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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq1_HTML.gif which are given by [5]
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ1_HTML.gif
(1.1)

The case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq2_HTML.gif is called the DSI equation, while http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq3_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq4_HTML.gif is the DSII equation. The parameter http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq8_HTML.gif , auxiliary ordinary equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq9_HTML.gif [7], first kind elliptic equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq10_HTML.gif , generalized Riccati equation http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq12_HTML.gif and dependent variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq13_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ3_HTML.gif
(2.1)
where http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ4_HTML.gif
(2.2)
where http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ5_HTML.gif
(2.3)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq16_HTML.gif is expanded into a polynomial in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq17_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ6_HTML.gif
(2.4)

where http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq22_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq23_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq24_HTML.gif ;   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq25_HTML.gif = http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq27_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq28_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq30_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq32_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq33_HTML.gif [7, 8, 10]. The superscripts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq34_HTML.gif , and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq37_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq38_HTML.gif , there may exist three parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq39_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq40_HTML.gif such that
http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq42_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ9_HTML.gif
(2.7)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq44_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ10_HTML.gif
(2.8)

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

Type 2.

When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq48_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ11_HTML.gif
(2.)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq50_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ12_HTML.gif
(2.10)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq51_HTML.gif denotes http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq54_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq55_HTML.gif . There may exist three parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq56_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq57_HTML.gif such that
http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq58_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq59_HTML.gif parameters:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq61_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ16_HTML.gif
(2.14)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq63_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ17_HTML.gif
(2.15)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq64_HTML.gif denotes http://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 http://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]
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq67_HTML.gif (2.16) has the solution
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ19_HTML.gif
(2.17)

Type 2.

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

Type 3.

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

Type 4.

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

Type 5.

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

Type 6.

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

Type 7.

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

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

Type 8.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq76_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ26_HTML.gif
(2.24)
When http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ28_HTML.gif
(2.26)

where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq80_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq83_HTML.gif of the Jacobi elliptic function satisfies ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq86_HTML.gif (see Table 3).

Table 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Case 5.

When http://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:
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ30_HTML.gif
(2.28)

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

Case 6.

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

Case 7.

when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq228_HTML.gif , the general elliptic equation have solution
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ33_HTML.gif
(3.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq229_HTML.gif are real functions, and the constants http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq231_HTML.gif and http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ34_HTML.gif
(3.2)
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq233_HTML.gif .

Integrating (3.3) w.r.t. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq234_HTML.gif and solving for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq235_HTML.gif we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ36_HTML.gif
(3.4)
where http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ37_HTML.gif
(3.5)
where
http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq237_HTML.gif ) with nonlinear term ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq238_HTML.gif ) in (3.5) gives leading http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ39_HTML.gif
(3.7)
where http://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 http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq242_HTML.gif , we obtain the following solutions:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ41_HTML.gif
(3.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq243_HTML.gif are arbitrary real constants, and http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ42_HTML.gif
(3.10)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ43_HTML.gif
(3.11)

and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq247_HTML.gif

Case 1.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq248_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq249_HTML.gif is one of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq250_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq251_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq252_HTML.gif that are real and http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ44_HTML.gif
(3.12)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ45_HTML.gif
(3.13)

Case 2.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq255_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq256_HTML.gif is one of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq257_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq258_HTML.gif contain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq259_HTML.gif are real and http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ46_HTML.gif
(3.14)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ47_HTML.gif
(3.15)

Case 3.

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

Type 1.

If http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ48_HTML.gif
(3.16)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ49_HTML.gif
(3.17)

Type 2.

If http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ50_HTML.gif
(3.18)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ51_HTML.gif
(3.19)

Case 4.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq268_HTML.gif are arbitrary constants, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq269_HTML.gif is one of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq270_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq271_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq272_HTML.gif that are real and one http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq274_HTML.gif , then http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ52_HTML.gif
(3.20)
where
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ54_HTML.gif
(3.22)
When http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ55_HTML.gif
(3.23)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ56_HTML.gif
(3.24)
If we select http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq278_HTML.gif , then http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ57_HTML.gif
(3.25)
where
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ59_HTML.gif
(3.27)
When http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ60_HTML.gif
(3.28)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ61_HTML.gif
(3.29)

Case 5.

If http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ62_HTML.gif
(3.30)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ63_HTML.gif
(3.31)

Case 7.

If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq285_HTML.gif in three variables http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq286_HTML.gif ,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ65_HTML.gif
(4.2)
where http://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
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ67_HTML.gif
(4.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq288_HTML.gif http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq290_HTML.gif satisfy the Riccati equation.

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

where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq293_HTML.gif http://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 http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ69_HTML.gif
(4.6)

are the following.

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

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

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

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq309_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq310_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq311_HTML.gif , and http://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,
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ77_HTML.gif
(5.2)
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq313_HTML.gif .

Integrating (5.3) w.r.t. http://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
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ80_HTML.gif
(5.5)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq316_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq317_HTML.gif setting the coefficients of these terms http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq319_HTML.gif , and http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ81_HTML.gif
(5.6)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq321_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq322_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq323_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq324_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq325_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq326_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq327_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq328_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq329_HTML.gif , http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ83_HTML.gif
(5.8)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_IEq331_HTML.gif and http://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
http://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
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F915721/MediaObjects/13661_2010_Article_966_Equ87_HTML.gif
(5.12)

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