symbolic computation and construction of soliton-like solutions to the (2 + 1)-dimensional...

International Journal of Engineering Science 42 (2004) 715–724www.elsevier.com/locate/ijengsci

Symbolic computation and construction ofsoliton-like solutions to the (2 + 1)-dimensional dispersive

long-wave equations

Yong Chen a,b,c,*, Biao Li b,c

a Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, 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

Received 29 January 2003; received in revised form 30 April 2003; accepted 2 June 2003

Abstract

By means of a new Riccati equation expansion method, we consider the (2 + 1)-dimensional dispersive

long-wave equations uyt þ gxx þ 12ðu2Þxy ¼ 0, gt þ ðugþ uþ uxyÞx ¼ 0. As a result, we not only can success-

fully recover the previously known formal solutions obtained by known tanh function methods but also

construct new and more general formal solutions. The solutions obtained include the nontravelling wave

and coefficient functions� soliton-like solutions, singular soliton-like solutions, triangular functions solu-

tions.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Generalized Riccati equation expansion method; Symbolic computation; (2 + 1)-dimensional dispersive

long-wave equation: soliton-like solutions; Solitons

1. Introduction

In recent years, the study of solitons and the related issue of the construction of solutions to awide variety of nonlinear evolution equations (NEEs) has become one of the most exciting andextremely active areas of research. There are a wealth of methods for finding special solutions ofNEEs, such as, inverse scattering method, B€acklund transformation, Darboux transformation,

*Corresponding author. Address: Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China.

Fax: +86-411-4707204.

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

0020-7225/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijengsci.2003.06.002

mail to: [email protected]

716 Y. Chen, B. Li / International Journal of Engineering Science 42 (2004) 715–724

Cole-Hopf transformation, Hirota method, Painlev�e method [1], tanh method [2–4], extendedtanh-function method [5–7], modified extended tanh-function method [8], generalized hyperbolic-function method [9–11], variable separation approach [12]. But finding more powerful methods isstill a significant subject in solving NEEs in soliton theory and its applications. With the rapiddevelopment of computerized symbolic computation, the application of symbolic computation tothe physical and mathematical sciences appears to have a bright future.

One of most effectively straightforward methods to construct exact solutions of NEEs is tanhmethod [2–4]. Recently, Fan [5,6] has proposed an extended tanh-function method. More re-cently, Fan [7], Yan [13,14], Li and Chen [15–17] further developed this idea and made it muchmore lucid and straightforward for a class of NEEs. Most recently, Elwakil et al. [8] modifiedextended tanh-function method and obtain some new exact solutions. Gao and Tian [9–11]presented the generalized tanh method and generalized hyperbolic-function method by intro-ducing coefficient functions. As is known, when applying direct method, the choice of anappropriate ans€atz is of great importance. In this paper, based on the above work [2–11,13–17]and with the aid of symbolic computation software Maple, by introducing a new more generalans€atz than the ans€atz in the above methods, we present the generalized Riccati equationexpansion method. To illustrate our algorithm, we take the (2+ 1)-dimensional dispersive long-wave equations (DLWS) [18–21] as a simple example, which reads

uyt þ gxx þ1

2ðu2Þxy ¼ 0; ð1:1aÞ

gt þ ðugþ uþ uxyÞx ¼ 0: ð1:1bÞ

The (2+ 1)-dimensional DLWS (1.1) was first derived by Boiti et al. [18] as a compatibility for a‘‘weak’’ Lax pair. Recently considerable effort has been devoted to the study of this system. In[19], Paquin and Winternitz showed that the symmetry algebra of (2+ 1)-dimensional DLWS (1.1)is infinite-dimensional and possesses a Kac–Moody–Virasoro structure. Some special similaritysolutions are also given in [19] by using symmetry algebra and the classical theoretical analysis.The more general symmetry algebra, w1 symmetry algebra, is given in [20]. Lou [21] has givennine types of the two-dimensional partial differential equation reductions and 13 types of theordinary differential equation reductions by means of the direct and nonclassical method. Thesystem (1.1) have no Painlev�e property though they are Lax or IST integrable [22]. More recently,Tang et al. [23], by means of the variable separation approach, the abundant localized coherentstructures of the system (1.1) are derived. In [24], the possible chaotic and fractal localizedstructures are revealed for the system (1.1). Zhang [25], starting from the homogeneous balancemethod, found that the richness of the localized coherent structures of the model is caused by theappearance of two variable-separated arbitrary functions.

The plan of the paper is as follows. In Section 2, we describe briefly the generalized Riccatiequation expansion method. In Section 3, we apply the method to Eq. (1.1) and bring out rich newfamilies of the exact solutions of system (1.1), including the nontravelling wave and coefficientfunctions� soliton-like solutions, singular soliton-like solutions, triangular functions solutions.Conclusions will be presented finally.

Y. Chen, B. Li / International Journal of Engineering Science 42 (2004) 715–724 717

2. Generalized Riccati equation expansion method

Let us simply describe the generalized Riccati equation expansion method, as follows:Consider a given system of NEEs in three independent variables x, y, t

E1ðu; v; ut; vt; ux; vx; uy ; vy; uxx; vxx; uxt; vxt; uxy; vxy; uyt; vyt; . . . ; Þ ¼ 0; ð2:1aÞ

E2ðu; v; ut; vt; ux; vx; uy ; vy; uxx; vxx; uxt; vxt; uxy; vxy; uyt; vyt; . . . ; Þ ¼ 0: ð2:1bÞ

We seek the following formal solutions of the given system by the new more general ans€atz

uðx; y; tÞ ¼ a0 þXmi¼1

ai/iðnÞ

�þ bi/

i�1ðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

qþ gi/

�iðnÞ�; ð2:2aÞ

vðx; y; tÞ ¼ A0 þXn

j¼1

Aj/iðnÞ

�þ Bj/

j�1ðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

qþ Gj/

�jðnÞ�; ð2:2bÞ

where m, n are integers to be determined by balancing the highest order derivative terms with thenonlinear terms in (2.1), R is a real constant, while a0 ¼ a0ðx; y; tÞ, A0 ¼ A0ðx; y; tÞ, ai ¼ aiðx; y; tÞ,bi ¼ biðx; y; tÞ, gi ¼ giðx; y; tÞ, Aj ¼ Aiðx; y; tÞ, Bj ¼ Bjðx; y; tÞ, Gj ¼ Gjðx; y; tÞ ði ¼ 1; . . . ;m; j ¼1; . . . ; nÞ, n ¼ nðx; y; tÞ are all differentiable functions and /ðnÞ satisfies

d/ðnÞdn

¼ Rþ /2ðnÞ: ð2:3Þ

It is easy to see that the ans€atz (2.2) is more general than the ans€atz in the generalized hyperbolic-function method [9–11], tanh method [2–4], extended tanh-function method [5–7,13–17], modifiedextended tanh-function method [8]. Firstly, compared with the tanh method, extended tanh-function, as well as the modified extended tanh-function method, the restriction on nðx; y; tÞ asmerely a linear function x, y, t and the restriction on the coefficients ai, bi, gi, Aj, Bj, Gj

ði ¼ 0; . . . ;m; j ¼ 0; . . . ; nÞ and n as constants are removed. Secondly, compared with the gen-eralized hyperbolic-function method [9–11], we cannot only recover the exact solutions for a givenNEE which are the superposition of different powers of the sechn function, tanh n function ortheir combinations, but also we can, with no extra effort, find other new and more general types ofsolutions, such as singular soliton-like solutions, coth-type solutions and triangular periodic-likesolutions, tan-type solutions, and these formal functions� combination, even rational solutions etc.More importantly, we add terms gi/

�iðnÞ in the new ans€atz (2.2), so more types of solutions wouldbe expected for some equations.

There exists the following steps to be considered further:

Step 1. Determined the values of m and n of system (2.2) by respectively balancing the highest-order partial derivative terms and the nonlinear terms in system (2.1).

Step 2. Substituting (2.2) along with (2.3) into (2.1), multiplying by the most simple common

denominator in the obtained system, setting the coefficients of /pðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

q� �q


ðp ¼ 0; 1; . . . ; q ¼ 0; 1Þ�Note: where /pðnÞ denotes p power of /ðnÞ and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

q� �q

denotes q power of


q �to zero, we obtain a set of over-determined partial dif-

ferential equations with regard to differentiable functions ai, bi, gi, Aj, Bj, Gj

ði ¼ 0; . . . ;m; j ¼ 0; . . . ; nÞ and n.Step 3. Solving the over-determined partial differential equations by use of the PDEtools package

of Maple, we would end up with the explicit expressions for ai, bi, gi, Aj, Bj, Gj

(i ¼ 0; . . . ;m; j ¼ 0; . . . ; n) and n or the constraints among them.Step 4. It is well-known that the general solutions of Riccati equation (2.3) are

/ðnÞ ¼

�ffiffiffiffiffiffiffi�R

ptanh

ffiffiffiffiffiffiffi�R

pn

� �; R < 0;

�ffiffiffiffiffiffiffi�R

pcoth


pn

� �; R < 0;ffiffiffi

Rp

tanffiffiffiR

pn

� �; R > 0;

�ffiffiffiR

pcot

ffiffiffiR

pn

� �; R > 0;

� 1n ; R ¼ 0:

8>>>>><>>>>>:

ð2:4Þ

Thus according to (2.2), (2.4) and the conclusions in Step 3, the soliton-like solutions of (2.1) canbe obtained.

For the generalization of the ans€atz, naturally more complicated computation is expected thanever before. Even if the availability of computer symbolic systems like Maple or Mathematicaallows us to perform the complicated and tedious algebraic calculation and differential calculationon a computer, in general, it is very difficult, sometime impossible, to solve the set of over-determined partial differential equations in Step 2. As the calculation goes on, in order to dras-tically simplify the work or make the work feasible, we often choose special function forms for ai,bi, gi, Aj, Bj, Gj ði ¼ 0; . . . ;m; j ¼ 0; . . . ; nÞ and n, on a trial-and-error basis.

3. Application

In this section, by use of the generalized Riccati equation expansion method, we investigate a(2+ 1)-dimensional dispersive long-wave system [18–25], i.e., Eq. (1.1). By balancing the highest-order contributions from both the linear and nonlinear terms in Eq. (1.1), we obtain m ¼ 1, n ¼ 2in (2.2). Therefore we assume the solutions of Eq. (1.1) in the form

uðx; y; tÞ ¼ a0 þ a1/ðnÞ þ b1


qþ g1/

�1ðnÞ; ð3:1aÞ

gðx; y; tÞ ¼ A0 þ A1/ðnÞ þ A2/2ðnÞ þ B1


qþ B2/ðnÞ


qþ G1/

�1ðnÞ þ G2/�2ðnÞ ð3:1bÞ

where a0 ¼ a0ðy; tÞ, a1 ¼ a1ðy; tÞ, b1 ¼ b1ðy; tÞ, g1 ¼ g1ðy; tÞ, A0 ¼ A0ðy; tÞ, A1 ¼ A1ðy; tÞ,A2 ¼ A2ðy; tÞ, B1 ¼ B1ðy; tÞ, B2 ¼ B2ðy; tÞ, G1 ¼ G1ðy; tÞ, G2 ¼ G2ðy; tÞ and n ¼ xpðy; tÞ þ qðy; tÞ alldifferentiable functions and /ðnÞ satisfies (2.3). The aim of choosing these functions to be specialforms, i.e., the x independence of a0, a1 etc., is to make calculation feasible.


Substituting (3.2) along with (2.3) into (1.1), multiplying by /ðnÞ4 and /4ðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /2ðnÞ

qin the

first equation and the second equation respectively, collecting coefficients of monomials of /ðnÞ,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRþ /ðwÞ2

qand x (Notice that a0, a1, b1, g1, A0, A1, A2, B1, B2, G1, G2, p and q are independent of

x.) with the aid of Maple, then setting each coefficients to zero, we can deduce a set of over-determined partial differential equations with respect to the unknown functions a0, a1, b1, g1, A0,A1, A2, B1, B2, G1, G2, p and q. Because the set includes 69 equations, for simplification, we do notlist them in the paper.

Using the powerful PDEtools package ofMaple, solving the set of partial differential equations,we can obtain the following nontrivial results. (Note: in the rest of this paper, c1, Ci ði ¼ 1; . . . ; 5Þare arbitrary constants, F3ðyÞ is an arbitrary function with respect to y, and so on.)

Case 1.

a0 ¼ � 1

C1

Z �F3ðyÞ2C1

dy�

þ F6ðtÞ�; q ¼

Z �� F3ðyÞt þ F4ðyÞ

2C1

�dy þ F6ðtÞ;

A2 ¼ F3ðyÞt þ F4ðyÞ; A0 ¼1

2C21

½�F3ðyÞ þ 2C21RF3ðyÞt þ 2C2

1RF4ðyÞ � 2C21 �;

a1 ¼ 2C1; p ¼ C1; g1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ G2 ¼ 0:

ð3:2Þ

Case 2.

g1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ G2 ¼ 0; q ¼Z

� F1ðyÞt þ F2ðyÞ2C1

dy þ F5ðtÞ;

a0 ¼ � 1

C1

Z�� 1

2C1

F1ðyÞdy þ F5ðtÞ�; A2 ¼ F1ðyÞt þ F2ðyÞ; a1 ¼ �2C1;

p ¼ C1; A0 ¼F1ðyÞ þ 2C2

1RF1ðyÞt þ 2C21RF2ðyÞ � 2C2

1

2C21

¼ 0:

ð3:3Þ

Case 3.

g1 ¼ 2RC1; G2 ¼ F3ðyÞt þ F4ðyÞ; q ¼Z

� F3ðyÞt þ F4ðyÞ2R2C1

dy þ F6ðtÞ;

A2 ¼ a1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ 0; a0 ¼ � 1

C1

Z�� F3ðyÞ

2R2C1

dy þ F6ðtÞ�;

A0 ¼F3ðyÞ � 2R2C2

1 þ 2C21RF3ðyÞt þ 2C2

1RF4ðyÞ2R2C2

1

; p ¼ C1:

ð3:4Þ

Case 4.

A2 ¼ a1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ 0; a0 ¼ � 1

C1

Z�� F1ðyÞ

2R2C1

dy þ F5ðtÞ�;

q ¼Z

� F1ðyÞt þ F2ðyÞ2R2C1

dy þ F5ðtÞ; G2 ¼ F1ðyÞt þ F2ðyÞ; g1 ¼ �2RC1;

A0 ¼�F1ðyÞ � 2R2C2

1 þ 2C21RF1ðyÞt þ 2C2

1RF2ðyÞ2R2C2

1

; p ¼ C1:

ð3:5Þ


Case 5.

a1 ¼ 2C1; g1 ¼ 2RC1; p ¼ C1; A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ 0;

a0 ¼ � F4ðtÞC1

; q ¼Z

� F2ðyÞ2C1

dy þ F4ðtÞ; G2 ¼ F2ðyÞR2; A0 ¼ �1; A2 ¼ F2ðyÞ:ð3:6Þ

Case 6.

g1 ¼ �2RC1; p ¼ C1; A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ 0; a1 ¼ �2C1;

a0 ¼ � F3ðtÞC1

; A2 ¼ F1ðyÞ; G2 ¼ F1ðyÞR2; q ¼Z

� F1ðyÞ2C1

dy þ F3ðtÞ; A0 ¼ �1:ð3:7Þ

Case 7.

a0 ¼ � 1

C1

Z�� F3ðyÞ

2C1

dy þ F6ðtÞ�; q ¼

Z� F3ðyÞt þ F4ðyÞ

2C1

dy þ F6ðtÞ;

A2 ¼ F3ðyÞt þ F4ðyÞ; g1 ¼ 2RC1; p ¼ C1; A1 ¼ B1 ¼ G1 ¼ B2 ¼ b1 ¼ 0; a1 ¼ �2C1;

G2 ¼ R2F3ðyÞt þ R2F4ðyÞ; A0 ¼4C2

1RF3ðyÞt þ 4C21RF4ðyÞ � 2C2

1 þ F3ðyÞ2C2

1

:

ð3:8Þ

Case 8.

a1 ¼ 2C1; g1 ¼�2RC1; p¼C1; A1 ¼ B1 ¼G1 ¼ B2 ¼ b1 ¼ 0;

a0 ¼� 1

C1

Z�� F1ðyÞ

2C1

dyþ F5ðtÞ�; A2 ¼ F1ðyÞtþ F2ðyÞ; q¼

Z�ðF1ðyÞtþF2ðyÞÞ

2C1

dyþ F5ðtÞ;

G2 ¼ R2F1ðyÞtþ F2ðyÞR2; A0 ¼4C2

1RF1ðyÞtþ 4C21RF2ðyÞ� 2C2

1 � F1ðyÞ2C2

1

:

ð3:9Þ

Case 9.

a1 ¼ g1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ G2 ¼ 0; p ¼ C1; a0 ¼ � F4ðtÞC1

;

q ¼Z

� F2ðyÞ2C1

dy þ F4ðtÞ; A2 ¼ F2ðyÞ; b1 ¼ 2C1; A0 ¼ �1þ 1

2F2ðyÞR:

ð3:10Þ

Case 10.

a1 ¼ g1 ¼ A1 ¼ B1 ¼ G1 ¼ B2 ¼ G2 ¼ 0; p ¼ C1; a0 ¼ � F3ðtÞC1

;

A2 ¼ F1ðyÞ; q ¼Z

� F1ðyÞ2C1

dy þ F3ðtÞ; A0 ¼ �1þ 1

2F1ðyÞR; b1 ¼ �2C1:

ð3:11Þ


Case 11.

g1 ¼ A1 ¼ B1 ¼ G1 ¼ G2 ¼ 0; p ¼ C1; q ¼Z

� F11ðyÞt þ F12ðyÞC1

dy þ F20ðtÞ;

b1 ¼ C1; a1 ¼ C1; A2 ¼ F11ðyÞt þ F12ðyÞ; a0 ¼ � 1

C1

Z�� F11ðyÞ

C1

dy þ F20ðtÞ�;

A0 ¼C2

1RF11ðyÞt þ C21RF12ðyÞ � F11ðyÞ � C2

1

C21

; B2 ¼ F11ðyÞt þ F12ðyÞ:

ð3:12Þ

Case 12.

g1 ¼ A1 ¼ B1 ¼ G1 ¼ G2 ¼ 0; p ¼ C1; b1 ¼ C1; a0 ¼ � 1

C1

ZF5ðyÞC1

dy�

þ F19ðtÞ�;

a1 ¼ �C1; A0 ¼ �C21RF5ðyÞt þ C2

1RF6ðyÞ þ F5ðyÞ þ C21

C21

; B2 ¼ F5ðyÞt þ F6ðyÞ;

A2 ¼ �F5ðyÞt � F6ðyÞ; q ¼Z

F5ðyÞt þ F6ðyÞC1

dy þ F19ðtÞ:

ð3:13Þ

Case 13.

g1 ¼ A1 ¼ B1 ¼ G1 ¼ G2 ¼ 0; p ¼ C1; a1 ¼ C1; a0 ¼ � 1

C1

ZF3ðyÞC1

dy�

þ F18ðtÞ�;

A0 ¼ �C21RF3ðyÞt þ C2

1RF4ðyÞ � F3ðyÞ þ C21

C21

; b1 ¼ �C1; A2 ¼ �F3ðyÞt � F4ðyÞ;

q ¼Z

F3ðyÞt þ F4ðyÞC1

dy þ F18ðtÞ; B2 ¼ F3ðyÞt þ F4ðyÞ:

ð3:14Þ

Case 14.

g1 ¼ A1 ¼ B1 ¼ G1 ¼ G2 ¼ 0; p ¼ C1; A2 ¼ F1ðyÞt þ F2ðyÞ; b1 ¼ �C1; a1 ¼ �C1;

a0 ¼ � 1

C1

Z�� F1ðyÞ

C1

dy þ F17ðtÞ�; q ¼

Z �ðF1ðyÞt þ F2ðyÞÞC1

dy þ F17ðtÞ;

A0 ¼C2

1RF1ðyÞt þ C21RF2ðyÞ þ F1ðyÞ � C2

1

C21

; B2 ¼ F1ðyÞt þ F2ðyÞ:

ð3:15Þ

From (3.1), (2.4) and (3.2)–(3.15), we can obtain the following solutions for the (2+ 1)-dimen-sional dispersive long-wave equation.


Type 1. From Case 1–2, we can obtain the following solutions

u11 ¼ a0 � a1ffiffiffiffiffiffiffi�R

ptanh


pðxp þ qÞ

� ;

g11 ¼ A0 þ A2R tanh2ffiffiffiffiffiffiffi�R

pðxp þ qÞ

� ; R < 0

(ð3:16Þ


pcoth


pðxp þ qÞ

� ;

g12 ¼ A0 þ A2R coth2ffiffiffiffiffiffiffi�R

pðxp þ qÞ

� ; R < 0

(ð3:17Þ

u13 ¼ a0 þ a1ffiffiffiR

ptan

ffiffiffiR

pðxp þ qÞ

� ;

g13 ¼ A0 þ A2R tan2ffiffiffiR

pðxp þ qÞ

� ; R > 0

(ð3:18Þ

u14 ¼ a0 � a1ffiffiffiR

pcot

ffiffiffiR

pðxp þ qÞ

� ;

g14 ¼ A0 þ A2R cot2ffiffiffiR

pðxp þ qÞ

� ; R > 0

(ð3:19Þ

where a0, a1, p, q, A0 and A2 are determined by (3.2) and (3.3), respectively. At the same time, dueto the arbitrariness of functions FiðyÞ ði ¼ 1; . . . ; 6Þ, F5ðtÞ, F6ðtÞ, the solutions obtained by Case 3–4are just the same as the solutions (3.16)–(3.19) .

Type 2. From Case 5–8, we can obtain the following solutions


ptanh


pðxp þ qÞ

� � coth


pðxp þ qÞ

� � ;


pðxp þ qÞ

� þ coth2½�Rðxp þ qÞ�

� ; R < 0

(ð3:20Þ


ptan

ffiffiffiR

pðxp þ qÞ

� � cot

ffiffiffiR

pðxp þ qÞ

� � ;

g22 ¼ A1 þ A2ÞR tan2ffiffiffiR

pðxp þ qÞ

� þ cot2½Rðxp þ qÞ�

� ; R > 0

(ð3:21Þ

where a0, A2, p, q are determined by (3.6)–(3.9), respectively.Type 3. From Cases 9–10, we can obtain the following solutions

u31 ¼ a0 þ b1ffiffiffiR

psech


pðxp þ qÞ

� ;


pðxp þ qÞ

� ; R < 0

(ð3:22Þ

u32 ¼ a0 þ b1ffiffiffiffiffiffiffi�R

pcsch


pðxp þ qÞ

� ;


pðxp þ qÞ

� ; R < 0

(ð3:23Þ

u33 ¼ a0 þ b1ffiffiffiR

psec

ffiffiffiR

pðxp þ qÞ

� ;


pðxp þ qÞ

� ; R > 0

(ð3:24Þ

u34 ¼ a0 � b1ffiffiffiR

pcsc

ffiffiffiR

pðxp þ qÞ

� ;


pðxp þ qÞ

� ; R < 0

(ð3:25Þ

where a0, b1, A0, A2, p and q are determined by (3.10), (3.11), respectively.


Type 4. From Cases 11–14, we can obtain the following solutions


ptanh


pn

� �þ b1

ffiffiffiR

psech


pn

� �;


pn

� �þ iB2R tanh


pn

� �sech


pn

� �; R < 0

(ð3:26Þ


pcoth


pn

� �þ b1


pcsch


pn

� �;


pn

� �þ B2R coth


pn

� �csch


pn

� �; R < 0

(ð3:27Þ


ptan

ffiffiffiR

pn

� �þ b1

ffiffiffiR

psec

ffiffiffiR

pn

� �;


pn

� �þ B2R tan

ffiffiffiR

pn

� �sec

ffiffiffiR

pn

� �; R > 0

(ð3:28Þ

u43 ¼ a0 � a1ffiffiffiR

pcot

ffiffiffiR

pn

� �þ b1

ffiffiffiR

pcsc

ffiffiffiR

pn

� �;


pn

� �þ B2R cot

ffiffiffiR

pn

� �csc

ffiffiffiR

pn

� �; R > 0

(ð3:29Þ

where n ¼ xp þ q and a0, a1, b1, A0, A2, B2, p, q are determined by (3.12)–(3.15), respectively.

4. Conclusions

In summary, based on the computerized symbolic computation and a Riccati equation, byintroducing a new more general ans€atz than the ans€atz in the extended tanh-function method,modified extended tanh-function method, and generalized hyperbolic-function method, we haveproposed the generalized Riccati equation expansion method for searching for exact solutions ofNEEs and implemented in computer symbolic systems. Making use of our method and with theaid of Maple, we study the (2+ 1)-dimensional dispersive long-wave equation and obtain somenew families of the exact solutions. In our obtained exact solutions the restriction on nðx; y; tÞ asmerely a linear function x, y, t and the restriction on the coefficients, such as a0, ai, bi, giði ¼ 1; . . . ;mÞ, etc., as constants are removed and, with no extra effort, the singular solitonicsolution and triangular function solutions, even rational formal solutions could be obtained. Tomake the work feasible, how to choose the forms for a0, ai, bi, gi ði ¼ 1; . . . ;mÞ, n, etc., in theans€atz would be the key step in the computation of our method. It is shown that the method,proposed in this paper for a system of NEEs, may be extended to find exact solutions of othermathematical and physical equation(s).

Acknowledgements

We would express our sincere thanks to the Referee for his valuable advice and corrections tothe original version. The work is supported by the National Natural Science Foundation of Chinaunder the Grant No. 10072013, the National Key Basic Research Development Project Programunder the Grant No. G998030600.


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symbolic computation and construction of soliton-like solutions to the (2 + 1)-dimensional...

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