Open Access

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

Boundary Value Problems20102010:915721

DOI: 10.1155/2010/915721

Received: 13 January 2010

Accepted: 26 March 2010

Published: 6 May 2010

Abstract

We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

1. Introduction

The investigation of the exact travelling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. These exact solutions when they exist can help the physicists to well understand the mechanism of the complicated physical phenomena and dynamically processes modeled by these NLPDEs. In recent years, large amounts of effort have been directed towards finding exact solutions. Many powerful method have been proposed, such as Darboux transformation [1], Hirota bilinear method [2], Lie group method [3], homogeneous balance method [4], tanh method. In this paper, we construct the exact travelling wave solutions for the Davey-Stewartson (DS) equations for the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq1_HTML.gif which are given by [5]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ1_HTML.gif
(1.1)

The case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq2_HTML.gif is called the DSI equation, while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq3_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq4_HTML.gif is the DSII equation. The parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq5_HTML.gif characterizes the focusing or defocusing case. The Davey-Stewartson I and II are two well-known examples of integrable equations in two-dimensional space, which arise as higher-dimensional generalizations of the nonlinear shrodinger (NLS) equation, from the point of physical view as well as from the study in [6]. Indeed, they appear in many applications, for example, in the description of gravity-capillarity surface wave packets and in the limit of the shallow water.

Davey and Stewartson first derived their model in the context of water waves, just purely physical considerations. In the context, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq6_HTML.gif is the amplitude of a surface wave packet, while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq7_HTML.gif is the velocity potential of the mean flow interacting with the surface wave [6].

The extended tanh-function method, the modified extended tanh-function method, and the F-expansion method belong to a class of methods called subequation methods for which they appear some basic relationships among the complicated NLPDEs under study and some simple solvable nonlinear ordinary equations. Thus by these subequation methods we seek for the solutions of the nonlinear partial differential equations in consideration as a polynomial in one variable satisfying equations (named subequation), for example, Riccati equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq8_HTML.gif , auxiliary ordinary equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq9_HTML.gif [7], first kind elliptic equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq10_HTML.gif , generalized Riccati equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq11_HTML.gif [8], and so on. Fan [9] developed a new algebraic method, belonging to the class of subequation methods, to seek for more new solutions of nonlinear partial differential equations that can be expressed as a polynomial in an elementary function which satisfies a more general subequation than other subequations like Riccati equation, first-kind elliptic equation, and generalized Riccati equation. Recently Yomba [10] and Soliman and Abdou [11] extended Fan's method to show that the general elliptic equation can be degenerated in some special conditions to Riccati equation, first-kind elliptic equation, and generalized Riccati equation. We will consider a general elliptic equation in the formal will through
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ2_HTML.gif
(1.2)

In addition, we apply a new compound Riccati equations rational expansion method [12] to the Davey-Stewartson (DS) equations and construct new complexion solutions. The rest of this paper is organized as follows. In Section 2, we simply provide the mathematical framework of Fan's subequation method. In Section 3, we apply the new presented method to the Davey-Stewartson (DS) equations. In Section 4, we briefly describe the new CRERE method. In Section 5, we obtain new complexion solutions of the Davey-Stewartson (DS) equations. In Section 6 and finally, some conclusions are given.

2. The Extended Fan Subequation Method

In the following we shall outline the main steps of our method.

For given nonlinear partial differential equations with independent variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq12_HTML.gif and dependent variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq13_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ3_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq14_HTML.gif is in general a polynomial function of its argument, and the subscripts denote the partial derivatives. We first consider its travelling wave solutions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ4_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq15_HTML.gif are all arbitrary constants. Substituting (2.2) into (2.1), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ5_HTML.gif
(2.3)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq16_HTML.gif is expanded into a polynomial in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq17_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ6_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq18_HTML.gif are constants to be determined later and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq19_HTML.gif satisfies (1.2). In order to determine https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq20_HTML.gif explicitly, one may take the following steps.

Step 1.

Determine https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq21_HTML.gif by balancing the linear term of the highest order with the nonlinear term in (2.3).

Step 2.

Substituting (2.4) with (1.2) into (2.3) and collecting all coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq22_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq23_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq24_HTML.gif ;   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq25_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq26_HTML.gif then, setting these coefficients, to zero we get a set of algebraic equations with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq27_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq29_HTML.gif

Step 3.

Solve the system of algebraic equations to obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq31_HTML.gif . Inserting these results into (2.4), we thus obtained the general form of travelling wave solutions.

Step 4.

By using the results obtained in the above steps, we can derive a series of fundamental solutions to (1.2) depending on the different values chosen for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq32_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq33_HTML.gif [7, 8, 10]. The superscripts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq34_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq35_HTML.gif determine the group of the solution while the subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq36_HTML.gif determines the rank of the solution. Those solutions are listed as follows.

Case 1.

In some special cases, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq37_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq38_HTML.gif , there may exist three parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq39_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq40_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ7_HTML.gif
(2.5)
Equation (2.5) is satisfied only if the following relations hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ8_HTML.gif
(2.6)

For example, if the conditions (1.2)–(2.5) are satisfied, the following solutions are obtained [8].

Type 1.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq42_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ9_HTML.gif
(2.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq43_HTML.gif are two nonzero real constants and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq44_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ10_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq45_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq46_HTML.gif .

Type 2.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq48_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ11_HTML.gif
(2.)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq49_HTML.gif are two nonzero real constants and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq50_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ12_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq51_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq52_HTML.gif .

Case 2.

Case 1 includes another special case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq54_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq55_HTML.gif . There may exist three parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq56_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq57_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ13_HTML.gif
(2.11)
Equation (2.2) requires for its existence the following relations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ14_HTML.gif
(2.12)
The following constraint should exist between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq58_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq59_HTML.gif parameters:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ15_HTML.gif
(2.13)

For example, if the conditions (1.2), (2.5), (2.12), and (2.13) are satisfied, the following solutions are obtained.

Type 1.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq61_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ16_HTML.gif
(2.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq62_HTML.gif are two nonzero real constants and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq63_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ17_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq64_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq65_HTML.gif . Thus for (2.5), (2.11), the general elliptic equation is reduced to the generalized Riccati Equation [8].

Case 3.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq66_HTML.gif , the general elliptic equation is reduced to the auxiliary ordinary equation [7]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ18_HTML.gif
(2.16)

For example, if the condition (2.16) is satisfied, the following solutions are obtained.

Type 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq67_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ19_HTML.gif
(2.17)

Type 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq68_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ20_HTML.gif
(2.18)

Type 3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq69_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ21_HTML.gif
(2.19)

Type 4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq70_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ22_HTML.gif
(2.20)

Type 5.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq71_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ23_HTML.gif
(2.21)

Type 6.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq72_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ24_HTML.gif
(2.22)

Type 7.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq73_HTML.gif (2.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ25_HTML.gif
(2.23)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq74_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq75_HTML.gif are arbitrary constants.

Type 8.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq76_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ26_HTML.gif
(2.24)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq77_HTML.gif , the general elliptic equation is reduced to the elliptic equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ27_HTML.gif
(2.25)
Equation (2.25) includes the Riccati equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ28_HTML.gif
(2.26)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq78_HTML.gif , and solutions of (2.26) can be deduced from those of (2.25) in the specific case where the modulus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq79_HTML.gif of the Jacobi elliptic functions is drived to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq81_HTML.gif .

Case 4.

Assume that the conditions of verification of (2.26) are fulfilled, then the general solutions are just the single solution and the combined nondegenerative Jacobi elliptic functions. The relations between the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq82_HTML.gif and the corresponding Jacobi elliptic function solution of the NODE (2.25) are given in Table 1.

where the modulus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq83_HTML.gif of the Jacobi elliptic function satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq84_HTML.gif

The Jacobi elliptic function degenerates as hyperbolic functions when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq85_HTML.gif (see Table 2).

The Jacobi elliptic function degenerates as hyperbolic functions when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq86_HTML.gif (see Table 3).

Table 1

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq87_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq88_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq89_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq90_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq91_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq92_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq93_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq94_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq95_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq96_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq97_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq98_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq99_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq100_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq101_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq102_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq103_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq104_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq105_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq106_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq107_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq108_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq109_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq110_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq111_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq112_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq113_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq114_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq115_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq116_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq117_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq118_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq119_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq120_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq121_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq122_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq123_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq124_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq125_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq126_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq127_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq128_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq129_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq130_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq131_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq132_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq133_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq134_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq135_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq136_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq137_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq138_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq139_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq140_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq141_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq143_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq144_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq145_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq146_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq147_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq148_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq149_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq150_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq151_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq152_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq153_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq154_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq155_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq156_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq157_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq158_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq159_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq160_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq161_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq162_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq163_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq164_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq165_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq166_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq167_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq168_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq169_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq170_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq172_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq173_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq174_HTML.gif

Table 2

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq175_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq176_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq177_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq178_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq179_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq180_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq181_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq182_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq183_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq184_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq185_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq186_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq187_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq188_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq189_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq190_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq191_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq192_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq193_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq194_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq195_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq196_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq197_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq198_HTML.gif

Table 3

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq199_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq200_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq201_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq202_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq203_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq204_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq205_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq206_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq207_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq208_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq209_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq210_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq211_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq212_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq213_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq214_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq215_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq216_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq217_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq218_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq219_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq220_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq221_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq222_HTML.gif

Case 5.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq223_HTML.gif , the general elliptic equation is reduced to the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ29_HTML.gif
(2.27)
For example, if the condition (2.27) holds, the solution is of Weierstrass elliptic doubly periodic type
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ30_HTML.gif
(2.28)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq224_HTML.gif

Case 6.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq225_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq226_HTML.gif the general elliptic equation admits solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ31_HTML.gif
(2.29)

Case 7.

when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq228_HTML.gif , the general elliptic equation have solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ32_HTML.gif
(2.30)

3. Exact Solutions of the Davey-Stewartson (DS) Equations

Now, we will construct the exact solutions to (DS) equations (1.1). Let us assume the travelling wave solutions of (1.1) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ33_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq229_HTML.gif are real functions, and the constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq230_HTML.gif are real which can be determined later. Substituting (3.1) into (1.1), we find that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq232_HTML.gif satisfy the following coupled ordinary differential system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ34_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ35_HTML.gif
(3.3)

where "the prime" denotes to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq233_HTML.gif .

Integrating (3.3) w.r.t. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq234_HTML.gif and solving for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq235_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ36_HTML.gif
(3.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq236_HTML.gif is an integration constant. Substituting (3.4) into (3.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ37_HTML.gif
(3.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ38_HTML.gif
(3.6)
Balancing the highest-order derivative term ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq237_HTML.gif ) with nonlinear term ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq238_HTML.gif ) in (3.5) gives leading https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq239_HTML.gif We thus suppose that (3.5) has the following formal solutions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ39_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq240_HTML.gif are to be determined later; substituting (3.7) along with (1.2) into (3.5) yields a polynomial equation in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq241_HTML.gif . Setting to zero their coefficients yields the following set of algebraic equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ40_HTML.gif
(3.8)
Substituting (3.6) into (3.8) and solving with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq242_HTML.gif , we obtain the following solutions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ41_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq243_HTML.gif are arbitrary real constants, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq244_HTML.gif are real constants so we choose the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq245_HTML.gif .

The exact travelling wave solutions of the DSII equations (1.1) are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ42_HTML.gif
(3.10)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ43_HTML.gif
(3.11)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq246_HTML.gif are arbitrary constants. We may have many kinds of solutions depending on the special values for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq247_HTML.gif

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq248_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq249_HTML.gif is one of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq250_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq251_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq252_HTML.gif that are real and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq253_HTML.gif that are complex. For example, if we select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq254_HTML.gif then one could write down explicitly the following soliton solutions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ44_HTML.gif
(3.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ45_HTML.gif
(3.13)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq255_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq256_HTML.gif is one of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq257_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq258_HTML.gif contain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq259_HTML.gif are real and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq260_HTML.gif are complex. For example, if we select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq261_HTML.gif then one may write down explicitly the following soliton solutions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ46_HTML.gif
(3.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ47_HTML.gif
(3.15)

Case 3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq262_HTML.gif are arbitrary constants, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq263_HTML.gif is one of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq264_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq265_HTML.gif as follows.

Type 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq266_HTML.gif then the travelling wave solutions are given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ48_HTML.gif
(3.16)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ49_HTML.gif
(3.17)

Type 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq267_HTML.gif then the travelling wave solutions are given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ50_HTML.gif
(3.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ51_HTML.gif
(3.19)

Case 4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq268_HTML.gif are arbitrary constants, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq269_HTML.gif is one of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq270_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq271_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq272_HTML.gif that are real and one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq273_HTML.gif that is complex. For example, if we select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq274_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq275_HTML.gif , and the travelling wave solutions are rediscovered:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ52_HTML.gif
(3.20)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ53_HTML.gif
(3.21)
In the limit case, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq276_HTML.gif then (3.20) admits the soliton wave solutions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ54_HTML.gif
(3.22)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq277_HTML.gif , then (3.20) admits the soliton wave solutions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ55_HTML.gif
(3.23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ56_HTML.gif
(3.24)
If we select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq278_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq279_HTML.gif , and the travelling wave solutions are rediscovered:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ57_HTML.gif
(3.25)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ58_HTML.gif
(3.26)
In the limit case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq280_HTML.gif then (3.25) admits the soliton wave solutions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ59_HTML.gif
(3.27)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq281_HTML.gif , then (3.25) admits the soliton wave solutions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ60_HTML.gif
(3.28)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ61_HTML.gif
(3.29)

Case 5.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq282_HTML.gif are arbitrary constants. The system does not admit solutions of this group.

Case 6.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq283_HTML.gif , the travelling wave solutions of (1.1) are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ62_HTML.gif
(3.30)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ63_HTML.gif
(3.31)

Case 7.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq284_HTML.gif is arbitrary constant. The system does not admit solutions of this group.

4. Summary of the New Compound Riccati Equations Rational Expansion Method

The key steps of our method are as follows.

Step 1.

For a given NLPDEs with some physical fields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq285_HTML.gif in three variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq286_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ64_HTML.gif
(4.1)
by using the wave transformation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ65_HTML.gif
(4.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq287_HTML.gif are constants to be determined later. Then (4.1) is reduced to an ordinary differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ66_HTML.gif
(4.3)

Step 2.

We introduce a solution of (4.3) in terms of finite rational formal expansion in the following forms:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ67_HTML.gif
(4.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq288_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq289_HTML.gif are constants to be determined later and the new variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq290_HTML.gif satisfy the Riccati equation.

That is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ68_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq291_HTML.gif are arbitrary constants.

Step 3.

The parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq292_HTML.gif can be found by balancing the highest nonlinear terms and the highest-order partial derivative terms in (4.1) or (4.3).

Step 4.

Substitute (4.3), (4.4) with (4.5) and then set all coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq293_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq294_HTML.gif of the resulting system's numerator to zero to get an overdetermined system of nonlinear algebraic system with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq295_HTML.gif

Step 5.

Solving the overdetermined system of nonlinear algebraic equations by use of Maple or Mathematica software, we would end up with the explicit expressions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq296_HTML.gif

Step 6.

It is well known that the general solutions of the Riccati equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ69_HTML.gif
(4.6)

are the following.

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq298_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ70_HTML.gif
(4.7)
() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq300_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ71_HTML.gif
(4.8)
() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq302_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ72_HTML.gif
(4.9)

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq304_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ73_HTML.gif
(4.10)
() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq306_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ74_HTML.gif
(4.11)
() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq308_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ75_HTML.gif
(4.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq309_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq310_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq311_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq312_HTML.gif is arbitrary constant.

5. Application of the New Compound Riccati Equations Rational Expansion Method to the Davey-Stewartson (DS) Equations

By considering the wave transformations,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ76_HTML.gif
(5.1)
Equation (1.1) reduces to the following ordinary differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ77_HTML.gif
(5.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ78_HTML.gif
(5.3)

where "the prime" denotes to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq313_HTML.gif .

Integrating (5.3) w.r.t. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq314_HTML.gif and setting the constant of integration to zero, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ79_HTML.gif
(5.4)
By balancing the highest nonlinear terms and the highest-Order partial derivative terms in (5.2) and (5.4), we suppose that (5.2) and (5.4) have the following solutions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ80_HTML.gif
(5.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq315_HTML.gif satisfiy (4.5), with the aid of Mathematica software; substituting (5.5) along with (4.5) into (5.2) and (5.4) yields a set of algebraic equations for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq316_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq317_HTML.gif setting the coefficients of these terms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq318_HTML.gif to zero yields a set of overdetermined algebraic equations with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq319_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq320_HTML.gif

By using the Maple software to solving the overdetermined algebraic equations, we get the following results:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ81_HTML.gif
(5.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq321_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq322_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq323_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq324_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq325_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq326_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq327_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq328_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq329_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq330_HTML.gif are arbitrary constants.

So we obtain the following solutions of (1.1).

Family 1

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ82_HTML.gif
(5.7)

Family 2

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ83_HTML.gif
(5.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq331_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq332_HTML.gif are arbitrary constants.

Family 3

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ84_HTML.gif
(5.9)

Family 4

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ85_HTML.gif
(5.10)

Family 5

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ86_HTML.gif
(5.11)

Family 6

Consider the following
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ87_HTML.gif
(5.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq333_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq334_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq335_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq336_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq337_HTML.gif are arbitrary constants.

6. Conclusion

In this paper, we have used the extended Fan's subequation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions and obtain many explicit solutions for the Davey-Stewartson equations.

We deduced a relation between the general elliptic equation involving five parameters and other subequations involving three parameter, like Riccati equation, auxiliary ordinary equation, first-kind elliptic equation, and generalized Riccati equation; many exact travelling wave solutions and new complexion solutions including more general soliton solutions, triangular solutions, double-periodic solutions, hyperbolic function solutions, and trigonometric function solutions are also given.

Authors’ Affiliations

(1)
Mathematics Department, Faculty of Science, Kafr El-Sheikh University
(2)
Mathematics Department, Faculty of Science, King Abdul Aziz University

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Copyright

© Hassan Zedan. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.