Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

J F Alzaidy

doi:10.1155/2012/597983

Abstract

We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Highlights

The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena
The main objective of this paper is using the extended mapping method to construct the exact solutions for nonlinear evolution equations in the mathematical physics via the Boussinesq system and the coupled KdV equations
Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case 4 to illustrate the effectiveness of the extended mapping method

Summary

Introduction

The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. The nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modeled by the bellshaped sech solutions and the kink-shaped tanh solutions. Many effective methods have been presented, such as inverse scattering transform method 1 , Backlund transformation 2 , Darboux transformation 3 , Hirota bilinear method 4 , variable separation approach 5 , various tanh methods 6–9 , homogeneous balance method 10 , similarity reductions method 11, , G /G -expansion method , the reduction mKdV equation method , the trifunction method 15, , the projective Riccati equation method , the Weierstrass elliptic function method , the Sine-Cosine method 19, 20 , the Jacobi elliptic function expansion 21, , the complex hyperbolic function method , the truncated Painleveexpansion , the F-expansion method , the rank analysis method , the ansatz method 27, , the exp-function expansion method , and the sub-ODE method. The main objective of this paper is using the extended mapping method to construct the exact solutions for nonlinear evolution equations in the mathematical physics via the Boussinesq system and the coupled KdV equations

Description of the Extended Mapping Method

Applications of the Method

Conclusion