Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations

Research ArticleClassification of Multiply Travelling WaveSolutions for Coupled Burgers Combined KdV-Modified KdVand Schroumldinger-KdV Equations

A R Seadawy12 and K El-Rashidy23

1 Mathematics Department Faculty of Science Taibah University Al-Ula Saudi Arabia2Mathematics Department Faculty of Science Beni-Suef University Egypt3Mathematics Department College of Arts and Science Taif University Ranyah Saudi Arabia

Correspondence should be addressed to A R Seadawy aly742001yahoocom

Received 24 August 2014 Accepted 1 November 2014

Academic Editor Yasir Khan

Copyright copy 2015 A R Seadawy and K El-Rashidy This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematicalphysics are presented By applying a theory of Frobenius decompositions and more precisely by using a transformation method tothe coupled Burgers combined Korteweg-de Vries- (KdV-) modified KdV and Schrodinger-KdV equation is written as bilinearordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated Theproperties of the multiple travelling wave solutions are shown by some figures All solutions are stable and have applications inphysics

1 Introduction

The investigation of traveling wave solutions of nonlinearevolution equations (NLEEs) plays a vital role in differentbranches of mathematical physics engineering sciences andother technical arenas such as plasma physics nonlinearoptics solid state physics fluid mechanics chemical physicsand chemistry

The Burgersrsquo equation has been found to describe variouskinds of phenomena such as a mathematical model ofturbulence [1] and the approximate theory of flow througha shock wave traveling in viscous fluid [2] Fletcher using theHopf-Cole transformation [3] gave an analytic solution forthe system of two-dimensional Burgersrsquo equations

The Korteweg-de Vries (KdV) equation which modelsshallow-water phenomena has been analyzed extensivelyusing the invariance properties that occur from the Liepoint symmetry generator that admits it In particulartravelling wave solutions arise from the combination oftranslations in space and time Also Galilean invariantsand scale-invariant solutions are dependent on first and

second Painleve transcendent [4] Further the modified KdV(mKdV) has attracted interest in a similar way and its Liepoint symmetry generators are known [5] Recently thecombinedKdV (cKdV) andmKdV equation has been studiedusing various methods with a special reference to soliton-type solutions For example simple soliton solutions to cKdV-mKdV used in plasma and fluid physics are obtained in [6]Here the particular form uses the fact that the equationadmits a scaling symmetry which is nonexistent for thegeneral cKdV [7 8]

The topic of solitons produced by nonlinear interactionsis a very fundamental topic in various fields includingoptical solitons in fibers [9] The one-dimensional solitoncan be considered as a localized wave pulse that propagatesalong one space direction undeformed that is dispersion iscompletely compensated by the nonlinear effects There isan enormous amount of literature about the integrability ofnonlinear equations related to scattering equations includingespecially inverse scattering theories in relation to solitons[10] In particular analysis related to NLS and KdV equationshas been studied [10ndash12]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 369294 7 pageshttpdxdoiorg1011552015369294

2 Abstract and Applied Analysis

In recent years various methods have been established toobtain exact traveling solutions of nonlinear partial differen-tial equations for example the Jacobi elliptic function expan-sion method [13] the generalized Riccati equation method[14] the Backlund transformation method [15] the Hirotarsquosbilinear transformationmethod [16] the variational iterationmethod [17ndash20] the tanh-coth method [21 22] the directalgebraic method [23 24] the Cole-Hopf transformationmethod [25 26] the Exp-function method [27ndash29] andothers [30] Recently Wang et al [31] introduced a methodto obtain traveling wave solutions of the nonlinear partialdifferential equations called the (1198661015840119866)- expansion method

This paper is organized as follows An introduction isgiven in Section 1 In Section 2 an analysis and theory ofmethod theory of transformation the partial differentialequations to a bilinear ordinary differential equations InSection 3 the multiple travelling wave solutions of coupledBurgersrsquo equations are obtained Two cases of traveling wavesolutions of the combined KdV-modified KdV equation aregiven in Section 4 In Section 5 multiple travelling wavesolutions of the coupled Schrodinger-KdV equations aregiven

2 An Analysis of the Method and Applications

Now we simply describe the generalized extended tanh-function method Consider a given system of NEEs say intwo variables 119909 and 119905

119867(119906 V 119906119905 V119905 119906119909 V119909 119906119909119905 V119909119905 ) = 0 (1)

119866 (119906 V 119906119905 V119905 119906119909 V119909 119906119909119905 V119909119905 ) = 0 (2)

We consider the following formal traveling wave solutions119906(119909 119905) = 119880(120585) V(119909 119905) = 119881(120585) and 120585 = 119909 minus 119888119905 where 119888 isa constant to be determined later Then (1) and (2) becomesystem of nonlinear ordinary differential equations as

119867(1198801198811198801015840 1198811015840 11988010158401015840 11988110158401015840 ) = 0 (3)

119866(1198801198811198801015840 1198811015840 11988010158401015840 11988110158401015840 ) = 0 (4)

In order to seek the traveling wave solutions of (3) and (4) weintroduce the following new ansatz

119880 (120585) = 11988610+

1198981

sum119895=1

1198861119895120601119895+ 1198871119895120601minus119895

+ 1198881119895120601119895minus1radic119887 + 1206012

+1198891119895

radic119887 + 1206012

120601119895

(5)

119881 (120585) = 11988620+

1198982

sum119895=1

1198862119895120601119895+ 1198872119895120601minus119895

+ 1198882119895120601119895minus1radic119887 + 1206012

+1198892119895

radic119887 + 1206012

120601119895

(6)

where 1198861198940 119886119894119895 119887119894119895 119888119894119895 119889119894119895(119895 = 1 2 119898

119894 119894 = 1 2) and 119887 are

constants to be determined laterThe value of119898119894in (5) and (6)

can be determined by balancing the highest-order derivativeterm with the nonlinear term in (5) and (6)The new variable120601 = 120601(120585) satisfies 1206011015840 = 119887 + 1206012

There exist the following steps to be considered further

Step 1 Determine the 1198981and 119898

2of (5) and (6) by respec-

tively balancing the highest order partial differential termsand the nonlinear terms in (3) and (4)

Step 2 Substituting (5) and (6) into (3) and (4) the corre-sponding ODEs then let all coefficients of 120601119901(radic119887 + 1206012)

119902(119902 =

0 1 119901 = 0 1 2 ) be zero to get an overdetermined systemof nonlinear algebraic equations with respect to 119886

1198940 119886119894119895 119887119894119895 119888119894119895

119889119894119895 119887 119888 (119895 = 1 2 119898

119894 119894 = 1 2)

Step 3 By solving the system we may determine the aboveparameters

Step 4 Substituting the parameters 1198861198940 119886119894119895 119887119894119895 119888119894119895 119889119894119895 119887 119888 (119895 =

1 2 119898119894 119894 = 1 2) obtained in Step 3 into (2) we can derive

the solutions of equation

3 Example I

The coupled Burgersrsquo equations [32] have applications in thequantum field theory plasma physics fluid mechanics andsolid state physics The usual system of equation is as follows

119906119905minus 119906119909119909

+ 2119906119906119909+ 120572 (119906V)

119909= 0 (7)

V119905minus V119909119909

+ 2VV119909+ 120573 (119906V)

119909= 0 (8)

Let us consider the traveling wave solutions 119906(119909 119905) = 119880(120585)V(119909 119905) = 119881(120585) and 120585 = 119909 minus 119888119905 and then (4) becomes

minus119888119880 minus 1198801015840+ 1198802+ 120572119880119881 = 0 (9)

minus119888119881 minus 1198811015840+ 1198812+ 120573119880119881 = 0 (10)

Balancing the nonlinear term 1198802 and the highest orderderivative 1198801015840 gives 119898 = 2 We suppose the solutions of (5)and (6) are of the forms

119880 (120585) = 1198860+ 1198861120593 + 1198862120593minus1

+ 1198863radic119887 + 1205932 + 119886

4

radic119887 + 1205932

120593 (11)

Abstract and Applied Analysis 3

119881 (120585) = 1198870+ 1198871120593 + 1198872120593minus1

+ 1198873radic119887 + 1205932 + 119887

4

radic119887 + 1205932

120593

+ 11988751205932+ 1198876120593minus2

+ 1198877120593radic119887 + 1205932 + 119887

8

radic119887 + 1205932

1205932

(12)

Substituting (11) and (12) into (9) and (10) with 1206011015840 = 119887 + 1206012

yields a set of algebraic equations for 1198860 1198861 1198862 1198863 1198864 1198870 1198871

1198872 1198873 1198874 1198875 1198876 1198877 1198878

By the solution of the above system of equations we canfind

1198860=

2119888

120573 119886

1= 0 119886

2=

21198882

120573 minus 1

1198863= minus

3119888radicminus119887 minus 21198882 minus 21198882120573

120573 (119887 + 21198882 + 21198882120573) 119886

4= 0

(13)

1198870= minus2119888 119887

1= minus1 119887

2=

212057321198882

1 minus 120573

1198873=


119887 + 21198882 + 21198882120573

1198874= plusmnradicminus119887 minus 21198882 minus 21198882120573 119887

5= 1198876= 1198877= 1198878= 0

(14)

Then combining (11) and (12) with (13) and (14) weobtain the traveling wave solutions of (7) and (8) as

119906 (119909 119905) =2119888

120573+

21198882

(120573 minus 1)radic119887cotradic119887 (119909 + 119888119905)

minus3119888radicminus119887 minus 21198882 minus 21198882120573

120573 (119887 + 21198882 + 21198882120573)radic119887secradic119887 (119909 + 119888119905)

(15)

V (119909 119905) = minus2119888 minus radic119887 tanradic119887 (119909 + 119888119905)

+212057321198882


+3119888radicminus119887 minus 21198882 minus 21198882120573

119887 + 21198882 + 21198882120573radic119887secradic119887 (119909 + 119888119905)

plusmn radicminus119887 minus 21198882 minus 21198882120573cscradic119887 (119909 + 119888119905)

(16)

The shape solutions (15) and (16) are shown in Figures1(a) 1(b) 1(c) and 1(d) with 120573 = minus2 119887 = 2 and 119888 = plusmn2

4 Example II

Consider the combined KdV-modified KdV equation

119906119905minus 120572119906119906

119909+ 1205731199062119906119909+ 120575119906119909119909119909

= 0 120573 = 0 (17)

where 120572 120573 and 120575 are constants Let us consider the travelingwave solutions 119906(119909 119905) = 119880(120585) and 120585 = 119909 + 119888119905 and then (17)becomes

119888119880 +120572

21198802+120573

31198803+ 11988010158401015840= 0 (18)

We suppose the solution of (18) is of the form (11) andsubstituting into (18) yields a set of algebraic equations for 119886

0

1198861 1198862 1198863 1198864 We have two cases for these equations that are

found

Case 1 By the solution of (11) we can find

1198860= minus

120572

2120573 119886

1= 0

1198862= plusmn119894radic

3

2120573119887 119886

3= 0


3119887

2120573

(19)

Substituting (11) into (19) we have obtained the followingsolution of (17)

119906 (119909 119905) = minus120572

2120573plusmn 119894radic

3119887

2120573cot [radic119887 (119909 + 119888119905)]

plusmn 119894radic3119887

2120573csc [radic119887 (119909 + 119888119905)]

(20)


1198860= minus130 119886

1= plusmn65radic

2

119887

1198862= plusmn65radic2119887 119886

3= 1198864= 0

(21)

Substituting from (11) and (21) we have obtained the followingsolution of (17)

119906 (119909 119905) = minus130 plusmn 65radic2 tan [radic119887 (119909 + 119888119905)]

plusmn 65radic2 cot [radic119887 (119909 + 119888119905)]

(22)

The shape solution (22) is shown in Figure 2(b) with 119887 = 2

and 119888 = minus16

5 Example III

Consider the coupled Schrodinger-KdV equations (Davey-Stewartson)

119894119906119905+ 119906119909119909

minus 119906119910119910

minus 2 |119906|2119906 minus 2119906V = 0 (23)

V119909119909

minus V119910119910

minus 2 (|119906|2)119909119909

= 0 (24)


20

0

00

05

1001

02

03

04

05

minus20

minus05

minus10

x

t

0

00

050 1

02

03

04

0

20

minus05

10

x

t

u(xt)

(a)

00 05 1001

02

03

04

05

minus10 minus05

(b)

50

minus50

0

00

0005

05

10

10

minus05

minus10

x

t

0 0

05

10

minus05

0 t

u(xt)

(c)

00 05 1000

02

04

06

08

10

minus10 minus05

(d)

Figure 1 Travelling waves solutions (15) with various different shapes are plotted periodic solitary waves in (a) and contour plot in (b)Travelling waves solutions (16) with various different shapes are plotted periodic solitary waves in (c) and contour plot in (d)

Let us consider the traveling wave solutions 119906(119909 119905) = 119880(120585)

and 120585 = (119896119909 + 119888119910 + 119889119905)(119909 + 119910 minus 119889119905) and then (23) and (24)become

(1199022minus 1199012minus 119903)119880 + (119896

2minus 1198882)11988010158401015840+ 1198963minus 2119880(3)

minus 2119880119881 = 0

(25)

(1198962+ 1198882)11988110158401015840+ (1198802)10158401015840

= 0 (26)

We suppose the solutions of (25) and (26) are of the forms (11)and (12) Substituting (11) and (12) into (25) and (26) yields aset of algebraic equations for 119886

0 1198861 1198862 1198863 1198864 1198870 1198871 1198872 1198873 1198874

1198875 1198876 1198877 1198878 The solution of the system of equations has two

cases

Case 1 Consider the following 1198962 = 14 1198882 = 34 and 119887 =

59

1198860= 0 119886

1= plusmn

10radic2

9 119886

2= minus1

1198863= 2 119886

4=

3

2

1198870=

1

36(minus17 plusmn 80radic2 minus 18119901

2+ 18119902

2minus 18119903)

1198871= minus

112

27 119887

2= minus

10

3

(27)


0

minus10

5

0

minus5

2

4

00

05

10

minus05

x

t0

2

4

00

0 5

minus05

x

t

u(xt)

(a)

00 05 100

1

2

3

4

5

minus10 minus05

(b)

1000

0

minus1000

minus10

minus05

00

05

10 000

005

010

x

tminus05

005t

u(xt)

(c)

00 05 10000

002

004

006

008

010

minus10 minus05

(d)


1198873= ∓

10radic2

3 119887

4= 4 119887

5= minus16

1198876= minus

9

4 119887

7=

20

9(1 plusmn 2radic2) 119887

8= 3

(28)

Substituting from (25) (26) (27) and (28) we haveobtained the following solutions of (23) and (24)

119906 (119909 119905) = plusmn10radic2119887

9tan 120585 minus 1

radic119887cot 120585 + 2radic119887 sec 120585 + 3

2csc 120585

(29)

V (119909 119905)

=1


2+ 18119902

2minus 18119903)

minus112radic119887

27tan 120585 minus 10

3radic119887cot 120585 ∓ 10radic2119887

3sec 120585 + 4 csc 120585

minus 16119887 tan2120585 minus 9

4119887cot2120585 + 20119887

9(1 plusmn 2radic2) tan 120585sec120585

+3

radic119887cot 120585 csc 120585

(30)

with 120585 = radic119887((119909 + radic3119910 + radic4119905)radic4(119909 + 119910 minus 119905))


5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)

Figure 3 Travelling waves solutions (29) and (30) with various different shapes are plotted multiple travelling wave solutions (a and b)Travelling waves solutions (33) and (34) with various different shapes are plotted multiple travelling wave solutions (c and d)

Case 2 Consider the following 1198962 = 19 1198882 = 89 and 119887 = 1

1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)

Substituting (31) and (32) into (25) and (26) we have obtainedthe following solutions of (23) and (24)

119906 (119909 119905) = 1 ∓radic14119887

3tan 120585 plusmn 1

3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)

V (119909 119905) = minus2 plusmn 2radic14119887

3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)

with 120585 = radic119887((119909 + radic8119910 + radic9119905)radic9(119909 + 119910 minus 119905))The shape solutions (29) and (30) are shown in Figures

3(a)and 3(b) with 119901 = 119902 = 119903 = 1 and 119905 = 001 The shapesolutions (33) and (34) are shown in Figures 3(c)and 3(d)with 119901 = 119902 = 119903 = 1 and 119905 = 001


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[2] C A J Fletcher ldquoGenerating exact solutions of the two-dimensional Burgersrsquo equationsrdquo International Journal forNumerical Methods in Fluids vol 3 no 3 pp 213ndash216 1983

[3] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

[4] P J Olver Applications of Lie Groups to Differential Equationsvol 107 Springer New York NY USA 1986

[5] G Bluman and S Kumei Symmetries and Differenti al Equa-tions Springer New York NY USA 1986

[6] J Zhang F Wu and J Shi ldquoSimple soliton solution method forthe combined KdV and MKdV equationrdquo International Journalof Theoretical Physics vol 39 no 6 pp 1697ndash1702 2000

[7] A R Seadawy ldquoNew exact solutions for the KdV equationwith higher order nonlinearity by using the variationalmethodrdquoComputers amp Mathematics with Applications vol 62 no 10 pp3741ndash3755 2011

[8] A R Seadawy ldquoStability analysis for Zakharov-Kuznetsovequation of weakly nonlinear ion-acoustic waves in a plasmardquoComputers amp Mathematics with Applications vol 67 no 1 pp172ndash180 2014

[9] G P Agrawal Applications of Nonlinear Fiber Optics Elsevier2008

[10] E Infeld andGRowlandsNonlinearWaves Solitons andChaosCambridge University Press 2000

[11] A H Khater M A Helal and A R Seadawy ldquoGeneral solitonsolutions of ndimensional nonlinear Schrodinger equationrdquo ILNuovo Cimento B vol 115 pp 1303ndash1312 2000

[12] A R Seadawy ldquoExact solutions of a two-dimensional nonlinearSchrodinger equationrdquoAppliedMathematics Letters vol 25 no4 pp 687ndash691 2012

[13] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[14] Z Yan and H Zhang ldquoNew explicit solitary wave solutions andperiodic wave solutions for Whitham-Broer-Kaup equation inshallow waterrdquo Physics Letters A vol 285 no 5-6 pp 355ndash3622001

[15] C Rogers and W F Shadwick Backlund Transformations andTheir Applications Academic Press New York NY USA 1982

[16] R Hirota ldquoExact solution of the korteweg-de vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[17] J H He ldquoVariational iteration method for delay differentialequationsrdquo Communications in Nonlinear Science and Numer-ical Simulation vol 2 no 4 pp 235ndash236 1997

[18] A S Arife and A Yildirim ldquoNewmodified variational iterationtransform method (MVITM) for solving eighth-order bound-ary value problems in one steprdquoWorld Applied Sciences Journalvol 13 no 10 pp 2186ndash2190 2011

[19] M A Helal and A R Seadawy ldquoVariational method for thederivative nonlinear Schrodinger equation with computationalapplicationsrdquo Physica Scripta vol 80 no 3 Article ID 035004pp 350ndash360 2009

[20] S T Mohyud-Din ldquoModified variational iteration method forintegro-differential equations and coupled systemsrdquo Zeitschriftfur Naturforschung A vol 65 no 4 pp 277ndash284 2010

[21] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo The American Journal of Physics vol 60 no 7 pp 650ndash654 1992

[22] A Bekir and A C Cevikel ldquoSolitary wave solutions of two non-linear physical models by tanh-coth methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 5 pp1804ndash1809 2009

[23] A R Seadawy and K El-Rashidy ldquoTraveling wave solutionsfor some coupled nonlinear evolution equationsrdquoMathematicaland Computer Modelling vol 57 no 5-6 pp 1371ndash1379 2013

[24] A R Seadawy ldquoTraveling wave solutions of the Boussinesqand generalized fifth order KdV equations by using the directalgebraic methodrdquo Applied Mathematical Sciences vol 6 pp4081ndash4090 2012

[25] A H Salas and C A Gomez ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[26] W X Ma and B Fuchssteiner ldquoExplicit and exact solutionsto a Kolmogorov-Petrovskii-PISkunov equationrdquo InternationalJournal of Non-Linear Mechanics vol 31 no 3 pp 329ndash3381996

[27] H Naher F A Abdullah and M A Akbar ldquoThe exp-functionmethod for new exact solutions of the nonlinear partial differ-ential equationsrdquo International Journal of Physical Sciences vol6 no 29 pp 6706ndash6716 2011

[28] I Aslan and V Marinakis ldquoSome remarks on exp-functionmethod and its applicationsrdquo Communications in TheoreticalPhysics vol 56 no 3 pp 397ndash403 2011

[29] A Yıldırım and Z Pınar ldquoApplication of the exp-functionmethod for solving nonlinear reaction-diffusion equationsarising in mathematical biologyrdquo Computers amp Mathematicswith Applications vol 60 no 7 pp 1873ndash1880 2010

[30] A-M Wazwaz ldquoA new (2+1)-dimensional Korteweg-de Vriesequation and its extension to a new (3+1)-dimensionalKadomtsev-Petviashvili equationrdquo Physica Scripta vol 84 no3 Article ID 035010 2011

[31] M Wang X Li and J Zhang ldquoThe (1198661015840

119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] D Kaya ldquoAn explicit solution of coupled viscous Burgersrsquoequation by the decomposition methodrdquo International Journalof Mathematics and Mathematical Sciences vol 27 no 11 pp675ndash680 2001

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In recent years various methods have been established toobtain exact traveling solutions of nonlinear partial differen-tial equations for example the Jacobi elliptic function expan-sion method [13] the generalized Riccati equation method[14] the Backlund transformation method [15] the Hirotarsquosbilinear transformationmethod [16] the variational iterationmethod [17ndash20] the tanh-coth method [21 22] the directalgebraic method [23 24] the Cole-Hopf transformationmethod [25 26] the Exp-function method [27ndash29] andothers [30] Recently Wang et al [31] introduced a methodto obtain traveling wave solutions of the nonlinear partialdifferential equations called the (1198661015840119866)- expansion method

This paper is organized as follows An introduction isgiven in Section 1 In Section 2 an analysis and theory ofmethod theory of transformation the partial differentialequations to a bilinear ordinary differential equations InSection 3 the multiple travelling wave solutions of coupledBurgersrsquo equations are obtained Two cases of traveling wavesolutions of the combined KdV-modified KdV equation aregiven in Section 4 In Section 5 multiple travelling wavesolutions of the coupled Schrodinger-KdV equations aregiven

2 An Analysis of the Method and Applications

Now we simply describe the generalized extended tanh-function method Consider a given system of NEEs say intwo variables 119909 and 119905

119867(119906 V 119906119905 V119905 119906119909 V119909 119906119909119905 V119909119905 ) = 0 (1)

119866 (119906 V 119906119905 V119905 119906119909 V119909 119906119909119905 V119909119905 ) = 0 (2)

We consider the following formal traveling wave solutions119906(119909 119905) = 119880(120585) V(119909 119905) = 119881(120585) and 120585 = 119909 minus 119888119905 where 119888 isa constant to be determined later Then (1) and (2) becomesystem of nonlinear ordinary differential equations as

119867(1198801198811198801015840 1198811015840 11988010158401015840 11988110158401015840 ) = 0 (3)

119866(1198801198811198801015840 1198811015840 11988010158401015840 11988110158401015840 ) = 0 (4)

In order to seek the traveling wave solutions of (3) and (4) weintroduce the following new ansatz

119880 (120585) = 11988610+

1198981

sum119895=1

1198861119895120601119895+ 1198871119895120601minus119895

+ 1198881119895120601119895minus1radic119887 + 1206012

+1198891119895

radic119887 + 1206012

120601119895

(5)

119881 (120585) = 11988620+

1198982

sum119895=1

1198862119895120601119895+ 1198872119895120601minus119895

+ 1198882119895120601119895minus1radic119887 + 1206012

+1198892119895

radic119887 + 1206012

120601119895

(6)

where 1198861198940 119886119894119895 119887119894119895 119888119894119895 119889119894119895(119895 = 1 2 119898

119894 119894 = 1 2) and 119887 are

constants to be determined laterThe value of119898119894in (5) and (6)

can be determined by balancing the highest-order derivativeterm with the nonlinear term in (5) and (6)The new variable120601 = 120601(120585) satisfies 1206011015840 = 119887 + 1206012

There exist the following steps to be considered further

Step 1 Determine the 1198981and 119898

2of (5) and (6) by respec-

tively balancing the highest order partial differential termsand the nonlinear terms in (3) and (4)

Step 2 Substituting (5) and (6) into (3) and (4) the corre-sponding ODEs then let all coefficients of 120601119901(radic119887 + 1206012)

119902(119902 =

0 1 119901 = 0 1 2 ) be zero to get an overdetermined systemof nonlinear algebraic equations with respect to 119886

1198940 119886119894119895 119887119894119895 119888119894119895

119889119894119895 119887 119888 (119895 = 1 2 119898

119894 119894 = 1 2)

Step 3 By solving the system we may determine the aboveparameters

Step 4 Substituting the parameters 1198861198940 119886119894119895 119887119894119895 119888119894119895 119889119894119895 119887 119888 (119895 =

1 2 119898119894 119894 = 1 2) obtained in Step 3 into (2) we can derive

the solutions of equation

3 Example I

The coupled Burgersrsquo equations [32] have applications in thequantum field theory plasma physics fluid mechanics andsolid state physics The usual system of equation is as follows

119906119905minus 119906119909119909

+ 2119906119906119909+ 120572 (119906V)

119909= 0 (7)

V119905minus V119909119909

+ 2VV119909+ 120573 (119906V)

119909= 0 (8)

Let us consider the traveling wave solutions 119906(119909 119905) = 119880(120585)V(119909 119905) = 119881(120585) and 120585 = 119909 minus 119888119905 and then (4) becomes

minus119888119880 minus 1198801015840+ 1198802+ 120572119880119881 = 0 (9)

minus119888119881 minus 1198811015840+ 1198812+ 120573119880119881 = 0 (10)

Balancing the nonlinear term 1198802 and the highest orderderivative 1198801015840 gives 119898 = 2 We suppose the solutions of (5)and (6) are of the forms

119880 (120585) = 1198860+ 1198861120593 + 1198862120593minus1

+ 1198863radic119887 + 1205932 + 119886

4

radic119887 + 1205932

120593 (11)


119881 (120585) = 1198870+ 1198871120593 + 1198872120593minus1

+ 1198873radic119887 + 1205932 + 119887

4

radic119887 + 1205932

120593

+ 11988751205932+ 1198876120593minus2

+ 1198877120593radic119887 + 1205932 + 119887

8

radic119887 + 1205932

1205932

(12)



1198872 1198873 1198874 1198875 1198876 1198877 1198878


1198860=

2119888

120573 119886

1= 0 119886

2=

21198882

120573 minus 1

1198863= minus


120573 (119887 + 21198882 + 21198882120573) 119886

4= 0

(13)

1198870= minus2119888 119887

1= minus1 119887

2=

212057321198882

1 minus 120573

1198873=


119887 + 21198882 + 21198882120573


5= 1198876= 1198877= 1198878= 0

(14)


119906 (119909 119905) =2119888

120573+

21198882



120573 (119887 + 21198882 + 21198882120573)radic119887secradic119887 (119909 + 119888119905)

(15)


+212057321198882



119887 + 21198882 + 21198882120573radic119887secradic119887 (119909 + 119888119905)


(16)


4 Example II


119906119905minus 120572119906119906

119909+ 1205731199062119906119909+ 120575119906119909119909119909

= 0 120573 = 0 (17)


119888119880 +120572

21198802+120573

31198803+ 11988010158401015840= 0 (18)


0


found


1198860= minus

120572

2120573 119886

1= 0


3

2120573119887 119886

3= 0


3119887

2120573

(19)


119906 (119909 119905) = minus120572


3119887

2120573cot [radic119887 (119909 + 119888119905)]


2120573csc [radic119887 (119909 + 119888119905)]

(20)


1198860= minus130 119886

1= plusmn65radic

2

119887

1198862= plusmn65radic2119887 119886

3= 1198864= 0

(21)




(22)



5 Example III


119894119906119905+ 119906119909119909

minus 119906119910119910

minus 2 |119906|2119906 minus 2119906V = 0 (23)

V119909119909

minus V119910119910

minus 2 (|119906|2)119909119909

= 0 (24)


20

0

00

05

1001

02

03

04

05

minus20

minus05

minus10

x

t

0

00

050 1

02

03

04

0

20

minus05

10

x

t

u(xt)

(a)

00 05 1001

02

03

04

05

minus10 minus05

(b)

50

minus50

0

00

0005

05

10

10

minus05

minus10

x

t

0 0

05

10

minus05

0 t

u(xt)

(c)

00 05 1000

02

04

06

08

10

minus10 minus05

(d)




(1199022minus 1199012minus 119903)119880 + (119896

2minus 1198882)11988010158401015840+ 1198963minus 2119880(3)

minus 2119880119881 = 0

(25)

(1198962+ 1198882)11988110158401015840+ (1198802)10158401015840

= 0 (26)


0 1198861 1198862 1198863 1198864 1198870 1198871 1198872 1198873 1198874


cases


59

1198860= 0 119886

1= plusmn

10radic2

9 119886

2= minus1

1198863= 2 119886

4=

3

2

1198870=

1


2+ 18119902

2minus 18119903)

1198871= minus

112

27 119887

2= minus

10

3

(27)


0

minus10

5

0

minus5

2

4

00

05

10

minus05

x

t0

2

4

00

0 5

minus05

x

t

u(xt)

(a)

00 05 100

1

2

3

4

5

minus10 minus05

(b)

1000

0

minus1000

minus10

minus05

00

05

10 000

005

010

x

tminus05

005t

u(xt)

(c)

00 05 10000

002

004

006

008

010

minus10 minus05

(d)


1198873= ∓

10radic2

3 119887

4= 4 119887

5= minus16

1198876= minus

9

4 119887

7=

20


8= 3

(28)


119906 (119909 119905) = plusmn10radic2119887

9tan 120585 minus 1


2csc 120585

(29)

V (119909 119905)

=1


2+ 18119902

2minus 18119903)

minus112radic119887

27tan 120585 minus 10

3radic119887cot 120585 ∓ 10radic2119887

3sec 120585 + 4 csc 120585

minus 16119887 tan2120585 minus 9

4119887cot2120585 + 20119887


+3

radic119887cot 120585 csc 120585

(30)



5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)



1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)


119906 (119909 119905) = 1 ∓radic14119887


3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)


3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)






References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119881 (120585) = 1198870+ 1198871120593 + 1198872120593minus1

+ 1198873radic119887 + 1205932 + 119887

4

radic119887 + 1205932

120593

+ 11988751205932+ 1198876120593minus2

+ 1198877120593radic119887 + 1205932 + 119887

8

radic119887 + 1205932

1205932

(12)



1198872 1198873 1198874 1198875 1198876 1198877 1198878


1198860=

2119888

120573 119886

1= 0 119886

2=

21198882

120573 minus 1

1198863= minus


120573 (119887 + 21198882 + 21198882120573) 119886

4= 0

(13)

1198870= minus2119888 119887

1= minus1 119887

2=

212057321198882

1 minus 120573

1198873=


119887 + 21198882 + 21198882120573


5= 1198876= 1198877= 1198878= 0

(14)


119906 (119909 119905) =2119888

120573+

21198882



120573 (119887 + 21198882 + 21198882120573)radic119887secradic119887 (119909 + 119888119905)

(15)


+212057321198882



119887 + 21198882 + 21198882120573radic119887secradic119887 (119909 + 119888119905)


(16)


4 Example II


119906119905minus 120572119906119906

119909+ 1205731199062119906119909+ 120575119906119909119909119909

= 0 120573 = 0 (17)


119888119880 +120572

21198802+120573

31198803+ 11988010158401015840= 0 (18)


0


found


1198860= minus

120572

2120573 119886

1= 0


3

2120573119887 119886

3= 0


3119887

2120573

(19)


119906 (119909 119905) = minus120572


3119887

2120573cot [radic119887 (119909 + 119888119905)]


2120573csc [radic119887 (119909 + 119888119905)]

(20)


1198860= minus130 119886

1= plusmn65radic

2

119887

1198862= plusmn65radic2119887 119886

3= 1198864= 0

(21)




(22)



5 Example III


119894119906119905+ 119906119909119909

minus 119906119910119910

minus 2 |119906|2119906 minus 2119906V = 0 (23)

V119909119909

minus V119910119910

minus 2 (|119906|2)119909119909

= 0 (24)


20

0

00

05

1001

02

03

04

05

minus20

minus05

minus10

x

t

0

00

050 1

02

03

04

0

20

minus05

10

x

t

u(xt)

(a)

00 05 1001

02

03

04

05

minus10 minus05

(b)

50

minus50

0

00

0005

05

10

10

minus05

minus10

x

t

0 0

05

10

minus05

0 t

u(xt)

(c)

00 05 1000

02

04

06

08

10

minus10 minus05

(d)




(1199022minus 1199012minus 119903)119880 + (119896

2minus 1198882)11988010158401015840+ 1198963minus 2119880(3)

minus 2119880119881 = 0

(25)

(1198962+ 1198882)11988110158401015840+ (1198802)10158401015840

= 0 (26)


0 1198861 1198862 1198863 1198864 1198870 1198871 1198872 1198873 1198874


cases


59

1198860= 0 119886

1= plusmn

10radic2

9 119886

2= minus1

1198863= 2 119886

4=

3

2

1198870=

1


2+ 18119902

2minus 18119903)

1198871= minus

112

27 119887

2= minus

10

3

(27)


0

minus10

5

0

minus5

2

4

00

05

10

minus05

x

t0

2

4

00

0 5

minus05

x

t

u(xt)

(a)

00 05 100

1

2

3

4

5

minus10 minus05

(b)

1000

0

minus1000

minus10

minus05

00

05

10 000

005

010

x

tminus05

005t

u(xt)

(c)

00 05 10000

002

004

006

008

010

minus10 minus05

(d)


1198873= ∓

10radic2

3 119887

4= 4 119887

5= minus16

1198876= minus

9

4 119887

7=

20


8= 3

(28)


119906 (119909 119905) = plusmn10radic2119887

9tan 120585 minus 1


2csc 120585

(29)

V (119909 119905)

=1


2+ 18119902

2minus 18119903)

minus112radic119887

27tan 120585 minus 10

3radic119887cot 120585 ∓ 10radic2119887

3sec 120585 + 4 csc 120585

minus 16119887 tan2120585 minus 9

4119887cot2120585 + 20119887


+3

radic119887cot 120585 csc 120585

(30)



5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)



1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)


119906 (119909 119905) = 1 ∓radic14119887


3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)


3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)






References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20

0

00

05

1001

02

03

04

05

minus20

minus05

minus10

x

t

0

00

050 1

02

03

04

0

20

minus05

10

x

t

u(xt)

(a)

00 05 1001

02

03

04

05

minus10 minus05

(b)

50

minus50

0

00

0005

05

10

10

minus05

minus10

x

t

0 0

05

10

minus05

0 t

u(xt)

(c)

00 05 1000

02

04

06

08

10

minus10 minus05

(d)




(1199022minus 1199012minus 119903)119880 + (119896

2minus 1198882)11988010158401015840+ 1198963minus 2119880(3)

minus 2119880119881 = 0

(25)

(1198962+ 1198882)11988110158401015840+ (1198802)10158401015840

= 0 (26)


0 1198861 1198862 1198863 1198864 1198870 1198871 1198872 1198873 1198874


cases


59

1198860= 0 119886

1= plusmn

10radic2

9 119886

2= minus1

1198863= 2 119886

4=

3

2

1198870=

1


2+ 18119902

2minus 18119903)

1198871= minus

112

27 119887

2= minus

10

3

(27)


0

minus10

5

0

minus5

2

4

00

05

10

minus05

x

t0

2

4

00

0 5

minus05

x

t

u(xt)

(a)

00 05 100

1

2

3

4

5

minus10 minus05

(b)

1000

0

minus1000

minus10

minus05

00

05

10 000

005

010

x

tminus05

005t

u(xt)

(c)

00 05 10000

002

004

006

008

010

minus10 minus05

(d)


1198873= ∓

10radic2

3 119887

4= 4 119887

5= minus16

1198876= minus

9

4 119887

7=

20


8= 3

(28)


119906 (119909 119905) = plusmn10radic2119887

9tan 120585 minus 1


2csc 120585

(29)

V (119909 119905)

=1


2+ 18119902

2minus 18119903)

minus112radic119887

27tan 120585 minus 10

3radic119887cot 120585 ∓ 10radic2119887

3sec 120585 + 4 csc 120585

minus 16119887 tan2120585 minus 9

4119887cot2120585 + 20119887


+3

radic119887cot 120585 csc 120585

(30)



5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)



1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)


119906 (119909 119905) = 1 ∓radic14119887


3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)


3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)






References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

minus10

5

0

minus5

2

4

00

05

10

minus05

x

t0

2

4

00

0 5

minus05

x

t

u(xt)

(a)

00 05 100

1

2

3

4

5

minus10 minus05

(b)

1000

0

minus1000

minus10

minus05

00

05

10 000

005

010

x

tminus05

005t

u(xt)

(c)

00 05 10000

002

004

006

008

010

minus10 minus05

(d)


1198873= ∓

10radic2

3 119887

4= 4 119887

5= minus16

1198876= minus

9

4 119887

7=

20


8= 3

(28)


119906 (119909 119905) = plusmn10radic2119887

9tan 120585 minus 1


2csc 120585

(29)

V (119909 119905)

=1


2+ 18119902

2minus 18119903)

minus112radic119887

27tan 120585 minus 10

3radic119887cot 120585 ∓ 10radic2119887

3sec 120585 + 4 csc 120585

minus 16119887 tan2120585 minus 9

4119887cot2120585 + 20119887


+3

radic119887cot 120585 csc 120585

(30)



5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)



1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)


119906 (119909 119905) = 1 ∓radic14119887


3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)


3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)






References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

4

3

2

minus10

minus05minus0500

00

05

05

10

10

minus10

x

0

minus05minus0500

00

05

05

1 0

x

y

u(xy)

(a)

0

minus10

minus20

minus10

minus05

00

05

10minus10

minus05

00

05

10

x

0

minus05

00

051 0

minus05

00

05

x

y

u(xy)

(b)

x

10

0

minus10

minus10

minus10

minus5minus50

0

5

5

10

10

x

0

minus5minus50

0

5

5

y

u(xy)

(c)

20

0

minus20

minus40

minus10

minus10

minus05minus0500

00

05

05

10

10

x

0

minus05minus0500

00

0 5

05

x

y

u(xy)

(d)



1198860= 1198864= 1 119886

1= 1198863= ∓

radic14

3 119886

2= plusmn

1

3radic7

2 (31)

1198870= minus2 119887

1= plusmn2radic

14

3 119887

2= plusmn

radic14

3

1198873= ∓

8radic14

3 119887

4=

1

54(minus10 + 9119901

2minus 91199022+ 9119903)

1198875= minus

14

9 119887

6=

1

108(minus36 + 9119901

2minus 91199022+ 9119903)

1198877=

28

9 119887

8= ∓

radic14

3

(32)


119906 (119909 119905) = 1 ∓radic14119887


3radic

7

2119887cot 120585 ∓

radic14119887

3sec + csc 120585

(33)


3tan 120585 plusmn

radic14

3119887cot 120585 ∓ 8radic14119887

3sec 120585

+1

54(minus10 + 9119901

2minus 91199022+ 9119903) csc 120585

minus14119887

9tan2120585

+1

108119887(minus36 + 9119901

2minus 91199022+ 9119903) cot2120585

+28119887

9tan 120585 sec 120585 ∓

radic14

3radic119887cot 120585 csc 120585

(34)






References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations