a unified rational expansion method to construct a series of explicit exact solutions to nonlinear...

Applied Mathematics and Computation 177 (2006) 396–409

www.elsevier.com/locate/amc

A unified rational expansion method to construct a seriesof explicit exact solutions to nonlinear evolution equations

Yong Chen a,c,*, Qi Wang b

a Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 315211, Chinab Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

c Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China

Abstract

A unified basic frame of rational expansion methods is presented, which leads to closed-form solutions of nonlinearevolution equations (NLEEs). The new unified algorithms are given to find exact rational formal polynomial solutionsof NLEEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccatiequation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choosesome physical significance NLEEs to illustrate the methods. As a consequence, we not only can successfully obtain thesolutions found by most existing Jacobi elliptic function methods and tanh methods, but also find other new and moregeneral solutions at the same time.� 2005 Elsevier Inc. All rights reserved.

Keywords: Symbolic computation; Exact solutions; Nonlinear evolution equations; Jacobi elliptic function rational expansion method;Riccati equation rational expansion method

1. Introduction

An exciting and extremely active area of research investigation during the past four years has been the studyof solitons and the related issue of the construction of solutions to a wide class of NLEEs. There has been agreat amount of activities aiming to find methods for exact solution of NLEEs, such as Backlund transforma-tion, Darboux transformation, variable separation approach, various tanh methods, Painleve method, general-ized hyperbolic-function method, homogeneous balance method, similarity reduction method and so on [1–14].The tanh method [5–7] is considered to be one of the most straightforward and effective algorithm to obtainsolitary wave solutions for a large of NLEEs. In line with the development of computerized symbolic compu-tation, much work has been concentrated on the various extensions and applications of the tanh method [8–16].

Recently, we present various rational function expansions methods [17] and various subequation expansionmethods [18] to construct new and more general solutions of NLEEs, the solutions obtained contain not only

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.11.018

* Corresponding author.E-mail address: [email protected] (Y. Chen).

mailto:[email protected]

Y. Chen, Q. Wang / Applied Mathematics and Computation 177 (2006) 396–409 397

the results obtained by using the various tanh methods [5–16] but also other types of solutions. The main pur-pose of this paper is to summary and extend our methods [17,18] to unified basic frame so that it can be usedto obtained more types and general formal solutions. For illustration, we apply the generalized method tosome NLEEs and successfully construct new and more general solutions including rational form Jacobi ellipticfunction solutions, solitary wave solutions, triangular periodic solutions. The related further discuss and com-parison with other tanh method will be given in Section 5.

The paper is organized as follows: in Section 2, we give the main steps of our algorithms for computingexact solutions of nonlinear polynomial NLEEs. In Section 3, we consider the Jacobi elliptic function rationalexpansion method. In Section 4, we show the Riccati equation rational expansion method. In Section 5, wefurther extend the Riccati equation rational expansion method to a generalized form. Conclusions will be pre-sented finally.

2. Summary of the rational expansion method

In the following we would like to outline the main steps of our method:

Step 1. Given a system of polynomial NLEEs with constant coefficients, with some physical fields ui(x,y, t) inthree variables x, y, t,

Dðui; uit; uix; uiy ; uitt; uixt; uiyt; uixx; uiyy ; uixy ; . . .Þ ¼ 0; ð2:1Þ
use the wave transformation
uiðx; y; tÞ ¼ U iðnÞ; n ¼ kðxþ ly þ ktÞ; ð2:2Þ
where k, l and k are constants to be determined later. Then the nonlinear partial differential system(2.1) is reduced to a nonlinear ordinary differential equation (ODE)
HðUi; U 0i; U 00i ; . . .Þ ¼ 0. ð2:3Þ
Step 2. We introduce a new ansatz in terms of finite rational formal expansion in the following forms:
UiðnÞ ¼ ai0 þXmi

j¼1

Pr1þr2¼j

aijr1r2

F r1ðnÞGr2ðnÞ

ðl1F ðnÞ þ l2GðnÞ þ 1Þj; ð2:4Þ

where ai0, aijr1r2

, l1 and l2 (i = 1,2, . . .) are constants to be determined later and the new variablesF = F(n) and G = G(n) satisfy

dFdn¼ K1ðF ;GÞ;

dGdn¼ K2ðF ;GÞ; ð2:5Þ

where K1 and K2 are polynomial of F and G.Step 3. Determine the mi of the rational formal polynomial solutions (2.4) by respectively balancing the high-

est nonlinear terms and the highest-order partial derivative terms in the given system equations (seeRef. [9–13] for details), and then give the formal solutions.

Step 4. Substitute (2.4) into (2.3) along with (2.5) and then set all coefficients of Fp(n)Gq(n), (p = 1,2, . . . ;q = 0,1) of the resulting system�s numerator to be zero to get an over-determined system of nonlinearalgebraic equations with respect to k, ai0, aij

r1r2, l1 and l2 (i = 1,2, . . .).

Step 5. By solving the over-determined system of nonlinear algebraic equations by use of symbolic computa-tion system Maple, we end up with the explicit expressions for k, ai0, aij

r1r2, l1 and l2 (i = 1,2, . . .).

Step 6. According to system (2.2) and (2.4), the general solutions of system (2.5) and the conclusions in Step 5,we can obtain rational formal exact solutions of system (2.1).

Remark 1. The method proposed here is more general than the various exist method for finding exactsolutions of NLEEs. We can easily draw this conclusion when F and G take particular variables in thefollowing sections. The appeal and success of the method lies in the fact that writing the exact solutions of

398 Y. Chen, Q. Wang / Applied Mathematics and Computation 177 (2006) 396–409

NLEEs as polynomials of F and G whose derivatives are in closed-form, the equations can changed into anonlinear system of algebraic equations. The system can be solved with help of symbolic computation.

3. Jacobi elliptic function rational expansion method

In this section we would like to apply our method to obtain rational formal Jacobi elliptic function solu-tions of NLEEs, i.e., restricting F and G in Jacobi elliptic functions.

Here snn, cnn, dnn, nsn, csn, scn, ncn, sdn and ndn are the Jacobian elliptic sine function, the Jacobianelliptic cosine function and the Jacobian elliptic function of the third kind and other Jacobian functions whichis denoted by Glaishers symbols and are generated by these three kinds of functions, namely [19],

nsn ¼ 1

snn; ncn ¼ 1

cnn; ndn ¼ 1

dnn; sdn ¼ snn

dnn; ð3:1:1Þ

scn ¼ snncnn

; csn ¼ cnnsnn

; dsn ¼ dnnsnn

; ð3:1:2Þ

which are double periodic and posses the following properties

1. Properties of triangular function

cn2 nþ sn2 n ¼ dn2 nþ m2sn2 n ¼ 1; ð3:2:1Þns2 n ¼ 1þ cs2 n; ns2 n ¼ m2 þ ds2 n; ð3:2:2Þsc2 nþ 1 ¼ nc2 n; m2sd2 nþ 1 ¼ nd2 n. ð3:2:3Þ

2. Derivatives of the Jacobi elliptic functions

sn0 n ¼ cnndnn; cn0 n ¼ �snndnn; dn0 n ¼ �m2snncnn; ð3:3:1Þns0 n ¼ �dsncsn; ds0 n ¼ �csnnsn; cs0 n ¼ �nsndsn; ð3:3:2Þsc0 n ¼ ncndcn; nc0 n ¼ scndcn; cd0 n ¼ cdnndn; nd0 n ¼ m2sdncdn; ð3:3:3Þ

where m is a modulus.3. Properties of limit

limm!1

snn ¼ tanh n; limm!1

cnn ¼ sechn; limm!1

dnn ¼ sechn; ð3:4:1Þ

limm!1

nsn ¼ coth n; limm!1

csn ¼ cschn; limm!1

dsn ¼ cschn; ð3:4:2Þ

limm!0

snn ¼ sin n; limm!0

cnn ¼ cos n; limm!0

dnn ¼ 1; ð3:4:3Þ

limm!0

nsn ¼ csc n; limm!0

csn ¼ cot n; limm!0

dsn ¼ csc n. ð3:4:4Þ

The Jacobi–Glaisher functions for elliptic function can be found in Ref. [19]. It can easily be seen that theJacobi elliptic functions satisfy system (2.5).

The six main steps of the Jacobi elliptic function (here just consider the condition F = snn and G = cnn)rational expansion method are illustrated with the (2 + 1)-dimensional dispersive long wave equation(DLWE), i.e.,

uyt þ vxx þ ðuuxÞy ¼ 0;

vt þ ux þ ðuvÞx þ uxxy ¼ 0.ð3:5Þ


According to the Step 1 in Section 2, we make the following travelling wave transformation:

uðx; y; tÞ ¼ UðnÞ; vðx; y; tÞ ¼ V ðnÞ; n ¼ kðxþ ly þ ktÞ; ð3:6Þ

where k, l and k are constants to be determined later, and thus system (3.5) becomes

klU 00 þ V 00 þ lU 02 þ lUU 00 ¼ 0;

kV 0 þ U 0 þ ðUV Þ0 þ k2lU 000 ¼ 0.ð3:7Þ

According to Step 2 in Section 2, we expand the solution of system (3.7) in the form

UðnÞ ¼ a0 þXmu

j¼1

ajsnjðnÞ þ bjsnj�1ðnÞcnðnÞðl1snðnÞ þ l2cnðnÞ þ 1Þj

;

V ðnÞ ¼ A0 þXmv

j¼1

AjsnjðnÞ þ Bjsnj�1ðnÞcnðnÞðl1snðnÞ þ l2cnðnÞ þ 1Þj

;

ð3:8Þ

According to Step 3 in Section 2, by balancing U000(n) and (V(n)U(n)) 0 in system (3.7) and by balancing V00(n)and U(n)U00(n) in system (3.7), we can obtain that mu = 1 and mv = 2. So we have

UðnÞ ¼ a0 þa1snðnÞ þ b1cnðnÞ

l1snðnÞ þ l2cnðnÞ þ 1;

V ðnÞ ¼ A0 þA1snðnÞ þ B1cnðnÞ

l1snðnÞ þ l2cnðnÞ þ 1þ A2sn2ðnÞ þ B2snðnÞcnðnÞðl1snðnÞ þ l2cnðnÞ þ 1Þ2

;

ð3:9Þ

where a0, a1, b1, A0, A1, A2, B1, B2, l1 and l2 are constants to be determined later.According to Step 4 in Section 2, with the aid of Maple, substituting (3.9) into (3.7), yields a set of algebraic

equations for sni(n)cnj(n) (i = 0,1, . . .; j = 0,1). Setting the coefficients of these terms sni(n)cnj(n) of the result-ing system�s numerator to be zero yields a set of over-determined algebraic equations with respect to a0, a1, b1,A0, A1, B1, A2, B2, k, l and k.

According to Step 5 in Section 2, by use of the Maple soft package ‘‘Charsets’’ by Dongming Wang, whichis based on the Wu-elimination method [20], solving the over-determined algebraic equations, we get explicitexpressions for a0, a1, b1, A0, A1, B1, A2, B2, l1, l2, k, l and k.

According to Step 6 in Section 2, we get following solutions of (2 + 1)-dimension DLWE:Family 1

u1 ¼�ð2kl3

2m2 � 2kl32 � 2kl2m2 þ kl2Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�l4

2m2 þ l42 þ 2m2l2

2 � l22 � m2

pkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�l42m2 þ l4

2 þ 2m2l22 � l2

2 � m2p

� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�l4

2m2 þ l42 þ 2m2l2

2 � l22 � m2

pkcnðkðxþ ly þ ktÞÞ

1þ l2cnðkðxþ ly þ ktÞÞ ; ð3:10:1Þ

v1 ¼l4

2 � l22 � l4

2m2 � m2 � 2lk2l22m2 þ lk2m2 þ 2m2l2

2 þ l42lk2m2 � l4

2lk2

l42m2 � l4

2 � 2m2l22 þ l2

2 þ m2

� 2lk2l2cnðkðxþ ly þ ktÞÞ1þ l2cnðkðxþ ly þ ktÞÞ þ

ð2lk2l22m2 � 2lk2l2

2 � 2lk2m2Þsn2ðkðxþ ly þ ktÞÞð1þ l2cnðkðxþ ly þ ktÞÞÞ2

; ð3:10:2Þ

where l2, k, l and k are arbitrary constants.


Family 2

u2 ¼�ðkl3

2 � l32m2k � kl2 þ kl2m2Þ � k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

2 � l42m2 � l2

2 þ 2m2l22 � m2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

2 � l42m2 � l2

2 þ 2m2l22 � m2

p

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2l2

2 þ l22 þ m2

pksnðkðxþ ly þ ktÞÞ

1þ l2cnðkðxþ ly þ ktÞÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

2 � l42m2 � l2

2 þ 2m2l22 � m2

pkcnðkðxþ ly þ ktÞÞ

1þ l2cnðkðxþ ly þ ktÞÞ ; ð3:11:1Þ

v2 ¼ ��l2

2 þ m2l22 � m2 þ lk2m2

m2l22 � l2

2 � m2� lk2l2cnðkðxþ ly þ ktÞÞ

1þ l2cnðkðxþ ly þ ktÞÞ

� lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2l2

2 þ l22 þ m2

pk2l2ð�l2

2 þ m2l22 þ 1� m2Þsnðkðxþ ly þ ktÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l42 � l4

2m2 � l22 þ 2m2l2

2 � m2p

ð1þ l2cnðkðxþ ly þ ktÞÞÞ

þ ð�lk2l22 þ lk2l2

2m2 � lk2m2Þsn2ðkðxþ ly þ ktÞÞð1þ l2cnðkðxþ ly þ ktÞÞÞ2

� lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2l2

2 þ l22 þ m2

q

� k2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

2 � l42m2 � l2

2 þ 2m2l22 � m2

psnðkðxþ ly þ ktÞÞcnðkðxþ ly þ ktÞÞ

ð1þ l2cnðkðxþ ly þ ktÞÞÞ2; ð3:11:2Þ

where l2, k, l and k are arbitrary constants.Family 3

u3 ¼�ðkl3

1 � kl1Þ � kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

1 � l21 � l2

1m2 þ m2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

1 � l21 � l2

1m2 þ m2p �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

1 � m2p

kcnðkðxþ ly þ ktÞÞl1snðkðxþ ly þ ktÞÞ þ 1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

1 � l21 � l2

1m2 þ m2p

ksnðkðxþ ly þ ktÞÞl1snðkðxþ ly þ ktÞÞ þ 1

; ð3:12:1Þ

v3 ¼lk2l4

1 � 2lk2l21m2 þ l2

1 � m2 þ lm2k2

�l21 þ m2

þ ð2lk2l31 � lk2l1 � lk2l1m2Þsnðkðxþ ly þ ktÞÞ

l1snðkðxþ ly þ ktÞÞ þ 1

� lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2 þ l2

1

pk2l1ðl2

1 � 1Þcnðkðxþ ly þ ktÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

1 � l21 � l2

1m2 þ m2p

ðl1snðkðxþ ly þ ktÞÞ þ 1Þ

þ ð�lk2l41 þ lk2l2

1 þ lk2l21m2 � lm2k2Þsn2ðkðxþ ly þ ktÞÞ

ðl1snðkðxþ ly þ ktÞÞ þ 1Þ2

� lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2 þ l2

1

pk2


1 � l21 � l2

1m2 þ m2p

snðkðxþ ly þ ktÞÞcnðkðxþ ly þ ktÞÞðl1snðkðxþ ly þ ktÞÞ þ 1Þ2

; ð3:12:2Þ

where l1, k, l and k are arbitrary constants.Family 4

u4 ¼�ðm2l1k þ kl1 � 2kl3

1Þ � kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil4

1 � l21m2 � l2

1 þ m2p


1 � l21m2 � l2

1 þ m2p � 2


1 � l21m2 � l2

1 þ m2p

ksnðkðxþ ly þ ktÞÞl1snðkðxþ ly þ ktÞÞ þ 1

;

ð3:13:1Þ

v4 ¼ �3lk2l4

1 � l41 � 6lk2l2

1m2 � m2 þ lm2k2 þ lk2m4 � 2lk2l61 þ 3lk2l4

1m2 þ l21m2 þ l2

1

�l41 þ l2

1m2 þ l21 � m2

þ ð�2lk2l1m2 � 2lk2l1 þ 4lk2l31Þsnðkðxþ ly þ ktÞÞ

l1snðkðxþ ly þ ktÞÞ þ 1

þ ð�2lk2l41 þ 2lk2l2

1m2 þ 2lk2l21 � 2lm2k2Þðsnðkðxþ ly þ ktÞÞÞ2

ðl1snðkðxþ ly þ ktÞÞ þ 1Þ2; ð3:13:2Þ

where l1, k, l and k are arbitrary constants.


Family 5

u5 ¼ �1

2a1 � kþ

a1sn � a1ðxþlyþktÞ2ffiffiffiffiffiffiffiffi1�m2p

� �

�sn � a1ðxþlyþktÞ2ffiffiffiffiffiffiffiffi1�m2p

� �� cn � a1ðxþlyþktÞ

2ffiffiffiffiffiffiffiffi1�m2p

� �þ 1

; ð3:14:1Þ

v5 ¼2A1 þ la2

1m2 þ 4� 4m2 � 2A1m2

4ðm2 � 1Þ �12la2

1sn2 � a1ðxþlyþktÞ2ffiffiffiffiffiffiffiffi1�m2p

� �

�sn�� a1ðxþlyþktÞ


�� cn � a1ðxþlyþktÞ


� �þ 1

� �2

þA1sn � a1ðxþlyþktÞ


� �




� �þ 1þ

� 12la2

1 þ A1

� �cn � a1ðxþlyþktÞ


� �




� �þ 1

þ12la2

1 � A1

� �sn � a1ðxþlyþktÞ


� �cn � a1ðxþlyþktÞ


� �

�snð� a1ðxþlyþktÞ2ffiffiffiffiffiffiffiffi1�m2p Þ � cn � a1ðxþlyþktÞ


� �þ 1

� �2; ð3:14:2Þ

where a1, A1, l and k are arbitrary constants.

Remark 2. The solutions (3.10.1) and (3.10.2) reproduce the solutions (16) in [10], when l2 = 0, k = �k* and

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2C1þ2þlk2

2lð1þm2Þ

q. The solutions (3.12.1) and (3.12.2) reproduce the solutions (17) in [10], when l1 = 0, k = �k*

and k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2C1þ2þlk2

lð2�m2Þ

q. The solutions (3.13.1) and (3.13.2) reproduce the solutions (15) in [10], when l1 = 0,

k = �k* and k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2C1þ2þlk2

2lð1�2m2Þ

q. The other solutions obtained here, to our knowledge, are all new families of

rational formal doubly periodic solution of the (2 + 1)-dimension DLWE.

Remark 3. If we replace the Jacobi elliptic functions sn(n), cn(n) in the ansatzes (3.8) with other pairs ofJacobi elliptic functions such as sn(n) and dn(n); ns(n) and cs(n); ns(n) and ds(n); sc(n) and nc(n); dc(n) andnc(n); sd(n) and nd(n); cd(n) and nd(n) (It is necessary to point out that they are satisfying system (2.5)),therefore other new double periodic wave solutions can be obtained for some system. For simplicity, we omitthem here.

4. Riccati equation rational expansion method

In this section we would like apply our method to obtain rational form soliton solutions of NLEEs, i.e.,restricting F and G in solutions of the Riccati equation.

The six main steps of the Riccati equation rational expansion method are illustrated with the Whitham–Broer–Kaup (WBK) system, i.e.,

ut þ uux þ vx þ buxx ¼ 0;

vt þ ðuvÞx þ auxxx � bvxx ¼ 0;ð4:1Þ

where a,b 5 0 are all constants.According to the Step 1 in Section 2, we make the following travelling wave transformation:

uðx; tÞ ¼ UðnÞ; vðx; tÞ ¼ V ðnÞ; n ¼ kðxþ ktÞ; ð4:2Þ

where k and k are constants to be determined later, and thus system (4.1) becomes

kU 0 þ UU 0 þ V 0 þ kbU 00 ¼ 0;

kV 0 þ ðUV Þ0 þ ak2U 000 � bkV 00 ¼ 0.ð4:3Þ


According to Step 2 in Section 2, We suppose that WBK system has the following formal travelling wavesolution:

UðnÞ ¼ a0 þXmu

i¼1

ai/iðnÞ þ bi/

i�1ðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

q

l1/ðnÞ þ l2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

qþ 1

� �i ;

V ðnÞ ¼ A0 þXmv

i¼1

Ai/iðnÞ þ Bi/

i�1ðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

q

l1/ðnÞ þ l2


qþ 1

� �i

ð4:4Þ

and the new variable / = /(n) satisfying the Riccati equation, i.e.,

/0 � ðRþ /2Þ ¼ d/dn� ðRþ /2Þ ¼ 0; ð4:5Þ

where R, a0, ai, bi, A0, Ai and Bi (i = 1,2, . . . ,mi) are constants to be determined later.For the WBK system, According to Step 3 in Section 2, by balancing the highest nonlinear terms and the

highest-order partial derivative terms in (4.3), gives mu = 1 and mv = 2. So we have

UðnÞ ¼ a0 þa1/ðnÞ þ b1


q

l1/ðnÞ þ l2


qþ 1

;

V ðnÞ ¼ A0 þA1/ðnÞ þ B1


q

l1/ðnÞ þ l2


qþ 1þ

A2/2ðnÞ þ B2/ðnÞ


q

l1/ðnÞ þ l2


qþ 1

� �2.

ð4:6Þ

According to Step 4 in Section 2, with the aid of Maple, substituting (4.6) along with (4.5) into (4.3), yields a

set of algebraic equations for /iðnÞðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

qÞj, (i = 0,1, . . . ; j = 0,1). Setting the coefficients of these terms

/iðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

q� �j

of the resulting system�s numerator to zero yields a set of over-determined algebraic

equations.

According to Step 5 in Section 2, by use of the Maple soft package ‘‘Charsets’’ by Dongming Wang, we getthe explicit expressions for a0, a1, b1, A0, A1, B1 A2, B2, l1, l2, k and k.

It is well known that the general solutions of Eq. (4.5) are

(1) when R < 0,

/ðnÞ ¼ �ffiffiffiffiffiffiffi�Rp

tanhðffiffiffiffiffiffiffi�Rp

nÞ; /ðnÞ ¼ �ffiffiffiffiffiffiffi�Rp

cothðffiffiffiffiffiffiffi�Rp

nÞ; ð4:7Þ
(2) when R = 0,
/ðnÞ ¼ � 1

n; ð4:8Þ

(3) when R > 0,

/ðnÞ ¼ffiffiffiRp

tanðffiffiffiRp

nÞ; /ðnÞ ¼ �ffiffiffiRp

cotðffiffiffiRp

nÞ. ð4:9Þ

Thus according to system (4.2), (4.7)–(4.9) and the conclusions in Step 5, we can obtain following rationalformal travelling-wave solutions of WBK system.

Note: Since tan- and cot-type solution appear in pairs with tanh- and coth-type solutions, respectively, weomit them in this paper, i.e., just considering the condition R < 0. In addition, some rational solutions are alsoomitted.


Family 1

u11 ¼ �kþ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ b2

qkffiffiffiffiffiffiffi�Rp


nÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2

2R� b2l22Rþ aþ b2


isechðffiffiffiffiffiffiffi�Rp

nÞ1� l2

ffiffiffiffiffiffiffi�Rp


nÞ; ð4:10:1Þ

v11 ¼k2R �a


q�


qb2 � ba� b3

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ b2

q þaþ b2 þ b


q� �k2Rtanh2ð


nÞ

ð1� l2



nÞÞ2

�k2Rl2aþ k2Rl2b

2 � bk2ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ b2

ql2R

� � ffiffiffiffiffiffiffi�Rp


nÞ

1� l2



nÞ

�


qk2 � bk2

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2


qtanhð


nÞRisechðffiffiffiffiffiffiffi�Rp

nÞ

ð1� l2



nÞÞ2; ð4:10:2Þ

u12 ¼ �kþ�




nÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2



cschðffiffiffiffiffiffiffi�Rp

nÞ1� l2



nÞ; ð4:11:1Þ

v12 ¼k2R �a


q�


qb2 � ba� b3

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ b2

q þaþ b2 þ b


q� �k2Rcoth2ð


nÞ

ð1� l2



nÞÞ2

�k2Rl2aþ k2Rl2b

2 � bk2ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ b2

ql2R

� � ffiffiffiffiffiffiffi�Rp


nÞ

1� l2



nÞ

�


qk2 � bk2

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2


qcothð


nÞRcschðffiffiffiffiffiffiffi�Rp

nÞ

ð1� l2



nÞÞ2; ð4:11:2Þ

where n = k(x + kt), R < 0, l2, k and k are arbitrary constants.Family 2

u21 ¼ a0 � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2




nÞ1� l2



nÞ; ð4:12:1Þ

v21 ¼ A0 �ð2k2Rl2aþ 2k2Rl2b

2Þffiffiffiffiffiffiffi�Rp


nÞ1� l2



nÞþ 2k2ðaþ b2ÞRtanh2ð


nÞð1� l2



nÞÞ2

�2bk2



qtanhð



nÞð1� l2



nÞÞ2; ð4:12:2Þ

u22 ¼ a0 � 2





nÞ1� l2



nÞ; ð4:13:1Þ

v22 ¼ A0 �ð2k2Rl2aþ 2k2Rl2b

2Þffiffiffiffiffiffiffi�Rp


nÞ1� l2



nÞþ 2k2ðaþ b2ÞRcoth2ð


nÞð1� l2



nÞÞ2

�2bk2



qcothð



nÞð1� l2



nÞÞ2; ð4:13:2Þ


where n = k(x + kt), a0 ¼ �kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�al2

2R�b2l2

2Rþaþb2

p�kal2R�kb2l2Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�al22R�b2l2

2Rþaþb2

p , A0 ¼ �ð2al2

2R�aþ2b2l2

2R�b2Þk2R

l22R�1

, R < 0, l2, k and k arearbitrary constants.

Family 3

u31 ¼ �k� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2Rl2

1 þ aþ b2 þ aRl21



nÞ�l1



nÞ þ 1; ð4:14:1Þ

v31 ¼ A0 þ�4ðb2Rl2

1 þ aþ b2 þ aRl21Þk

2l1Rffiffiffiffiffiffiffi�Rp


nÞ�l1



nÞ þ 1

�2bk2



qRffiffiffiffiffiffiffi�Rp

il1sechðffiffiffiffiffiffiffi�Rp

nÞ�l1



nÞ þ 1� A2Rtanh2ð


nÞð�l1



nÞ þ 1Þ2

�2bk2



qðRl2

1 þ 1Þ� �



nÞ

ð�l1



nÞ þ 1Þ2; ð4:14:2Þ

u32 ¼ �k� 2





nÞ�l1



nÞ þ 1; ð4:15:1Þ

v32 ¼ A0 þ�4ðb2Rl2

1 þ aþ b2 þ aRl21Þk

2l1Rffiffiffiffiffiffiffi�Rp


nÞ�l1



nÞ þ 1

�2bk2



qRl1


icschðffiffiffiffiffiffiffi�Rp

nÞ�l1



nÞ þ 1� A2Rcoth2ð


nÞð�l1



nÞ þ 1Þ2

�2bk2



qðRl2

1 þ 1Þ� �



nÞ

ð�l1



nÞ þ 1Þ2; ð4:15:2Þ

where n = k(x + kt), A0 ¼ �ð2b2Rl21 þ 2aRl2

1 þ b2 þ aÞk2R, A2 ¼ �4b2k2Rl21 � 2ak2 � 2b2k2 � 4ak2Rl2

1�2b2k2R2l4

1 � 2l41ak2R2, R < 0, l1, k and k are arbitrary constants.

Family 4

u41 ¼�2b2kl1R� 2akRl1 � k

ffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

qffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q � 2


qð1þ l2

1RÞkffiffiffiffiffiffiffi�Rp


nÞ�l1



nÞ þ 1; ð4:16:1Þ

v41 ¼ A0 �4k2Rl1ð1þ l2

1RÞ aþ b2 � bffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q� � ffiffiffiffiffiffiffi�Rp


nÞ

�l1



nÞ þ 1

þ2k2ð1þ l2

1RÞ2 aþ b2 � bffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q� �Rtanh2ð


nÞ

ð�l1



nÞ þ 1Þ2; ð4:16:2Þ

u42 ¼�2b2kl1R� 2akRl1 � k


qffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q � 2


qð1þ l2

1RÞkffiffiffiffiffiffiffi�Rp


nÞ�l1



nÞ þ 1; ð4:17:1Þ

v42 ¼ A0 �4k2Rl1ð1þ l2

1RÞ aþ b2 � bffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q� � ffiffiffiffiffiffiffi�Rp


nÞ

�l1



nÞ þ 1

þ2k2ð1þ l2

1RÞ2 aþ b2 � bffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

q� �Rcoth2ð


nÞ

ð�l1



nÞ þ 1Þ2; ð4:17:2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiq� �
where n = k(x + kt), A0 ¼ �2 aþ b2 � b b2 þ a ðRl2
1 þ 1Þk2R, R < 0, l1, k and k are arbitrary constants.


Family 5

u51 ¼ a0 þ�


qðRþ 1Þk



nÞ � �ffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

qR�


q� �kffiffiffiffiffiffiffi�Rp


nÞ

�ffiffiffiffiffiffiffi�Rp


nÞ �ffiffiffiffiffiffiffi�Rp


nÞ þ 1;

ð4:18:1Þ

v51 ¼ A0 þ2Rk2 �b2 � a� b


q� �ðRþ 1Þ



nÞ

ð�ffiffiffiffiffiffiffi�Rp




nÞ þ 1ÞðR� 1Þ

�2Rk2 �b2 � a� b


q� �ðRþ 1Þ



nÞ





nÞ þ 1ÞðR� 1Þ

�ðRþ 1Þ2k2 �b2 � a� b


q� �Rtanh2ð


nÞ





nÞ þ 1Þ2ðR� 1Þ

�ðRþ 1Þ2k2 �b2 � a� b


q� �tanhð



nÞ





nÞ þ 1Þ2ðR� 1Þ; ð4:18:2Þ

u52 ¼ a0 þ�


qðRþ 1Þk



nÞ � �ffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a

qR�


q� �kffiffiffiffiffiffiffi�Rp


nÞ





nÞ þ 1;

ð4:19:1Þ

v52 ¼ A0 þ2Rk2 �b2 � a� b


q� �ðRþ 1Þ



nÞ





nÞ þ 1� �

ðR� 1Þ

�2Rk2 �b2 � a� b


q� �ðRþ 1Þ



nÞ





nÞ þ 1� �

ðR� 1Þ

�ðRþ 1Þ2k2 �b2 � a� b


q� �Rcoth2ð


nÞ





nÞ þ 1� �2ðR� 1Þ

�ðRþ 1Þ2k2 �b2 � a� b


q� �cothð



nÞ





nÞ þ 1� �2ðR� 1Þ

; ð4:19:2Þ

where n = k(x + kt), A0 ¼ ðð�13R�7Þðb2þaÞ32b�ð1�RÞðabþb3Þ

ffiffiffiffiffiffiffiffib2þap

�ð2ab2þa2þb4Þð5Rþ1ÞÞk2R

�ð5Rþ1ÞðR�1Þðb2þaÞðRþ1Þ

, a0 ¼ �kRa�kb2R�ffiffiffiffiffiffiffiffib2þap

kffiffiffiffiffiffiffiffib2þap , R < 0, k

and k are arbitrary constants.

Remark 4. The solution (4.10) can be reduced to the solution 4 in [11], when a + b2 > 0, k ¼ffiffiffiffiffiffiffi�Rp

and l2 = 0.The solution (4.11) can be reduced to the solution 2 in [11], when a + b2 > 0, k ¼


and l2 = 0. Thesolution (4.14) can be reduced to the solution 2 in [11], when a + b2 < 0, k ¼


and l1 = 0. The solution(4.15) can be reduced to the solution 4 in [11], the solution (4.16) can be reduced to the solution 1 in [11], thesolution (4.17) can be reduced to the solution 3 in [11], when a + b2 > 0, k ¼


and l1 = 0. The othersolutions obtained here, to our knowledge, are all new families of exact solutions of the WBK system.


5. Generalized Riccati equation rational expansion method

In this section we would like extend the Riccati equation rational expansion method, i.e., using generalizedansatz to replace (2.4) and solutions of generalized Riccati equation to replace the variables in (2.4).

Consider the (2 + 1)-dimension Burgers system, i.e.,

� ut þ uuy þ avux þ buyy þ abuxx ¼ 0;

ux � vy ¼ 0.ð5:1Þ

According to the Step 1 in Section 2, we make the following travelling wave transformation:

uðx; y; tÞ ¼ UðnÞ; vðx; y; tÞ ¼ V ðnÞ; n ¼ kðxþ ly þ ktÞ; ð5:2Þ
where k and k are constants to be determined later, and thus system (5.1) becomes
� kU 0 þ lUU 0 þ aVU 0 þ bkl2U 00 þ abkU 00 ¼ 0;

U 0 � lV 0 ¼ 0.ð5:3Þ

According to Step 2 in Section 2, We suppose that (2 + 1)-dimension Burgers system has the following for-mal travelling wave solution:

UðnÞ ¼ a0 þXmu

i¼1

ai/iðnÞ þ bi/

i�1ðnÞ/0ðnÞ þ ci1

/iðnÞ

ðl1/ðnÞ þ l2/0ðnÞ þ l3

1/þ 1Þi

;

V ðnÞ ¼ A0 þXmv

i¼1

Ai/iðnÞ þ Bi/

i�1ðnÞ/0ðnÞ þ Ci1

/iðnÞ

l1/ðnÞ þ l2/0ðnÞ þ l3

1/þ 1

� �i ;

ð5:4Þ

and the new variable / = /(n) satisfying

/0 � ðh1 þ h2/2Þ ¼ d/

dn� ðh1 þ h2/

2Þ ¼ 0; ð5:5Þ

where R, a0, ai, bi, ci, A0, Ai, Bi and Ci (i = 1,2, . . . ,mi) are constants to be determined later.For the (2 + 1)-dimension Burgers system, according to Step 3 in Section 2, by balancing the highest non-

linear terms and the highest-order partial derivative terms in (5.3), gives mu = 1 and mv = 1. So we have

UðnÞ ¼ a0 þa1/

iðnÞ þ b1/0ðnÞ þ ci

1/ðnÞ


1/ðnÞ þ 1

;

V ðnÞ ¼ A0 þA1/

iðnÞ þ B1/0ðnÞ þ Ci

1/ðnÞ


1/ðnÞ þ 1

.

ð5:6Þ

According to Step 4 in Section 2, with the aid of Maple, substituting (5.6) along with (5.5) into (5.3), yields aset of algebraic equations for /i(n) (i = 0,1, . . .). Setting the coefficients of these terms /i(n) of the resultingsystem�s numerator to zero yields a set of over-determined algebraic equations.

According to Step 5 in Section 2, by use of the Maple soft package ‘‘Charsets’’ by Dongming Wang, we getthe explicit expressions for a0, a1, b1, c1, A0, A1, B1, C1, l1, l2, l3, k, l and k.

We know that the general solutions of Eq. (5.5) are

(1) when h1 ¼ 12

and h2 ¼ � 12,

/ðnÞ ¼ tanhðnÞ � isechðnÞ; /ðnÞ ¼ cothðnÞ � cschðnÞ; ð5:7Þ
(2) when h1 ¼ h2 ¼ � 1
2,

/ðnÞ ¼ secðnÞ � tanðnÞ; /ðnÞ ¼ cscðnÞ � cotðnÞ; ð5:8Þ
(3) when h1 = 1 and h2 = �1,
/ðnÞ ¼ tanhðnÞ; /ðnÞ ¼ cothðnÞ. ð5:9Þ


(4) when h1 = h2 = 1,

/ðnÞ ¼ tanðnÞ; ð5:10Þ
(5) when h1 = h2 = �1,
/ðnÞ ¼ cotðnÞ; ð5:11Þ
(6) when h1 = 0 and h2 5 0,
/ðnÞ ¼ � 1

h2nþ c0

. ð5:12Þ

Thus according to system (5.2), (5.7)–(5.12) and the conclusions in Step 5, we can obtain following rationalformal travelling-wave solutions of (2 + 1)-dimension Burgers system.

Note: Here like Section 4, we omit tan-, cot-type and rational solutions and just list new solutions comparedwith the result in Section 4.

Family 1

u11¼ a0þlbk2l2

2

ð4l21�l2

2Þð4l21þl2ÞðtanhðnÞ� isechðnÞÞ2

l1ðtanhðnÞ� isechðnÞÞ2þl2ðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞ tanhðnÞÞþ tanhðnÞ� isechðnÞþl3

þ lbkl1ð8l21�2l2

2ÞðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞtanhðnÞÞ�l2c1

l2ðl1ðtanhðnÞ� isechðnÞÞ2þl2ðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞ tanhðnÞÞþ tanhðnÞ� isechðnÞþl3Þ;

ð5:13:1Þ

v11¼A0þbk2l2

2

ð4l21�l2

2Þð4l21þl2ÞðtanhðnÞ� isechðnÞÞ2

l1ðtanhðnÞ� isechðnÞÞ2þl2ðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞtanhðnÞÞþ tanhðnÞ� isechðnÞþl3

þbkl1ð8l2

1�2l22ÞðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞ tanhðnÞÞ� l2

l c1

l2ðl1ðtanhðnÞ� isechðnÞÞ2þl2ðtanhðnÞ� isechðnÞÞðsech2ðnÞ� isechðnÞ tanhðnÞÞþ tanhðnÞ� isechðnÞþl3Þ;

ð5:13:2Þ

u12¼ a0þlbk2l2

2

ð4l21�l2

2Þð4l21þl2ÞðcothðnÞ� cschðnÞÞ2

l1ðcothðnÞ�cschðnÞÞ2þl2ðcothðnÞ�cschðnÞÞðcsch2ðnÞ�cschðnÞcothðnÞÞþ cothðnÞ� cschðnÞþl3

þ lbkl1ð8l21�2l2

2ÞðcothðnÞ�cschðnÞÞðcsch2ðnÞ� icschðnÞcothðnÞÞ�l2c1

l2ðl1ðcothðnÞ� cschðnÞÞ2þl2ðcothðnÞ� cschðnÞÞðcsch2ðnÞ� cschðnÞcothðnÞÞþcothðnÞ�cschðnÞþl3Þ;

ð5:14:1Þ

v12¼A0þbk2l2

2

ð4l21�l2

2Þð4l21þl2ÞðcothðnÞ� cschðnÞÞ2

l1ðcothðnÞ�cschðnÞÞ2þl2ðcothðnÞ� cschðnÞÞðcsch2ðnÞ�cschðnÞcothðnÞÞþ cothðnÞ� cschðnÞþl3

þbkl1ð8l2

1�2l22ÞðcothðnÞ� cschðnÞÞðcsch2ðnÞ� icschðnÞcothðnÞÞ� l2

l c1

l2ðl1ðcothðnÞ� cschðnÞÞ2þl2ðcothðnÞ� cschðnÞÞðcsch2ðnÞ� cschðnÞcothðnÞÞþcothðnÞ�cschðnÞþl3Þ;

ð5:14:2Þ

where n = k(x + ly + kt), a0 ¼2bkl2l1l

22þ2bkl2l2l1�8l3

1abk�8l3

1bkl2þ2abkl2l1þ2abkl1l

22�aA0l2

2þkl2

2

l22l

, l3 ¼ �l1ð8l2

1�l2

2�2l2Þ

l22

,

c1 ¼ �lbkð4l2

1�l2Þð8l2

1�l2

2�2l2Þð8l2

1�l2

2Þ

2l42

, l1, l2, l and k are arbitrary constants.


Family 2

u21 ¼ a0 þ2lbkð�l2

2 þ l21Þð2l2

1 þ l2Þtanh2ðnÞl2

2ðl1tanh2ðnÞ þ l2 tanhðnÞsech2ðnÞ þ l3 þ tanhðnÞÞ

þ 4lbkl1l32ð�l2

2 þ l21Þ tanhðnÞsech2ðnÞ � 2lbkð2l2

1 � l2Þð�l22 þ 2l2

1 � l2Þð�l22 þ l2

1Þl4

2ðl1tanh2ðnÞ þ l2 tanhðnÞsech2ðnÞ þ l3 þ tanhðnÞÞ; ð5:15:1Þ

v21 ¼ A0 þ2bkð�l2

2 þ l21Þð2l2

1 þ l2Þtanh2ðnÞl2

2ðl1tanh2ðnÞ þ l2 tanhðnÞsech2ðnÞ þ l3 þ tanhðnÞÞ

þ 4bkl1l32ð�l2

2 þ l21Þ tanhðnÞsech2ðnÞ � 2bkð2l2

1 � l2Þð�l22 þ 2l2

1 � l2Þð�l22 þ l2

1Þl4

2ðl1tanh2ðnÞ þ l2 tanhðnÞsech2ðnÞ þ l3 þ tanhðnÞÞ; ð5:15:2Þ

u22 ¼ a0 þ2lbkð�l2

2 þ l21Þð2l2

1 þ l2Þcoth2ðnÞl2

2ðl1coth2ðnÞ þ l2 cothðnÞcsch2ðnÞ þ l3 þ cothðnÞÞ

þ 4lbkl1l32ð�l2

2 þ l21Þ cothðnÞcsch2ðnÞ � 2lbkð2l2

1 � l2Þð�l22 þ 2l2

1 � l2Þð�l22 þ l2

1Þl4

2ðl1coth2ðnÞ þ l2 cothðnÞcsch2ðnÞ þ l3 þ cothðnÞÞ; ð5:16:1Þ

v22 ¼ A0 þ2bkð�l2

2 þ l21Þð2l2

1 þ l2Þcoth2ðnÞl2

2ðl1coth2ðnÞ þ l2 cothðnÞcsch2ðnÞ þ l3 þ cothðnÞÞ

þ 4bkl1l32ð�l2

2 þ l21Þ cothðnÞcsch2ðnÞ � 2bkð2l2

1 � l2Þð�l22 þ 2l2

1 � l2Þð�l22 þ l2

1Þl4

2ðl1coth2ðnÞ þ l2 cothðnÞcsch2ðnÞ þ l3 þ cothðnÞÞ; ð5:16:2Þ

where n = k(x + ly + kt), a0 ¼4bkl2l1l

22þ2bkl2l2l1�4l3

1abk�4l3

1bkl2þ2abkl2l1þ4abkl1l

22�aA0l2

2þkl2

2

l22l

, l3 ¼ �l1ð�l2

2þ2l2

1�l2Þ

l22

, l1, l2, l

and k are arbitrary constants.

6. Summary and conclusions

In summary, we have proposed a unified algebraic method: rational expansion method with symbolic com-putation, which greatly exceeds the applicability of the existing tanh method in obtaining multiple travellingwave solutions of the NLEEs.

In Section 3, when l1 = l2 = 0, our method just reduces to the Jacobi elliptic function expansion method[10].

In Section 4, when l1 = l2 = 0, our method just reduces to the tanh-method [6,7,9,10]; when l1 = 0, ourmethod just reduce to the projective Riccati equation expansion method [14–16].

As we can see in Section 5, we can naturally extend our method to generalized form, such as: we can replacesystem (2.4) by

U iðnÞ ¼ ai0 þXmi

j¼1

Pr1þ��þrn¼ja

jr1��rn

F r11 � � � F rn

nPl1þ��þln¼jl

jl1��ln

F l11 � � � F ln

n þ 1; ð6:1Þ

where ai0, ajr1��rn

, ljl1��ln

and n are differentiable function to be determined and dF idn ¼ KiðF 1; . . . ; F nÞ, where Ki are

polynomial of Fi. It is clearly to see that (6.1) is also satisfying solving the recurrent relation or derivative rela-tion for the terms of polynomial for computation closed.

The feature of our proposed method is that, we firstly propose a more general ansatz, in which the closed-form solutions of NLEEs that can be expressed as a finite series of some Jacobi elliptic function or solutions ofsubequation (like the Riccati equation) in rational form. The method provides us with new and more generaltravelling wave solutions that cannot be obtained by the tanh method, Jacobi elliptic function expansion meth-ods, the homogeneous balance method and projective Riccati equation expansion method. This method is alsocomputerizable. The key idea is to decompose NLEE system into a system of algebraic equations or an ordin-ary differential equation, which allows us to perform complicated and tedious algebraic computation by usingsymbolic computation. Finally, we point out that our method also can be applied to a large class of either


integrable or non-integrable nonlinear coupled systems. The details for these cases will be investigated in ourfuture works.

Acknowledgements

The work was supported by the China Postdoctoral Science Foundation, Nature Foundation of ZhejiangProvince of China (Y604056) and Ningbo Doctoral Foundation of China (2005A610030).

References

[1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, NewYork, 1991.

[2] V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991.[3] Y. Chen, Z.Y. Yan, H.Q. Zhang, Theor. Math. Phys. 132 (1) (2002) 970;

Y. Chen, B. Li, H.Q. Zhang, Commun. Theor. Phys. 39 (2003) 135;Y. Chen, B. Li, H.Q. Zhang, Chaos Solitons Fractals 17 (2003) 693;B. Li, Y. Chen, H.Q. Zhang, Phys. Lett. A 305 (6) (2002) 377;B. Li, Y. Chen, H.Q. Zhang, Z. Naturforsch. A 58 (7–8) (2003) 464.

[4] S.Y. Lou, J.Z. Lu, J. Phys. A 29 (1996) 4029;S.Y. Lou, Phys. Lett. A 277 (2000) 94.

[5] R. Hirta, J. Satsuma, Phys. Lett. A 85 (1981) 407.[6] S.Y. Lou, G.X. Huang, H.Y. Ruan, J. Phys. A 24 (1991) L587.[7] E.J. Parkes, B.R. Duffy, Comput. Phys. Commun. 98 (1996) 288;

E.J. Parkes, B.R. Duffy, Phys. Lett. A 229 (1997) 217.[8] W. Hereman, Comput. Phys. Commun. 65 (1991) 143.[9] E. Fan, Phys. Lett. A 277 (2000) 212;

E. Fan, Comput. Math. Appl. 43 (2002) 671.[10] Z.Y. Yan, Comput. Phys. Commun. 153 (2003) 145.[11] Z.Y. Yan, H.Q. Zhang, Phys. Lett. A 285 (2001) 355.[12] Y. Chen, Y. Zheng, Int. J. Mod. Phys. C 14 (5) (2003) 601;

Y. Chen, B. Li, H.Q. Zhang, Int. J. Mod. Phys. C 13 (2003) 99;Y. Chen, X.D. Zheng, B. Li, H.Q. Zhang, Appl. Math. Comput. 149 (1) (2004) 277;Q. Wang, Y. Chen, B. Li, H.Q. Zhang, Appl. Math. Comput. 160 (2005) 77.

[13] B. Li, Y. Chen, H.Q. Zhang, Appl. Math. Comput. 146 (2–3) (2003) 653;B. Li, Y. Chen, H.Q. Zhang, Z. Naturforsch. A 57 (11) (2002) 874;B. Li, Y. Chen, H.Q. Zhang, Appl. Math. Comput. 146 (2003) 653.

[14] R. Conte, M. Musette, J. Phys. A: Math. Gen. 25 (1992) 5609.[15] Z.Y. Yan, Chaos Solitons Fractals 16 (2003) 759.[16] Y. Chen, B. Li, Chaos Solitons Fractals 19 (4) (2004) 977;

B. Li, Y. Chen, Z. Naturforsch. A 58 (9–10) (2003) 511.[17] Y. Chen, Q. Wang, B. Li, Z. Naturforsch. A 59a (2004) 529.[18] Y. Chen, Q. Wang, Z. Naturforsch. A. 60a (2005) 1.[19] K. Chandrasekharan, Elliptic Function, Springer, Berlin, 1978;

D.V. Patrick, Elliptic Function and Elliptic Curves, Cambridge University Press, Cambridge, 1973.[20] W.J. Wu, Algorithams and Computation Berli, Springer, Berlin, 1994, p. 1.

a unified rational expansion method to construct a series of explicit exact solutions to nonlinear...

Documents