Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations
© Hassan Zedan. 2010
Received: 13 January 2010
Accepted: 26 March 2010
Published: 6 May 2010
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.
The case is called the DSI equation, while is the DSII equation. The parameter 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 . 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, is the amplitude of a surface wave packet, while is the velocity potential of the mean flow interacting with the surface wave .
In addition, we apply a new compound Riccati equations rational expansion method  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.
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 , and [7, 8, 10]. The superscripts , and determine the group of the solution while the subscript determines the rank of the solution. Those solutions are listed as follows.
For example, if the conditions (1.2)–(2.5) are satisfied, the following solutions are obtained .
For example, if the conditions (1.2), (2.5), (2.12), and (2.13) are satisfied, the following solutions are obtained.
where denotes . Thus for (2.5), (2.11), the general elliptic equation is reduced to the generalized Riccati Equation .
For example, if the condition (2.16) is satisfied, the following solutions are obtained.
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 and the corresponding Jacobi elliptic function solution of the NODE (2.25) are given in Table 1.
The Jacobi elliptic function degenerates as hyperbolic functions when (see Table 2).
3. Exact Solutions of the Davey-Stewartson (DS) Equations
4. Summary of the New Compound Riccati Equations Rational Expansion Method
The key steps of our method are as follows.
are the following.
5. Application of the New Compound Riccati Equations Rational Expansion Method to the Davey-Stewartson (DS) Equations
where 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 setting the coefficients of these terms to zero yields a set of overdetermined algebraic equations with respect to , and
So we obtain the following solutions of (1.1).
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.
- Fan EG: Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. Journal of Physics A 2000,33(39):6925-6933. 10.1088/0305-4470/33/39/308MATHMathSciNetView ArticleGoogle Scholar
- Tam H-W, Ma W-X, Hu X-B, Wang D-L: The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited. Journal of the Physical Society of Japan 2000,69(1):45-52. 10.1143/JPSJ.69.45MATHMathSciNetView ArticleGoogle Scholar
- Zayed EME, Zedan HA, Gepreel KA: Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations. International Journal of Nonlinear Sciences and Numerical Simulation 2004, 5: 221-234. 10.1515/IJNSNS.2004.5.3.221MathSciNetView ArticleGoogle Scholar
- Lei Y, Fajiang Z, Yinghai W: The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation. Chaos, Solitons and Fractals 2002,13(2):337-340. 10.1016/S0960-0779(00)00274-5MATHMathSciNetView ArticleGoogle Scholar
- McConnell M, Fokas AS, Pelloni B: Localised coherent solutions of the DSI and DSII equations—a numerical study. Mathematics and Computers in Simulation 2005,69(5-6):424-438. 10.1016/j.matcom.2005.03.007MATHMathSciNetView ArticleGoogle Scholar
- Davey A, Stewartson K: On three-dimensional packets of surface waves. Proceedings of the Royal Society A 1974, 338: 101-110. 10.1098/rspa.1974.0076MATHMathSciNetView ArticleGoogle Scholar
- Sirendaoreji , Jiong S: Auxiliary equation method for solving nonlinear partial differential equations. Physics Letters A 2003,309(5-6):387-396. 10.1016/S0375-9601(03)00196-8MATHMathSciNetView ArticleGoogle Scholar
- Xie F, Zhang Y, Lü Z:Symbolic computation in non-linear evolution equation: application to -dimensional Kadomtsev-Petviashvili equation. Chaos, Solitons and Fractals 2005,24(1):257-263.MATHMathSciNetView ArticleGoogle Scholar
- Fan EG: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos, Solitons and Fractals 2003,16(5):819-839. 10.1016/S0960-0779(02)00472-1MATHMathSciNetView ArticleGoogle Scholar
- Yomba E: The extended Fan's sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations. Physics Letters A 2005,336(6):463-476. 10.1016/j.physleta.2005.01.027MATHMathSciNetView ArticleGoogle Scholar
- Soliman AA, Abdou MA: Exact travelling wave solutions of nonlinear partial differential equations. Chaos, Solitons and Fractals 2007,32(2):808-815. 10.1016/j.chaos.2005.11.053MATHMathSciNetView ArticleGoogle Scholar
- Song L-N, Wang Q, Zhang H-Q:A new compound Riccati equations rational expansion method and its application to the -dimensional asymmetric Nizhnik-Novikov-Vesselov system. Chaos, Solitons and Fractals 2008,36(5):1348-1356. 10.1016/j.chaos.2006.09.001MATHMathSciNetView ArticleGoogle Scholar
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.