Exact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized $(\frac{G'}{G})$-Expansion Method

Mahmoud A M Abdelaziz,A E Moussa,D M Alrahal

doi:10.5539/jmr.v6n2p91

Abstract

In this article, we construct the exact traveling wave solutions of the nonlinear (2+1)-dimensional Davey-Stewartson equation (D-S) using the generalized $(\frac{G'}{G})$-expansion method which play an important role in mathematical physics. As a result, hyperbolic, trigonometric and rational function solutions with parameters are obtained. When these parameters are taken special values, the solitary and periodic solutions are derived from the hyperbolic and trigonometric function solutions respectively. New complex type traveling wave solutions to the nonlinear (2+1)-dimensional Davey-Stewartson equation were obtained with Liu's theorem.

Highlights

Many important phenomena in various fields can be described by the nonlinear partial differential equations (NLPDEs)
)-expansion method and they demonstrated that it was a powerful technique for seeking analytic solutions of NLPDEs
The Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear Schrodinger equation (NLSE)

Summary

Introduction

Many important phenomena in various fields can be described by the nonlinear partial differential equations (NLPDEs). Searching and constructing exact solutions for NLPDEs is interesting and important. Zhang et al (2008) first proposed this method to construct exact solutions of the mKdV equation with variable coefficients. The Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear Schrodinger equation (NLSE). They appear in many applications, for example in the description of gravity-capillarity surface wave packets in the limit of the shallow water. Many powerful methods have been established and developed to obtain analytic solutions of Equation (2), such as the homotopy analysis method, the sine-cosine method and the variational iteration method (Davey et al, 1974; Zedan et al, 2010; Jafari et al, 2012)

Method

Conclusions