 lt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198611 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radic2120572119888119898radic(120599119898minus 120601119898) (120601119898minus 120578119898) (120601119898minus 120574119898)

120601119898minus1

(112)

where 120599119898 120578119898 120574119898 are roots of equation 120573119888119898 + ℎ(120601119898

)2

minus

(2120572119888119898)(120601119898

)3

= 0 and 120599119898 gt 120578119898

gt 0 gt 120574119898 Taking (0 120599)

as the initial values of the variables (120585 120601) substituting (112)into the 119889120601119889120585 = 119910 of (11) and then integrating it yield

V (119909 119905) = 120601 (120585) = [120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20120585 11989620)]

1119898

(113)

where Ω20

= 119898radic120572(120599119898minus 120574119898)2119888119898 119896

20=

radic(120599119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (113) we obtain periodic wavesolutions of (1) as follows

119906 (119909 119905) =

1

119898

ln 10038161003816100381610038161003816120599119898

minus (120599119898

minus 120578119898

) sn2 (Ω20(119909 minus 119888119905) 119896

20)

10038161003816100381610038161003816

(114)

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

(c) When 119899 = minus2119898 119898 isin Z+ and 120572120573 gt 0 120573119888 lt 0 120572119888 gt 0ℎ gt ℎ1198612 (14) can be reduced to

119910 = plusmn

radic120573119888119898 + ℎ(120601119898)2

minus (2120572119888119898) (120601119898)3

120601119898minus1

= plusmn

radicminus2120572119888119898radic(120599119898minus 120601119898) (120578119898minus 120601119898) (120601119898minus 120574119898)

120601119898minus1

(115)

where 120599119898 gt 0 gt 120578119898 gt 120574119898 Taking (0 120578) as the initial values ofthe variables (120585 120601) substituting (115) into the 119889120601119889120585 = 119910 of(11) and then integrating it yield

V (119909 119905) = [120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

]

1119898

(116)

where Ω21

= 119898radicminus120572(120599119898minus 120574119898)2119888119898 119896

21=

radic(120578119898minus 120578119898)(120599119898minus 120574119898) By using the transformations

119906(119909 119905) = ln |V(119909 119905)| and (116) we obtain a periodic wavesolution of (1) as follows

119906 (119909 119905) =

1

119898

ln100381610038161003816100381610038161003816100381610038161003816

120578119898

minus 120599119898

1198962

21sn2 (Ω

21120585 11989621)

dn2 (Ω21120585 11989621)

100381610038161003816100381610038161003816100381610038161003816

(117)

4 Exact Traveling Wave Solutions of (1) andTheir Properties under the Conditions119898 isin Zminus 119899 isin Zminus

In this sections we consider the case 119898 isin Zminus 119899 isin Zminus of (1)that is we will investigate different kinds of exact travelingwave solutions of (1) under the conditions 119898 isin Zminus 119899 isin ZminusLetting119898 = minus119902 119899 = minus119901 119901 isin Z+ 119902 isin Z+ (11) becomes

119889120601

119889120585

= 119910

119889119910

119889120585

=

1198881199102

minus 1205721206012minus119902

minus 1205731206012minus119901

119888120601

(118)

First we consider a special case 119901 = 119902 = 1 When 119901 = 119902 =1 under the transformation 119889120585 = 119888120601119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus (120572 + 120573) 120601 (119)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2 (120572 + 120573)

119888

120601 (120)

where ℎ is integral constant When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2 (120572 + 120573)

119888120601

=ℎ (121)

Obviously when 120572 = minus 120573 system (119) has only one equilib-rium point 119874(0 0) in the 120601-axis

Second we consider the special case 119901 = 2 119902 = 1 andanother case 119901 = 1 119902 = 2 is very similar to the front case so

we only discuss the front case hereWhen 119901 = 2 119902 = 1 underthe transformation 119889120585 = 119888120601119889120591 system (118) can be reduced tothe following regular system

119889120601

119889120591

= 119888120601119910

119889119910

119889120591

= 1198881199102

minus 120572120601 minus 120573 (122)

which is an integrable system and it has the following firstintegral

1199102

=ℎ1206012

+

2120572

119888

120601 +

120573

119888

(123)

also we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888120601

minus

120573

1198881206012=ℎ (124)

Indeed system (122) has only one equilibrium point 119860(1206010 0)

in the 120601-axis where 1206010= minus120573120572 When 119888120573 gt 0 system (122)

has two equilibrium points 119878plusmn(0 119884plusmn) in the singular line 120601 =

0 where 119884plusmn= plusmnradic120573119888

Finally we consider the general case 119901 gt 2 When 119901 gt 2under the transformation 119889120585 = 119888120601119901minus1119889120591 system (118) can bereduced to the following regular system

119889120601

119889120591

= 119888120601119901minus1

119910

119889119910

119889120591

= 119888120601119901minus2

1199102

minus 120572120601119901minus119902

minus 120573 (125)

which is an integrable system and it has the following firstintegral

1199102

= 1206012ℎ +

2120572

119888119902

1206012minus119902

+

2120573

119888119901

1206012minus119901

(126)

When 120601 = 0 we define

119867(120601 119910) =

1199102

1206012minus

2120572

119888119902

120601minus119902

minus

2120573

119888119901

120601minus119901

=ℎ (127)

Obviously when 119901 = 119902 system (125) has not any equilibriumpoint in the 120601-axis When 119901 minus 119902 is even number and 120572120573 lt 0system (125) has two equilibrium points 119861

12(plusmn120601+ 0) in the 120601-

axis where 120601+= (minus120573120572)

1(119901minus119902) When 119901 minus 119902 is odd numbersystem (125) has only one equilibrium point at 119861

1(120601+ 0) in

the 120601-axisRespectively substituting the above equilibrium points

into (121) (124) and (127) we have

ℎ119860= 119867 (120601

0 0) = minus

1205722

119888120573

(128)

ℎ1= 119867 (120601

+ 0) = minus

2120572

119888119902

120601minus119902

+minus

2120573

119888119901

120601minus119901

+ (129)

ℎ2= 119867 (minus120601

+ 0) = (minus1)

119902+12120572

119888119902

120601minus119902

++ (minus1)

119901+12120573

119888119901

120601minus119901

+

(130)

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

41 Nonsmooth Peakon Solutions and Periodic Blow-UpWaveSolutions under Two Special Cases 119901= 119902= 1 and 119901= 2 119902 = 1(1) When 119901 = 119902 = 1 and 120572119888 gt 0 ℎ lt 0 (120) can be reducedto

119910 = plusmnradicℎ1206012minus

2 (120572 + 120573)

119888

120601(131)

Substituting (131) into the first equation of (118) taking(0 119888

ℎ(2(120572 + 120573))) as initial value conditions and integrating

it we obtain

V (119909 119905) = 120601 (120585) =120572 + 120573

119888ℎ

(cosradicminusℎ120585 minus 1) (132)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (132) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln10038161003816100381610038161003816100381610038161003816

120572 + 120573

119888ℎ

[cos(radicminusℎ (119909 minus 119888119905)) minus 1]10038161003816100381610038161003816100381610038161003816

(133)

where (119909 minus 119888119905) = 2119870120587radicminusℎ 119870 isin Z

(2) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ = ℎ

119860= 1205722

120573119888 (123) can be reduced to

119910 = plusmn

120572

radic120573119888

10038161003816100381610038161003816100381610038161003816

120601 +

120573

120572

10038161003816100381610038161003816100381610038161003816

(134)

Similarly substituting (134) into the first equation of (118) andthen integrating it and setting the integral constant as zerowe have

V (119909 119905) = 120601 (120585) = minus120573

120572

[1 minus exp(minus10038161003816100381610038161205721205851003816100381610038161003816

radic120573119888

)] (135)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (135) weobtain a broken solitary cusp wave solution of (1) as follows

119906 (119909 119905) = ln100381610038161003816100381610038161003816100381610038161003816

minus

120573

120572

[1 minus exp(minus|120572 (119909 minus 119888119905)|radic120573119888

)]

100381610038161003816100381610038161003816100381610038161003816

(119909 minus 119888119905) = 0

(136)

(3) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 gt 0 (or 120572120573 lt 0120573119888 gt 0) ℎ lt 0 (123) can be reduced to

119910 = plusmnradicminusℎradicminus

120573

119888ℎ

minus

2120572

119888ℎ

120601 minus 1206012

= plusmnradicminusℎradic(120601119872minus 120601) (120601 minus 120601

119898)

(137)

where 120601119872= (minus120572+radic120572

2minus 119888ℎ120573)119888

ℎ 120601119898= (minus120572minusradic120572

2minus 119888ℎ120573)119888

ℎ

are roots of equation minus120573119888ℎminus (2120572119888ℎ)120601minus1206012 = 0 Substituting(137) into the first equation of (118) taking (0 120601

119872) as initial

value conditions and then integrating it we have

V (119909 119905) = 120601 (120585) =1

2

[ (120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos (radicminusℎ120585)]

120585 isin (minus1198791 1198791)

(138)

where 1198791= (1

radicminusℎ)arccos((120601

119872+ 120601119898)(120601119872minus 120601119898)) From the

transformation 119906(119909 119905) = ln |V(119909 119905)| and (138) we obtain aperiodic blow-up wave solution of (1) as follows

119906 (119909 119905)

= ln10038161003816100381610038161003816100381610038161003816

1

2

[(120601119872+ 120601119898) minus (120601

119872minus 120601119898) cos(radicminusℎ (119909 minus 119888119905))]

10038161003816100381610038161003816100381610038161003816

(139)

where 119909 minus 119888119905 = 1198791+ 119870120587 119870 isin Z

(4) When 119901 = 2 119902 = 1 and 120572120573 gt 0 120573119888 lt 0 (or 120572120573 lt 0120573119888 lt 0) ℎ gt minus1205722119888120573 (123) can be reduced to

119910 = plusmnradicℎ1206012+

2120572

119888

120601 +

120572

119888

(140)

Similarly we have

V (119909 119905) = 120601 (120585) = minus120572

119888ℎ

minus

radic1205722minus 120573119888

ℎ

119888ℎ

cos(radicminusℎ120585) (141)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (141) weobtain a periodic blow-up wave solution of (1) as follows

119906 (119909 119905) = ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572

119888ℎ

+

radic1205722minus 120573119888

ℎ

119888ℎ

cosradicminusℎ (119909 minus 119888119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(142)

42 Periodic Blow-Up Wave Solutions in the Special Case119901= 2119902 gt 1 When 119901 = 2119902 gt 2 and 120572120573 lt 0 120572119888 gt 0 (or 120572120573 gt 0120572119888 lt 0) ℎ isin (ℎ

1 0) (118) becomes

119910 = plusmnradic1206012(ℎ +

2120572

119888119902

120601minus119902+

120573

119888119902

120601minus2119902)

= plusmn

radicminusℎradicminus120573119888119902

ℎ minus (2120572119888119902

ℎ) 120601119902minus (120601119902)2

120601119902minus1

(143)

Substituting (143) into the first equation of (118) and thenintegrating it we have

V (119909 119905) = 120601 (120585) = [

[

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ120585)]

]

1119902

(144)

From the transformation 119906(119909 119905) = ln |V(119909 119905)| and (144) weobtain periodic blow-up wave solutions of (1) as follows

119906 (119909 119905)=

1

119902

ln

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus

120572

119888119902ℎ

+ radic

1205722

minus 119888119902ℎ120573

11988821199022ℎ2

cos(119902radicminusℎ (119909 minus 119888119905))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(145)

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 13

5 Conclusion

In this work by using the integral bifurcation methoda generalized Tzitzeica-Dodd-Bullough-Mikhailov (TDBM)equation is studied In some special cases we obtainedmany exact traveling wave solutions of TDBM equationThese exact solutions include smooth solitary wave solutionssingular periodic wave solutions of blow-up form singularsolitary wave solutions of blow-up form broken solitary wavesolutions broken kinkwave solutions and some unboundarywave solutions Comparing with the results in many ref-erences (such as [24ndash27]) these types of solutions in thispaper are very different than those types of solutions whichare obtained by Exp-function method Though the types ofsolutions derived from two methods are different they areall useful in nonlinear science The smooth solitary wavesolutions and singular solitary wave solutions obtained inthis paper especially can be explained phenomenon of solitonsurface that the total curvature at each point is proportionalto the fourth power of the distance from a fixed point to thetangent plane

Acknowledgments

The authors would like to thank the reviewers very muchfor their useful comments and helpful suggestions Thiswork is supported by the Natural Science Foundation ofYunnan Province (no 2011FZ193) and the National NaturalScience Foundation of China (no 11161020) The e-mailaddress ldquoweiguorhhualiyuncomrdquo in the authorrsquos formerpublications (published articles) has been replaced by newe-mail address ldquoweiguorhhualiyuncomrdquo because YahooCompany in China will be closed on August 19 2013 Pleaseuse the new E-mail address ldquoweiguorhhualiyuncomrdquo if thereaders contact him

References

[1] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 144pp 1257ndash1259 1907

[2] G Tzitzeica ldquoGeometric infinitesimale-sur une nouvelle classesde surfacesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 150pp 955ndash956 1910

[3] G Tzitzeica ldquoSur certaines courbes gauchesrdquo Annales Scien-tifiques de lrsquoEcole Normale Superieure Troisieme Serie vol 28pp 9ndash32 1911

[4] R K Bullough and R K Dodd ldquoPolynomial conserved den-sities for the sine-Gordon equationsrdquo Proceedings of the RoyalSociety A vol 352 no 1671 pp 481ndash503 1977

[5] A V Ziber and A B Sabat ldquoThe Klein-Gordon equation withnontrivial grouprdquoDoklady Akademii Nauk SSSR vol 247 no 5pp 1103ndash1107 1979

[6] R Abazari ldquoThe 1198661015840119866-expansion method for Tzitzeica typenonlinear evolution equationsrdquo Mathematical and ComputerModelling vol 52 no 9-10 pp 1834ndash1845 2010

[7] A-M Wazwaz ldquoThe variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics and Com-putation vol 177 no 2 pp 745ndash754 2006

[8] F Natali ldquoOn periodic waves for sine- and sinh-Gordonequationsrdquo Journal of Mathematical Analysis and Applicationsvol 379 no 1 pp 334ndash350 2011

[9] Y Zeng ldquoConstructing hierarchy of sinh-Gordon-type equa-tions from soliton hierarchy with self-consistent sourcerdquo Phys-ica A vol 259 no 3-4 pp 278ndash290 1998

[10] J Teschner ldquoOn the spectrum of the sinh-Gordon model infinite volumerdquo Nuclear Physics B vol 799 no 3 pp 403ndash4292008

[11] A H Salas and J E Castillo H ldquoNew exact solutions to sinh-cosh-Gordon equation by using techniques based on projectiveRiccati equationsrdquoComputers ampMathematics with Applicationsvol 61 no 2 pp 470ndash481 2011

[12] X Fan S Yang J Yin and L Tian ldquoTraveling wave solutions tothe (119899 + 1)-dimensional sinh-cosh-Gordon equationrdquo Comput-ers ampMathematics with Applications vol 61 no 3 pp 699ndash7072011

[13] A-M Wazwaz ldquoThe tanh method solitons and periodicsolutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquoChaos Solitons and Fractals vol 25no 1 pp 55ndash63 2005

[14] V A Andreev ldquoBacklund transformations of the Bullough-Dodd-Zhiber-Shabat equation and symmetries of integrableequationsrdquo Akademiya Nauk SSSR Teoreticheskaya i Matem-aticheskaya Fizika vol 79 no 1 pp 151ndash154 1989 (Russian)

[15] I Yu Cherdantsev and R A Sharipov ldquoFinite-gap solutions ofthe Bullough-Dodd-Zhiber-Shabat equationrdquoAkademiya NaukSSSR Teoreticheskaya i Matematicheskaya Fizika vol 82 no 1pp 155ndash160 1990 (Russian)

[16] I Y Cherdantzev and R A Sharipov ldquoSolitons on a finite-gapbackground in Bullough-Dodd-Jiber-Shabat modelrdquo Interna-tional Journal of Modern Physics A vol 5 no 15 pp 3021ndash30271990

[17] Y V Brezhnev ldquoDarboux transformation and somemulti-phasesolutions of the Dodd-Bullough-Tzitzeica equation 119880

119909119905= 119890119880

minus

119890minus2119880rdquo Physics Letters A vol 211 no 2 pp 94ndash100 1996

[18] W K Schief ldquoSelf-dual Einstein spaces via a permutabilitytheorem for the Tzitzeica equationrdquo Physics Letters A vol 223no 1-2 pp 55ndash62 1996

[19] S V Meleshko and W K Schief ldquoA truncated Painleve expan-sion associated with the Tzitzeica equation consistency andgeneral solutionrdquo Physics Letters A vol 299 no 4 pp 349ndash3522002

[20] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-order dispersive PDEsrdquo Nonlinear Analysis Theory Methods ampApplications vol 69 no 4 pp 1256ndash1267 2008

[21] R Weiguo L Yao H Bin and L Zhenyang ldquoIntegral bifurca-tionmethod combinedwith computer for solving a higher orderwave equation of KdV typerdquo International Journal of ComputerMathematics vol 87 no 1-3 pp 119ndash128 2010

[22] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012

[23] W Rui ldquoDifferent kinds of exact solutions with two-loopcharacter of the two-component short pulse equations of thefirst kindrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 10 pp 2667ndash2678 2013

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Journal of Applied Mathematics

[24] F Khani and S Hamedi-Nezhad ldquoSome new exact solutions ofthe (2 + 1)-dimensional variable coefficient Broer-Kaup systemusing the Exp-function methodrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2325ndash2329 2009

[25] F Khani ldquoAnalytic study on the higher order Ito equations newsolitary wave solutions using the Exp-functionmethodrdquo ChaosSolitons and Fractals vol 41 no 4 pp 2128ndash2134 2009

[26] F Khani F Samadi and S Hamedi-Nezhad ldquoNew exactsolutions of the Brusselator reaction diffusion model using theexp-function methodrdquo Mathematical Problems in Engineeringvol 2009 Article ID 346461 9 pages 2009

[27] F Khani S Hamedi-Nezhad and A Molabahrami ldquoA reliabletreatment for nonlinear Schrodinger equationsrdquo Physics LettersA vol 371 no 3 pp 234ndash240 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of