The Riccati equation method with variable expansion coefficients. II. Solving the KdV equation

The Riccati equation method

with variable expansion coefficients.

II. Solving the KdV equation.

Solomon M. Antoniou

SKEMSYS Scientific Knowledge Engineering

and Management Systems

37 Κoliatsou Street, Corinthos 20100, Greece [email protected]

Abstract

The Riccati equation method with variable expansion coefficients, introduced in a

previous paper, is used to find travelling wave solutions to the KdV equation

0uauuu xxxxt =++ . The −ξ dependent coefficients A and B of the Riccati

equation 2BYAY +=′ satisfy their own nonlinear ODEs, which can be further

solved by one of the known methods, like Jacobi's elliptic equation method. The

KdV equation is also solved by the −′ )G/G( expansion method with variable

expansion coefficients.

Keywords: Riccati method, nonlinear evolution equations, traveling wave

solutions, KdV equation, exact solutions.

2

1. Introduction .

Nonlinear partial differential equations arise in a number of areas of Mathematics

and Physics in an attempt to model physical processes, like Chemical Kinetics

(Gray and Scott [37]), Fluid Mechanics (Whitham [94]), or biological processes

like Population Dynamics (Murray [65]). In the recent past there are a number of

new methods which have been invented in solving these equations. Among the

new methods are the inverse scattering method (Ablowitz and Segur [13],

Ablowitz and Clarkson [12], Novikov, Manakov, Pitaevskii and Zakharov [69]),

Hirota’s bilinear method (Hirota [43] and [44), the Bäcklund transformation

method (Rogers and Shadwick [77]), the Painleve truncated method (Weiss,

Tabor and Carnevale [92] and [93]), the Cole-Hopf transformation method (Salas

and Gomez [80]), the algebro-geometric approach (Belokolos et al [19]), the Lie

symmetry method (Lie point symmetries, potential symmetries, nonclassical

symmetries, the direct method) (Bluman and Kumei [20], Hydon [45], Olver [70],

Ovsiannikov [71], Stephani [82]), the tanh-coth method (Malfliet [58] and [59],

Malfliet and Hereman [60] and [61], El-Wakil and Abdou [26], Fan [30], Griffiths

and Sciesser [38], Fan and Hon [31], Parkes and Duffy [73], Parkes, Zhou, Duffy

and Huang [75], Wazwaz [90] ), the sn-cn method (Baldwin et al [17]), the F-

expansion method (Abdou [4] and [7], Wang and Li [87]), the Jacobi elliptic

function method (Abbott, Parkes and Duffy [1], Abdou and Elhanbaly [10], Chen

and Zhang [22], Chen and Wang [23], Fan and Zhang [32], Inc and Ergüt [46],

Liu, Fu, Liu and Zhao [55], [56], Lu and Shi [57], Parkes, Duffy and Abbott [74]),

the Riccati equation method (Zhang and Zhang [101], Abdou [3], Antoniou [15]),

the Weierstrass elliptic function method (Kudryashov [51], [53]), the exp-

function method (He and Wu [41], Abdou [8], Aslan [16], Bekir and Boz [18],

Ebaid [24], El-Wakil, Abdou and Hendi [27], He and Abdou [40], Naher,

Abdullah and Akbar [66] and [67]), the Adomian decomposition method

(Adomian [14], Abdou [2], Wazwaz [89] and [91]), the −′ G/G expansion

3

method (Borhanibar and Moghanlu [21], Feng, Li and Wan [33], Jabbari, Kheiri

and Bekir [47], Naher, Abdullah and Akbar [68], Ozis and Aslan [72], Wang, Li

and Zhang [88], Zayed [98], Zayed and Gepreel [100]), the homogeneous

balance method (Fan [29], Wang, Zhou and Li [86], El-Wakil, Abulwafa,

Elhanbaly and Abdou [25], Wang, Zhou and Li [86]), the direct algebraic method

(Soliman and Abdou [81]), the basic equation method (Kudryashov [52]) and its

variants, like the simplest equation method (Abdou [6], Jawad, Petkovich and

Biswas [48], Vitanov [85], Yefimova [96], Zayed [99]), the first integral method

(Feng [34], Raslan [76]), the integral bifurcation method (Rui, Xie, Long and He

[78]), the reduced differential transform method (Keskin and Oturanc [49]), the

homotopy perturbation method (Taghizadeh, Akbari and Ghelichzadeh [83],

Yahya et al [95], Liao [54], El-Wakil and Abdou [28]), the variational iteration

method (He [39], Abdou [5], Abdou and Soliman [9], Wazwaz [91]). A more

detailed (but not complete) set of references of the above methods appears in

Antoniou [15]. The implementation of most of these methods was made possible

only using Symbolic Languages like Mathematica, Macsyma, Maple, etc.

In this paper we introduce the Riccati equation method with variable expansion

coefficients and we find traveling wave solutions of the KdV equation. The paper

is organized as follows: In Section 2 we introduce the basic ingredients of the

method used. In Section 3 we consider KdV equation and Riccati’s equation

method of solution where the expansion coefficients depend on the variable ξ . In

Section 4 the KdV equation is solved using the −′ )G/G( expansion method with

variable expansion coefficients.

2. The Method.

We consider an evolution equation of the general form

),u,u,u(Gu xxxt L= or ),u,u,u(Gu xxxtt L= (2.1)

where u is a sufficiently smooth function.

We introduce a new variable ξ given by

4

)tx(k ω−=ξ (2.2)

where k and ω are constants. Changing variables, since

)(u)k(ut ξ′ω−= , )(ukux ξ′= , )(uku 2xx ξ′′= , … (2.3)

equation (2.1) becomes an ordinary nonlinear differential equation

ξξ=

ξω− L,

d

udk,

ddu

k,uGddu

)k(2

22 (2.4)

or

ξξ=

ξL,

d

udk,

ddu

k,uGd

udk

2

22

2

22 (2.5)

Equations (2.4) or (2.5) will be solved considering expansions of the form

∑=

=ξn

0k

kkYa)(u (2.6)

or

∑∑==

+=ξn

1kkk

n

0k

kk

Y

bYa)(u (2.7)

where all the expansion coefficients depend on the variable ξ ,

)(aa kk ξ≡ , )(bb kk ξ≡ for every n,,2,1,0k L=

contrary to the previously considered cases where the expansion coefficients were

considered as constants. The function )(YY ξ≡ satisfy Riccati’s equations

2YBA)(Y ⋅+=ξ′ (2.8)

where again the coefficients A and B depend on the variable ξ .

In solving equations (2.4) or (2.5), we consider the expansions (2.6) or (2.7) and

then we balance the nonlinear term with the highest derivative of the function

)(u ξ which determines n (the number of the expansion terms). Equating similar

powers of the function )(Yξ we can determine the various coefficients and thus

find the solution of the equation considered.

5

3. The KdV equation and its solutions.

The KdV equation was introduced by D. J. Korteweg and G. de Vries (Korteweg

and de Vries [50]) in modeling the motion of surface waves in shallow, narrow

canals, according to an observation made by J. S. Russell (Russell [79]). The

interest to Russell's observation was triggered by a computer experiment at the Los

Alamos Laboratory by Fermi, Pasta and Ulam (Fermi, Pasta and Ulam [35]). In

the above list it is fair to include the name of Mary Tsingou (see also Tuck and

Menzel [84], where Mary Tsingou appears under the name M. T. Menzel). Ten

years later, N. J. Zabusky and M. D. Kruskal (Zabusky and Kruskal [97]),

published a paper in which they provided a satisfactory interpretation of Fermi-

Pasta-Ulam experiment and introduced the term "soliton" to name the solitary-

wave solutions of the KdV equation. The KdV equation can be solved through the

famous Inverse Scattering Transform (AKNS [11], Gardner, Greene, Kruskal and

Miura [36], Miura [62] and [64], Miura, Gardner and Kruskal [63]), and the

algebro-geometric approach (Belokolos et al [19]).

Some solutions have also been found using the tanh-method (Hereman and

Malfliet [42]):

))tx(k(tanhka12ka8)t,x(u 222 ω−−+ω=

We consider the KdV equation in the form

0uauuu xxxxt =++ (3.1)

and try to find traveling wave solutions of this equation. We introduce a new

variable ξ given by

)tx(k ω−=ξ (3.2)

where k and ω are constants. Changing variables, since

)(u)k(ut ξ′ω−= , )(ukux ξ′= and )(uku 3xxx ξ′′=

equation (3.1) becomes an ordinary nonlinear differential equation

0)(uka)(u)(uk)(u)k( 3 =ξ′′′+ξ′ξ+ξ′ω− (3.3)

6

Integrating the above equation once, we obtain

022 c)(uka)(u

21

)(u)( =ξ′′+ξ+ξω− (3.4)

where 0c is a constant.

We consider the following cases in solving equation (3.4).

3.1. First Case. The Riccati Method.

We consider the solution of equation (3.4) to be of the form

∑=

=ξn

0k

kkYa)(u (3.5)

where )(aa kk ξ≡ , for every n,,2,1,0k L= .

3.1.1. Method I. We first consider that )(YY ξ≡ satisfies Riccati’s equation

2BYA)(Y +=ξ′ (3.6)

where A and B depend on the variable ξ : )(AA ξ≡ and )(BB ξ≡ .

We substitute (3.5) into (3.4) and take into account Riccati’s equation (3.6). We

then balance the second order derivative term with that of the nonlinear term. The

order of the nonlinear term )(u2 ξ is n2 and that of the second order derivative

term is 2n + . We thus get the equation 2nn2 += from which we obtain 2n = .

Therefore

2210 YaYaa)(u ++=ξ (3.7)

We calculate )(u ξ′′ from the above equation, taking into account Riccati’s

equation (3.6) and that the various coefficients 0a , 1a , 2a , A and B depend on

ξ . We find

++′+′+′=ξ′′ }A)Aa2a()Aaa{()(u 2110

+′++′+′+ Y}A)aBa(2)Aa2a{( 2121

++′′+++′+ 222121 Y}BAa6)aBa(B)Aa2a{(

422

3221 Y)Ba(6Y})Ba(B)aBa({2 +′+′++ (3.8)

7

Because of (3.7) and (3.8), equation (3.4) becomes

++++++ω− 22210

2210 )YaYaa(

21

)YaYaa()(

++′+′+′+ }A)Aa2a()Aaa[{(ka 21102

+′++′+′+ Y}A)aBa(2)Aa2a{( 2121

++′′+++′+ 222121 Y}BAa6)aBa(B)Aa2a{(

042

23

221 c]Y)Ba(6Y})Ba(B)aBa({2 =+′+′++

We first arrange the above equation in powers of Y, and then equate the

coefficients of each one power to zero. We obtain

coefficient of 0Y :

0211022

00 c]A)Aa2a()Aaa([kaa21

a)( =+′+′+′++ω− (3.9)

coefficient of Y :

0]A)aBa(2)Aa2a([kaaaa)( 21212

101 =′++′+′++ω− (3.10)

coefficient of 2Y :

++′+++ω− B)Aa2a([ka)aaa2(21

a)( 2122

1202

0]BAa6)aBa( 221 =+′′++ (3.11)

coefficient of 3Y :

0])Ba(B)aBa([ka2aa 2212

21 =′+′++ (3.12)

coefficient of 4Y :

0Baka6a21 2

222

2 =+ (3.13)

We now have to solve the system of simultaneous equations (3.9)-(3.13).

From equation (3.13), ignoring the trivial case, we determine the coefficient 2a :

222 Bka12a −= (3.14)

From equation (3.12), because of (3.14), we obtain

8

Bka12a 21 ′−= (3.15)

From equation (3.11), using the values of 1a and 2a , given by (3.15) and (3.14)

respectively, we obtain the equation

BAka8B

Bka4

B

Bka3a 22

22

0 −′′

−

′+ω= (3.16)

From equation (3.10), using (3.14) and (3.15), we derive another expression for

the coefficient 0a :

BAka10BBA

ka2BB

kaa 22

220 −

′′

−′′′′

−ω= (3.17)

Finally from equation (3.9), using again (3.14) and (3.15), we obtain the equation

0]BA2)BA(BA[ka12aa21

aka 22420

200

2 =+′′+′′−ω−+′′ (3.18)

Equating the two different expressions of 0a , given by (3.16) and (3.17), we

obtain the equation

0B

BA2

B

BBA2

B

B4

B

B3

22

=′

′+

′′′′

++′′

−

′

The above equation can be written as

0)AB(2BB

BB

BB

BB

2B

BB3

BB 23

2=′+

′−

′′

′−

′+

′′′−

′′′ (3.19)

and can easily be integrated, considering the substitution

BB

F′

= (3.20)

Equation (3.19) thus becomes 0)AB(2FFF =′+′−′′ which can be integrated

12 cAB2F

21

F =+−′ (3.21)

From the above equation we get

MBB

41

BB

21

AB2

+

′+

′

′−= (3.22)

9

where M is an arbitrary constant.

Equations (3.16) or (3.17), give because of (3.22):

Mka8B

Bka3a 2

22

0 −

′−ω= (3.23)

Finally, from equation (3.18), because of (3.23) and (3.22), we derive (after some

lengthy calculations) the equation

02422 cMka8

21 =+ω− (3.24)

which is a consistency condition between the different parameters and constants. It

can also be considered as an equation which determines ω .

Riccati’s equation 2BYA)(Y +=ξ′ under the substitution

)(w)(w

)(B1

)(Yξξ′

⋅ξ

−=ξ (3.25)

takes on the form of a linear second order ordinary differential equation

0)(wBA)(wBB

)(w =ξ+ξ′

′−ξ′′ (3.26)

with unknown function )(w ξ .

We now transform equation (3.26) under the substitution

)(y)(B)(w ξ⋅ξ=ξ (3.27)

The derivatives of the function )(uξ transform as

′+′=ξ′ yBBy21

B

1)(w (3.28)

′′+′′+

′⋅−′′=ξ′′ yByBy

B)B(

41

B21

B

1)(w

2 (3.29)

Equation (3.26), because of (3.22) and (3.27)-(3.29), takes on the form

+

′+′

′−

′′+′′+

′⋅−′′ yBBy

21

BB

B

1yByBy

B)B(

41

B21

B

1 2

10

0yBMBB

41

BB

21 2

=

+

′+

′

′−+

which gives us upon multiplying by B , the simple equation (some miraculous

cancellations take place)

0)(yM)(y =ξ⋅+ξ′′ (3.30)

Considering various forms of the constant M, we can determine )(y ξ and then

)(w ξ from (3.27). For 2mM = , equation (3.30) admits the general solution

)msin(C)mcos(C)(y 21 ξ+ξ=ξ , whereas for 2mM −= admits the general

solution )msinh(C)mcosh(C)(y 21 ξ+ξ=ξ . If 0M = then 21 CC)(y +ξ=ξ .

The function )(Yξ can be determined from (3.25) and then )(uξ from (3.7), using

the expressions 0a , 1a and 2a from (3.23), (3.15) and (3.14) respectively.

We thus find that

=++=ξ 2210 YaYaa)(u

+

ξξ′

⋅ξ

−′−+

−

′−ω=

)(w)(w

)(B1

)Bka12(Mka8BB

ka3 222

2

2

22

)(w)(w

)(B1

)Bka12(

ξξ′

⋅ξ

−−+

or

−

ξξ′′

+

−

′−ω=ξ

)(w)(w

BB

ka12Mka8BB

ka3)(u 222

2

2

2

)(w)(w

ka12

ξξ′

− (3.31)

Since )(y)(B)(w ξ⋅ξ=ξ , we obtain that

)(y)(y

)(B)(B

21

)(w)(w

ξξ′

+ξξ′

⋅=ξξ′

(3.32)

11

Using the above expression, we obtain from (3.31) that

2

22

)(y)(y

ka12Mka8)(u

ξξ′

−−ω=ξ (3.33)

This is quite a remarkable result : no matter what the coefficients of Riccati’s

equation are, we arrive at the equation (3.33) where )(y ξ satisfies equation (3.30).

We thus obtain the following three solutions, depending on the values of the

constant M (see equation (3.30))

(I) 2

2222

)mtan(C)mtan(C1

mka12mka8)(u

ξ+ξ⋅−−−ω=ξ for 2mM = (3.34)

where ω satisfies equation (3.24): 04422 cmka8

21 =+ω−

(II) 2

2222

)mtanh(C)mtanh(C1

mka12mka8)(u

ξ+ξ⋅+−+ω=ξ for 2mM −= (3.35)

where ω satisfies equation (3.24): 04422 cmka8

21 =+ω−

(III) 2

2

C1C

ka12)(u

ξ+−ω=ξ for 0M = (3.36)

where ω satisfies equation (3.24): 02 c

21 =ω−

C is an arbitrary constant, the ratio of the two constants appearing in the general

solution of equation (3.30).

3.2. Second Case. The Extended Riccati Method.

In this case we consider the expansion

∑∑==

+=ξn

1kkk

n

0k

kk

Y

bYa)(u

and balance the second order derivative term with the second order nonlinear term

of (3.4). We then find 2n = and thus

12

2212

210Y

b

Y

bYaYaa)(u ++++=ξ (3.37)

where again all the coefficients 0a , 1a , 2a and 1b , 2b depend on ξ , and Y

satisfies Riccati’s equation 2BYAY +=′ . From equation (3.37) we obtain, taking

into account 2BYAY +=′

+−′−+′+′−+′=ξ′′ }B)bB2b(A)aA2a()BbAaa({)(u 2121110

++′+′+′+ Y}A)Baa(2)aA2a({ 1221

++′+′++′+ 221221 Y}BAa6)Baa(B)aA2a({

422

3212 Y)Ba(6Y})Ba(2B)Baa(2{ +′++′+

−′−+′−′

+Y

B)bbA(2)bB2b( 2121

222121

Y

)bBA(6)bbA(A)bB2b( −′′−+−′−

42

2

3221

Y

bA6

Y

)bA(2A)bbA(2 +′−′−+ (3.38)

Therefore equation (3.4), under the substitution (3.42) and (3.43), becomes

+

+++++

++++ω−2

2212

2102212

210Y

b

Y

bYaYaa

21

Y

b

Y

bYaYaa)(

+−′−+′+′−+′+ }B)bB2b(A)aA2a()BbAaa([{ka 21211102

++′+′+′+ Y}A)Baa(2)aA2a({ 1221

++′+′++′+ 221221 Y}BAa6)Baa(B)aA2a({

422

3212 Y)Ba(6Y})Ba(2B)Baa(2{ +′++′+

−′−+′−′

+Y

B)bbA(2)bB2b( 2121

222121

Y

)bBA(6)bbA(A)bB2b( −′′−+−′−

13

042

2

3221 c]

Y

bA6

Y

)bA(2A)bbA(2 =+′−′−+

Equating the coefficients of Y to zero, we obtain a system of differential

equations from which we can determine the various expansion coefficients.

coefficient of 0Y :

++++ω− )ba2ba2a(21

a 2211200

021211102 c]B)bB2b(A)aA2a()BbAaa([ka =−′−+′+′−+′+ (3.39)

coefficient of Y:

+++ω− 10121 aabaa)(

0]A)Baa(2)aA2a([ka 12212 =+′+′+′+ (3.40)

coefficient of 2Y :

+++ω− )aa2a(21

a)( 20212

0]BAa6)Baa(B)aA2a([ka 212212 =+′+′++′+ (3.41)

coefficient of 3Y :

0])Ba(B)Baa[(ka2aa 2122

21 =′++′+ (3.42)

coefficient of 4Y :

0)Ba6(kaa21 2

222

2 =+ (3.43)

coefficient of 1Y − :

0]B)bbA(2)bB2b[(kababab)( 21212

21101 =′−+′−′+++ω− (3.44)


−++ω− )ba2b(21

b)( 20212

0)]bBA(6)bbA(A)bB2b[(ka 221212 =−′′−+−′− (3.45)

14


0])bA(A)bbA[(ka2bb 2212

21 =′−′−+ (3.46)


0)Ab6(kab21 2

222

2 =+ (3.47)

Solving the system of equations (3.47) and (3.46) (ignoring the trivial solutions)

we obtain

222 Aka12b −= and Aka12b 2

1 ′= (3.48)

Solving the system of equations (3.43) and (3.42) (ignoring the trivial solutions)

we obtain

222 Bka12a −= and Bka12a 2

1 ′−= (3.49)

From (3.41), using equations (3.49), we obtain

+

′−

′′−ω= AB8

BB

3BB

4kaa2

20 (3.50)

From (3.45), using equations (3.48), we obtain

+

′−

′′−ω= AB8

AA

3AA

4kaa2

20 (3.51)

From (3.44), using the coefficients (3.48) and (3.49), we get

+′

′+

′′′′

−ω= AB10A

BA14

AA

kaa2

20 (3.52)

From (3.40), using the coefficients (3.48) and (3.49), we get

+′

′+

′′′′

−ω= AB10BAB

14BB

kaa2

20 (3.53)

Finally from equation (3.39), using again (3.48) and (3.49), we get

+−′′−+ω−+ )BABA()ka144(aa21

ka 22420

20

2

15

02242 c)BA2ABBABA()ka24( =+′′+′′+′′−+ (3.54)

Equating the two expressions (3.50) and (3.51) for 0a we conclude that A and B

are proportional each other:

AsB 2−= (3.55)

with 2s− being the proportionality factor where s is real. The choice AsB 2=

leads to imaginary solutions (see below, eqns (3.59) and (3.60)). Let us now prove

that the functions B and A are proportional each other. Equating the two

expressions (3.50) and (3.51) for 0a , we obtain the relation

22

A

A3

A

A4

B

B3

B

B4

′−

′′=

′−

′′, which is equivalent to

′+

′

′−

′=

′′−

′′BB

AA

BB

AA

3BB

AA

4 and under the substitution BABAF ′−′= ,

taking into account the identity BABA)BABA( ′′−′′=′′−′ , we obtain the relation

AB)AB(

3FF

4′

=′

which upon integration gives 0AB

FF

3

=

or

0B

B

A

A)BABA(

3

=

′−

′′−′ , where we have set the integration constant equal to

zero. From the last relation, equating to zero either factor, i.e. 0BABA =′−′ or

0BB

AA =

′−

′, we conclude that A and B are proportional each other.

Equating again the two expressions (3.51) and (3.52) for 0a and taking into

account (3.55), we obtain the equation

0)(A)(As16)(A))(A(2))(A(3)(A)(A 42232 =ξ′ξ⋅−ξ′′′ξ−ξ′+ξ′′′⋅ξ (3.56)

This equation has been solved in Appendix A.

We have to determine the function )(w ξ from equation (3.16)

0)(wBA)(wBB

)(w =ξ+ξ′

′−ξ′′ (3.57)

16

This equation written in terms of A, using (3.47), takes on the form

0)(wAs)(wAA

)(w 22 =ξ−ξ′

′−ξ′′ (3.58)

The previous equation can be written as 0wAsA

AwAw 2

2=−

′′−

′′ which is

equivalent to 0wAsAw 2 =−

′

′. Multiplying by

Aw′

, we obtain

0wwsAw

Aw 2 =′−

′

′′, which is equivalent to 0)w(s

Aw 22

2

=′−′

′

and from this, by integration,

0wsA

w 222

=−

′ (3.59)

where we have put the constant of integration equal to zero. We thus get

)(As)(w)(w ξ±=

ξξ′

(3.60)

Another method of solution of equation (3.51) is provided in Paper I.

Therefore

s1

)(w)(w

)(As

1)(w)(w

)(B1

)(Y2

±=ξξ′

⋅ξ

=ξξ′

⋅ξ

−=ξ (3.61)

We then obtain the following expression for the function )(u ξ using (3.42), (3.46),

(3.47) and (3.53):

)(Aska4)(A)(A

ka)(u ξ±ξξ′

+ω=ξ (3.62)

Using the solution of the function )(Aξ calculated in Appendix A, we are able to

find the function )(u ξ . Using (A.15), i.e.

D4]eKa)ama(2[

eDaK4)(A

2Ds4101

Ds41

−++ρ=ξ

ξ⋅−

ξ⋅−

we obtain from (3.62) that

17

−++ρ

⋅++ρ⋅−−ω=ξξ⋅−

ξ⋅−ξ⋅−

D4]eKa)ama(2[

e]eKa)ama(2[K21Dska4)(u

2Ds4101

Ds4Ds4101

−++ρ±

ξ⋅−

ξ⋅−

D4]eKa)ama(2[

eaDKska16

2Ds4101

Ds41 (3.63)

where K is an arbitrary constant, and D, ρ , m, n are given by

na)ama(D 221

201 ρ−+ρ= ,

130

2 aas64

1=ρ , 40

22

204

21 as16babam +−=

21

2042

41

24

40

22

21

404

2602

280

4 aabb2ababaabs224abs32as256n −++−−=

The solution (3.63) contains 0a , 1a , 2b , 4b as free parameters.

4. The G'/G method with variable expansion coefficients

Using the expansion

2

210 G

G)(a

G

G)(a)(a)(u

′ξ+

′ξ+ξ=ξ (4.1)

equation (3.4) becomes

22

210

2

210 GG

)(aGG

)(a)(a21

GG

)(aGG

)(a)(a)(

′ξ+

′ξ+ξ+

′ξ+

′ξ+ξω−

′′

′−

′′′+

′′−′′′+

′′′+′′+

GG

GG

a3GG

aGG

a2GG

a2GG

aaka 11

2

11102

2

2

3

22

2

2

3

1 G

Ga2

G

Ga4

G

G

G

Ga4

G

Ga

G

Ga2

′′+

′′−

′′

′′+

′′′+

′+

0

4

2

2

22 cGG

a6GG

GG

a10GG

GG

a2 =

′+

′′

′−

′′′

′+ (4.2)

We expand the previous equation and we equate the coefficients of G to zero. We

find

Coefficient of 0G :

18

0022

00 cakaa21

a =′′++ω− (4.3)

Coefficient of 1G− :

0Ga)GaGa2Ga(kaGaa 11112

10 =′ω−′′′+′′′+′′′+′ (4.4)


})G(aGGa2)G(a2)G(a2GGa3GGa4{ka 222

21

2212

2 ′′′+′′′′+′′−′′+′′′−′′′′

0)G(a)G(a21

)G(aa 22

221

220 =′ω−′+′+ (4.5)


0)G(aa}G)G(a10)G(a2)G(a4{ka 321

22

31

32

2 =′+′′′−′+′′− (4.6)


0)G(a21

)G(aka6 422

42

2 =′+′ (4.7)

From equation (4.7) we find

22 ka12a −= (4.8)

Using the value of 2a into (4.6), we determine the coefficient 1a :

GG

ka12a 21 ′

′′= (4.9)

The values of 1a and 2a are substituted into (4.5) and we obtain the equation

′

′′′

+′′′′

−

′′′

+ω=GG

GG

ka2GG

kaa 22

20 (4.10)

The value of 1a is substituted into (4.4) and we obtain the equation

′′′′

+′

′′′

+

′′′

″

′′′

−ω=GG

GG

2

GGGG

kaa 20 (4.11)

Combining (4.10) and (4.11), we derive the equation

19

0

GGGG

GG

GG 2

=

′′′

″

′′′

+′′′′

−

′′′

(4.12)

The above equation determines the function )(G ξ . Using the substitution

GG

F′′′

= (4.13)

equation (4.13) transforms into 0FFF =′−′′ , which by integration gives

MF21

F 2 =−′ (4.14)

where M is the constant of integration.

(I) If 22M λ= , λ real, then the solution of (4.14) is given by

)tan(2F µ+ξλλ= , where µ is a constant. The differential equation FG

G =′′′

admits the solution [ ]

µ+ξλ−µ+λξ

λ+ξ+= )()(tan

1CCG 21 . We also obtain

])tan([CC

)](tan1[C

GG

21

22

µ−µ+ξλ+λµ+ξλλ+λ=

′, an expression reserved for later use in (4.15).

(II) If 22M λ−= , λ real, then the solution of (4.14) is given by

ξλ

ξλ

µ+µ−λ=

2

2

e1

e12F where µ is a constant. The differential equation F

GG =

′′′

admits the solution ξλµ++=

22

1e1

CCG . We also obtain

)]e1(CC[)e1(

eC2

GG

212

2

22

ξλξλ

ξλ

µ++µ+µλ=

′, an expression reserved for later use in

(4.15). (III) If 0M = , then the solution of (4.14) is given by ξ−µ

= 2F . The

differential equation FGG =

′′′

admits the solution ξ−µ

+=ξ 21

CC)(G where µ is a

20

constant. We also obtain )]B(CC[)B(

C

GG

12

2

ξ−+ξ−=

′, an expression reserved for

later use in (4.15).

Using the values of 1a and 2a into (4.1) we obtain

′−

′′+=ξ

22

0 GG

GG

ka12a)(u

and using (4.10), we get

′−

′′+

′

′′′

+′′′′

−

′′′

+ω=ξ2

222

2

GG

GG

ka12GG

GG

ka2GG

ka)(u

or, in simplified form,

′

′+

′

′′′

+

′′′

−ω=ξGG

ka12GG

4GG

ka)(u 22

2 (4.15)

So far we have not used equation (4.3). This equation, combined with (4.10), gives

us a compatibility condition between the constants and the various parameters. It

might also be considered as the equation determining the ω parameter.

Using the three solutions for G, we can find the following three expressions of

)(u ξ , using (4.15). We obtain

(I) +λ+µ+ξλλ−ω=ξ ]2)(tan3[ka4)(u 2222

×µ+ξλ+µ−λ

µ+ξλ+λ+2

221

2222

)](tanC)CC([

)](tan1[Cka12

}C)(tan)](tanC)CC(2[{ 2221 −µ+ξλµ+ξλ+µ−λλ× (4.16)

The compatibility condition, equation (4.3), reads

04422 cka8

21 =λ+ω− (4.17)

(II) +

µ+µ−

µ+µ−λ−ω=ξ ξλ

ξλ

ξλ

ξλ

22

22

2

222

)e1(

e8

e1

e1ka4)(u

21

22121

22

41

221

222

]eCCC[)e1(

]eCCC[Cka48 ξλξλ

ξλ

µ++µ+µ−+µλ+ (4.18)


04422 cka8

21 =λ+ω− (4.19)

(III)

ξ−µ+ξ−µ+−

ξ−µ−ω=ξ

212

1222

2

)](CC[

)](C2C[C1

)(

1ka12)(u (4.20)


02 c

21 =ω− (4.21)

Appendix A.

In this Appendix we solve equation (3.56):

0)(A)(As16)(A))(A(2))(A(3)(A)(A 42232 =ξ′ξ⋅−ξ′′′ξ−ξ′+ξ′′′⋅ξ (A.1)

We consider an expansion of the form

∑=

ξϕ=ξn

0k

kk )(a)(A (A.2)

where )(ξϕ satisfies Jacobi's differential equation

40

31

2234 bbbbb)(

dd ϕ+ϕ+ϕ+ϕ+=ξϕξ

(A.3)

Upon substitution of (A.2) into (A.1) and balancing )(A)(A 2 ξ′′′⋅ξ with

)(A)(A 4 ξ′ξ , and taking into account (A.3), we obtain 2n = . We thus substitute

)(a)(aa)(A 2210 ξϕ+ξϕ+=ξ (A.4)

Since

=ξϕ′ξϕ+=ξ′ )()](a2a[)(A 21

40

31

223421 bbbbb)](a2a[ ϕ+ϕ+ϕ+ϕ+ξϕ+= (A.5)

22

+ξϕ′ϕ+ϕ+ϕ+ϕ+=ξ′′ )(bbbbba2)(A 40

31

22342

)(bbbbb2

)b4b3b2b()a2a(4

03

12

234

30

212321 ξϕ′

ϕ+ϕ+ϕ+ϕ+

ϕ+ϕ+ϕ+ϕ++

+ϕ+ϕ+ϕ+ϕ+= )bbbbb(a2 40

31

22342

)b4b3b2b()a2a(21 3

02

12321 ϕ+ϕ+ϕ+ϕ+⋅+ (A.6)

+ξϕ′ϕ+ϕ+ϕ+=ξ′′′ )()b4b3b2b(a2)(A 30

21232

)()b4b3b2b(a 30

21232 ξϕ′ϕ+ϕ+ϕ++

)()b12b6b2()a2a(21 2

01221 ξϕ′ϕ+ϕ+ϕ+⋅+

+ϕ+ϕ+ϕ+= )b4b3b2b(a2{ 30

21232

)b4b3b2b(a 30

21232 ϕ+ϕ+ϕ++

×ϕ+ϕ+ϕ+⋅+ })b12b6b2()a2a(21 2

01221

40

31

2234 bbbbb ϕ+ϕ+ϕ+ϕ+× (A.7)

)()a2a()aaa(2))(A( 212

2102 ξϕ′ϕ+ϕ+ϕ+=′ξ

×ϕ+ϕ+ϕ+= )a2a()aaa(2 212

210

40

31

2234 bbbbb ϕ+ϕ+ϕ+ϕ+× (A.8)

Upon substituting (A.4)-(A.8) into (A.1) we obtain an equation which when

multiplied by 40

31

2234 bbbbb ϕ+ϕ+ϕ+ϕ+ , results in an equation which can

be brought into an equation of 13-th degree in ϕ . Equating all the coefficients to

zero, we obtain fourteen equations. Solving this system of simultaneous equations,

we obtain four solutions, from which we select only the non-trivial one:

00 aa = , 11 aa = , 0a2 = , 21

20 as16b = ,

30

421

40

22

201

1a2

)baas48ba(ab

−+=

23

22 bb = , 10

2204

21

40

2

3 aa2

baba3as16b

−−−= , 44 bb = (A.9)

Equation (A.3) then becomes

−ϕ+ϕ−+

+ϕ=

ξϕξ

22

330

421

40

22

20142

12

2

ba2

)baas48ba(aas16)(

dd

410

2204

21

40

2

baa2

baba3as16 +ϕ−−−

The above equation can be brought into the form

)()(a

aas16)(

dd

21

2

1

021

22

θρ−ϕθρ−ϕ

+ϕ=

ξϕξ

(A.10)

where

130

2 aas64

1=ρ , nm2,1 ±=θ , 40

22

204

21 as16babam +−=

21

2042

41

24

40

22

21

404

2602

280

4 aabb2ababaabs224abs32as256n −++−−= (A.11)

We then have (taking the plus sign only)

)()(a

aas4)(

dd

211

01 θρ−ϕθρ−ϕ

+ϕ=ξϕ

ξ

and by integration

=

+

+ϕ−

+ϕ+

+ϕ⋅−

+ϕ211

02

1

0

11

021

1

0 a

Da

ap

a

a

aD2

a

ap

a

D2

a

a1

ξ⋅−⋅= Ds4eK (A.12)

where

na)ama(D 221

201 ρ−+ρ= ,

1

01

a

)ama(2p

+ρ= (A.13)

and K is an arbitrary constant.

Equation (A.12), fortunately enough, can be solved with respect to ϕ , giving

24

1

02Ds42

1

Ds4

a

a

D4)eKp(a

eDK4)( −

−+=ξϕ

ξ⋅−

ξ⋅− (A.14)

We thus obtain that, using the above expression and (A.4),

D4)eKp(a

eDaK4)(A

2Ds421

Ds41

−+=ξ

ξ⋅−

ξ⋅− (A.15)

References

[1] P. C. Abbott, E. J. Parkes and B. R. Duffy:“The Jacobi elliptic-function

method for finding periodic-wave solutions to nonlinear evolution

equations”. Available online at

http//physics.uwa.edu.au/pub/Mathematica/Solitons

[2] M. A. Abdou: “Adomian decomposition method for solving the telegraph

equation in charged particle transport”.

J. Quant. Spectro. Rad. Trans 95 (2005) 407-414

[3] M. A. Abdou: “Exact solutions for nonlinear evolution equations via

the extended projective Riccati equation expansion method”.

Electron. J. Theor. Physics 4 (2007) 17-30.

[4] M. A. Abdou: “The extended F-expansion method and its applications

for a class of nonlinear evolution equations”.

Chaos, Solitons and Fractals 31 (2007) 95-104.

[5] M. A. Abdou: “On the variational iteration method”.

Phys. Lett. A 366 (2007) 61-68

[6] M. A. Abdou:”A generalized auxiliary equation method and its

applications”. J. Nonlin. Dyn. 52 (2008) 95-102

[7] M. A. Abdou: “An improved generalized F-expansion method and its

applicatuions”. J. Comput. and Appl. Math. 214 (2008) 202-208

[8] M. A. Abdou: “Generalized solitary and periodic solutions for nonlinear

partial differential equations by the Exp-function method”.

J. Nonlin. Dyn. 52 (2008) 1-9

25

[9] M. A. Abdou and A. A. Soliman: “New applications of Variational

Iteration Method”. Physica D 211 (2005) 1-8

[10] M. A. Abdou and A. Elhanbaly: “Construction of periodic and solitary

wave solutions by the extended Jacobi elliptic function expansion method".

Comm. in Nonlin. Science and Num. Sim. 12 (2007) 1229-1241

[11] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur: “The Inverse

Scattering Transform. Fourier Analysis for Nonlinear Problems”.

Stud. Appl. Math. 53 (1974) 249-315

[12] M. J. Ablowitz and P. A. Clarkson: “Solitons, Nonlinear Evolution

Equations and Inverse Scattering Transform”.

Cambridge University Press, Cambridge, 1991.

[13] M. J. Ablowitz and H. Segur: “Solitons and the Inverse Scattering

Transform”. SIAM, 1981.

[14] G. Adomian: “Nonlinear Stochastic Operator Equations”.

Academic Press, San Diego, (1986).

[15] S. Antoniou: “The Riccati equation method with variable expansion

coefficients. I. Solving the Burgers equation”. submitted for publication

[16] I. Aslan: “Application of the exp-function method to nonlinear lattice

differential equations for multi-wave and rational solutions”.

Mathem. Methods in the Applied Sciences 60 (2011) 1707-1710.

[17] D. Baldwin, Ü. Göktaş, W. Hereman, L. Hong, R.S. Martino and

J.C. Miller : “Symbolic computation of exact solutions expressible in

hyperbolic and elliptic functions for nonlinear PDEs”.

J. Symb. Comp. 37 (2004) 669-705

[18] A. Bekir and A. Boz: “Exact Solutions for Nonlinear Evolution Equations

using Exp-Function Method”. Phys. Lett. A 372 (2008) 1619-1625.

[19] E. D. Belokolos, A. . Bobenko, V. Z. Enolskii, A. R. Its and V. Matveev:

"Algebro-Geometric Approach to Nonlinear Integral Equations"

Springer-Verlag 1994

26

[20] G. Bluman and S. Kumei: “Symmetries and Differential Equations”.

Springer-Verlag 1989

[21] A. Borhanifar and A. Z. Moghanlu: “Application of the −′ )G/G(

expansion method for the Zhiber-Sabat equation and other related

equations”. Mathem. and Comp. Mod. 54 (2011) 2109-2116.

[22] H. T. Chen and H. Q. Zhang: “Improved Jacobian elliptic function

method and its applications”.

Chaos, Solitons and Fractals 15 (2003) 585-591

[23] Y. Chen and Q. Wang: “Extended Jacobi elliptic function rational

expansion method and abundant families of Jacobi elliptic function

solutions to (1+1)-dimensional dispersive long wave equation”.


[24] A. E. Ebaid: “Generalization of He’s Exp-Function Method and New

Exact Solutions for Burgers Equation”. Z. Naturforsch. 64a (2009) 604–608

[25] S. A. El-Wakil, E. M. Abulwafa, A. Elhanbaly and M. A. Abdou:“The

extended homogeneous balance method and its applications”.


[26] S. A. El-Wakil and M. A. Abdou: “New exact travelling wave solutions

using modified extended tanh-function method”.


[27] S. A. El-Wakil, M. A. Abdou and A. Hendi: “New periodic wave

solutions via Exp-function method”. Physics Letters A 372 (2008) 830-840

[28] S. A. El-Wakil and M. A. Abdou: "New applications of the homotopy

analysis method". Zeitschrift für Naturforschung A (2008)

[29] E. Fan: “Two new applications of the homogeneous balance method”

Physics Letters A 265 (2000) 353-357

[30] E. Fan: “Extended tanh-function method and its applications to nonlinear

equations” . Physics Letters A 277 (2000) 212-218.

27

[31] E. Fan and Y. C. Hon: “Applications of extended tanh-method to

“special” types of nonlinear equations”.

Appl. Math. and Comp. 141 (2003) 351-358.

[32] E. Fan and H. Zhang: “Applications of the Jacobi elliptic function method

to special-type nonlinear equations”. Phys. Lett. A 305 (2002) 383-392.

[33] J. Feng, W. Li and Q. Wan: “Using −′ )G/G( expansion method to seek

traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation”

Appl. Math. Comp. 217 (2011) 5860-5865.

[34] Z. S. Feng: "The first integral method to study the Burgers-Korteweg de

Vries equation". Phys. Lett. A: Math Gen. A 302 (2002) 343-349

[35] E. Fermi, J. Pasta, S. Ulam and M. Tsingou: “Studies of Nonlinear

Problems. I". Los Alamos report LA-1940. May 1955

[36] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura:

“Method for solving the Korteweg-de Vries equation”.

Phys. Rev. Lett. 19 (1967) 1095-1097

[37] P. Gray and S. Scott: “Chemical Oscillations and Instabilities”

Clarendon Press, Oxford, 1990.

[38] P. Griffiths and W. E. Sciesser: “Traveling Wave Analysis of Partial

Differential Equations”. Academic Press 2012.

[39] J. H. He: “A variational iteration method-a kind of nonlinear analytical

technique: Some examples”. Int. J. Nonlin. Mech. 34 (1999) 699–708

[40] J. H. He and M. A. Abdou: “New periodic solutions for nonlinear

evolution equations using Exp function method”.


[41] J. H. He and X. H. Wu: “Exp-Function method for nonlinear wave

equations”. Chaos, Solitons and Fractals 30 (2006) 700-708.

[42] W. Hereman and W. Malfliet: “The tanh method: A Tool to Solve

Nonlinear Partial Differential Equations with Symbolic Software”.

Available online, Colorado School of Mines.

28

[43] R. Hirota: “The Direct Method in Soliton Theory”

Cambridge University Press 2004.

[44] R. Hirota: “Exact Solution of the KdV Equation for Multiple Collisions of

Solitons”. Physics Review Letters 27 (1971) 1192-1194.

[45] P. E. Hydon: “Symmetry Methods for Differential Equations”

Cambridge University Press 2000

[46] M. Inc and M. Ergüt : “Periodic wave solutions for the generalized

shallow water wave equation by the improved Jacobi elliptic function

method”. Appl. Math. E-Notes 5 (2005) 89-96

[47] A. Jabbari, H. Kheiri and A. Bekir : “Exact solutions of the coupled

Higgs equation and the Maccari system using He’s semi-inverse method

and −′ )G/G( expansion method”.

Comp. and Math. with Applic. 62 (2011) 2177-2186.

[48] A. J. M. Jawad, M. D. Petkovich and A. Biswas : “Modified simple

equation method for nonlinear evolution equations”.

Appl. Math. Comput. 217 (2010) 869–877

[49] Y. Keskin and G. Oturanc: “Reduced Differential Transform Method

for Partial Differential Equations".

Int. J. Nonl. Sci. and Num. Simul. 10 (2009) 741-749

[50] D. J. Korteweg and G. de Vries: "On the change of form of long waves

advancing in a rectangular channel and on a new type of long stationary

waves". Phil. Mag. 39 (1895) 422-443

[51] N. A. Kudryashov: “Exact Solutions of the Generalized Kuramoto-

Sivashinsky Equation”. Physics Letters A 147 (1990) 287-291.

[52] N. A. Kudryashov: “Simplest equation method to look for exact solutions

of nonlinear differential equations”. arXiv:nlin/0406007v1, 4 Jun 2004

[53] N. A. Kudryashov: “Nonlinear differential equations with exact solutions

expressed via the Weirstrass function”. arXiv:nlin/0312035v1,16 Dec 2003

29

[54] S. Liao: “Homotopy Analysis Method in Nonlinear Differential

Equations”. Springer 2012

[55] S. K. Liu, Z. Fu, S. Liu and Q. Zhao : “Jacobi elliptic function

expansion method and periodic wave solutions of nonlinear wave

equations”. Physics Letters A 289 (2001) 69-74.

[56] S. K. Liu, Z. Fu, S. Liu and Q. Zhao : “Expansion about the Jacobi

Elliptic Function and its applications to Nonlinear Wave Equations”.

Acta Phys. Sinica 50 (2001) 2068-2072.

[57] D. Lu and Q. Shi: “New Jacobi elliptic functions solutions for the

combined KdV-mKdV Equation”. Int. J. Nonlin. Sci. 10 (2010) 320-325

[58] W. Malfliet :“Solitary wave solutions of nonlinear wave equations”.

American Journal of Physics 60 (1992) 650-654.

[59] W. Malfliet : “The tanh method: a tool for solving certain classes of

nonlinear evolution and wave equations”

J. Comp. Appl. Math. 164-165 (2004) 529-541

[60] W. Malfliet and W. Hereman: “The tanh method: I. Exact solutions of

nonlinear evolution and wave equations”.

Physica Scripta 54 (1996) 563-568

[61] W. Malfliet and W. Hereman: “The tanh method: II. Perturbation

technique for conservative systems”. Physica Scripta 54 (1996) 569-575.

[62] R. M. Miura : “Korteweg-de Vries equations and generalizations. I.

A remarkable explicit nonlinear transformation”.

J. Math. Phys. 9 (1968) 1202-1204

[63] R. M. Miura, C. S. Gardner and M.D. Kruskal: “Korteweg-de Vries

equations and generalizations. II. Existence of conservation laws and

constants of motion”. J. Math. Phys. 9 (1968) 1204-1209.

[64] R. M. Miura : “The Korteweg de Vries equation: a survey of results".

SIAM Rev. 18 (1976) 412-459.

30

[65] J. Murray : “Mathematical Biology”. Springer-Verlag, Berlin, 1989.

[66] H. Naher, F. Abdullah and M. A. Akbar: “The Exp-function method for

new exact solutions of the nonlinear partial differential equations”.

Intern. J. Phys. Sciences 6 (2011) 6706- 6716.

[67] H. Naher, F. A. Abdullah and M. A. Akbar: “New travelling wave

solutions of the higher dimensional nonlinear partial differential

equation by the Exp-function method”. J. Appl. Math. (2012)

[68] H. Naher, F. A. Abdullah and M. A. Akbar:“The −′ )G/G( expansion

method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon

equation”. Math. Prob. in Engin. (2011)

[69] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov:

“Theory of Solitons: The Inverse Scattering Method”. Plenum, NY 1984

[70] P. J. Olver: “Applications of Lie Groups to Differential Equations ”

Graduate Texts in Mathematics, vol.107, Springer Verlag, N.Y. 1993

[71] L. V. Ovsiannikov: “Group Analysis of Differential Equations”

Academic Press, New York 1982

[72] T. Ozis and I. Aslan: “Application of the −′ )G/G( expansion method

to Kawahara type equations using symbolic computation”.

Applied Mathematics and Computation 216 (2010) 2360-2365.

[73] E. J. Parkes and B. R. Duffy: “An automated tanh-function method for

finding solitary wave solutions to nonlinear evolution equations”.

Comp. Phys. Comm. 98 (1996) 288-300

[74] E. J. Parkes, B. R. Duffy and P. C. Abbott: “The Jacobi elliptic function

method for finding periodic-wave solutions to nonlinear evolution

equations”. Phys. Lett. A 295 (2002) 280-286

[75] E. J. Parkes, E. J. Zhu, B. R. Duffy and H. C. Huang: “Sech-polynomial

traveling solitary-wave solutions of odd-order generalized KdV equations”

Phys. Lett. A 248 (1998) 219-224

31

[76] K. R. Raslan: “The first integral method for solving some important

nonlinear partial differential equations”. Nonlin. Dynamics 2007

[77] C. Rogers and W. F. Shadwick: “Bäcklund Transformations”.

Academic Press, New York, 1982.

[78] W. Rui, S. Xie, Y. Long and B. He: "Integral Bifurcation Method and its

Applications for solving the modified Equal Width Wave equation and its

variants". Rostock Math. Kolloq. 62 (2007) 87-106

[79] J. S. Russell:"Report on Waves"

The British Association for the Advancement of Science, London 1845

[80] A. H. Salas, and C. A. Gomez : “Application of the Cole-Hopf

transformation for finding exact solutions to several forms of the

seventh-order KdV equation”. Math. Prob. in Eng. (2010)

[81] A. A. Soliman and H. A. Abdou: “New exact solutions of nonlinear

variants of the RLW, the phi-four and Boussinesq equations based on

modified extended direct algebraic method”.

Intern. Journ. of Nonl. Sci. 7 (2009) 274-282.

[82] H. Stephani: “Differential Equations: Their Solutions Using Symmetries”.

Cambridge University Press, 1989

[83] N. Taghizadeh, M. Akbari and A. Ghelichzadeh: “Exact solution of

Burgers equations by homotopy perturbation method and reduced

differential transformation method”.

Austr. J. of Basic and Applied Sciences 5 (2011) 580-589

[84] J. L. Tuck and M. T. Menzel: “The Superposition of the Nonlinear

Weighted String (FPU) Problem”.

Advances in Mathematics 9 (1972) 399-407

[85] N. K. Vitanov: “Application of simplest equations of Bernoulli and Riccati

kind for obtaining exact traveling-wave solutions for a class of PDEs

with polynomial nonlinearity”.

Comm. in Nonlin. Sci. and Num. Simulation 15 (2010) 2050–2060

32

[86] M. L. Wang, Y. B. Zhou and Z. B. Li: “Application of a homogeneous

balance method to exact solutions of nonlinear equations in

Mathematical Physics”. Physics Letters A 216 (1996) 67-75.

[87] M. L. Wang and X. Z. Li: “Applications of F-Expansion to periodic wave

solutions for a new Hamiltonian amplitude equation”.


[88] M. Wang, X. Li and J. Zhang: “The −′ )G/G( expansion method and

travelling wave solutions of nonlinear evolution equations in

[89] A.M. Wazwaz: “A reliable modification of Adomian’s decomposition

Method”. Appl. Math.Comput. 92 (1998) 1–7.

[90] A. M. Wazwaz: “The tanh-coth method for solitons and kink solutions

for nonlinear parabolic equations”.

Appl. Math. Comput. 188 (2007) 1467-1475.

[91] A. M. Wazwaz: “Partial Differential Equations and Solitary Waves

Theory”. Springer-Verlag Berlin Heidelberg 2009

[92] J. Weiss, M. Tabor and G. Carnevale: “The Painleve Property for Partial

Differential Equations”. Journal of Math. Physics 24 (1982) 522-526.

[93] J. Weiss, M. Tabor and G. Carnevale: “The Painleve Property ”.

J. of Math. Phys. 24 (1983) 1405-

[94] G. Whitham: “Linear and Nonlinear Waves”. Wiley, NY 1974

[95] K. Yahya, J. Biafar, H. Azari and P. R. Fard: “Homotopy Perturbation

Method for Image Restoration and Denoising”. Available online.

[96] O. Yu. Yefimova: “The modified simplest equation method to look for

exact solutions of nonlinear partial differential equations”.

arXiv:1011.4606v1 [nlin.SI] 20 Nov 2010

[97] N. J. Zabusky and M. D. Kruskal: "Interaction of "solitons" in a

collisionless plasma and the recurrence of initial states"

Phys. Rev. Letters 15 (1965) 240-243

33

[98] E. M. E. Zayed: “Traveling wave solutions for higher dimensional

nonlinear evolution equations using the −′ )G/G( expansion method”.

J. of Appl. Math. & Inform. 28 (2010) 383- 395

[99] E. M. E. Zayed: “A note on the modified simple equation method applied

to Sharma-Tasso-Olver equation”.

Appl. Math. Comp. 218 (2011) 3962-3964

[100] E. M. E. Zayed and K. A. Gepreel: “The (G′/G)-expansion method for

finding traveling wave solutions of nonlinear partial differential equations

in mathematical physics”. J. Math. Phys. 50 (2009) 013502

[101] X. L. Zhang and H. Q. Zhang: “A new generalized Riccati equation

rational expansion method to a class of nonlinear evolution equations with

nonlinear terms of any order”. Appl. Math. and Comp.186 (2007) 705-714

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