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

Hassan Zedan

doi:10.1155/2010/915721

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.

Highlights

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
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

Summary

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. 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 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 φ A φ2, auxiliary ordinary equation φ 2 Bφ2 Cφ3 Dφ4 7 , first kind elliptic equation φ 2 A Bφ2 Dφ4, generalized Riccati equation φ r pφ qφ2 8 , and so on.

The Extended Fan Subequation Method

Summary of the New Compound Riccati Equations Rational Expansion Method

Conclusion