Application of Symbolic Computation in Nonlinear Differential-Difference Equations

Fuding Xie,Min Ji,Zhen Wang

doi:10.1155/2009/158142

Abstract

A method is proposed to construct closed-form solutions of nonlinear differential-difference equations. For the variety of nonlinearities, this method only deals with such equations which are written in polynomials in function and its derivative. Some closed-form solutions of Hybrid lattice, Discrete mKdV lattice, and modified Volterra lattice are obtained by using the proposed method. The travelling wave solutions of nonlinear differential-difference equations in polynomial in function tanh are included in these solutions. This implies that the proposed method is more powerful than the one introduced by Baldwin et al. The results obtained in this paper show the validity of the proposal.

Highlights

Wadati 1 introduced the following nonlinear differential-difference equation NDDE : dun t dt α βun γ u2n un−1 − un 1, 1.1 where α, β, and γ / 0 are constants
Baldwin et al 3 recently presented an adaptation of the tanh-method to solve NDDEs
Suppose the NDDE we study in this work is in the following form

Summary

Introduction

Wadati 1 introduced the following nonlinear differential-difference equation NDDE : dun t dt α βun γ u2n un−1 − un 1 , 1.1 where α, β, and γ / 0 are constants. In this study, searching for the closed-form solutions, especially solitary wave solutions and periodic solutions of 1.4 and 1.5 , is considered. Baldwin et al 3 recently presented an adaptation of the tanh-method to solve NDDEs. Some analytical closed-form solutions of several lattices in polynomial in function tanh have been obtained 3. The method proposed in 3 where tanh-solutions are only considered is firstly generalized, and is applied to solve 1.1. The solutions presented here cover the known one presented by Baldwin et al 3 , and introduce new solutions for some NDDEs. The rest of the paper is organized as follows: an improved method is proposed and how to obtain the solitary wave solutions and periodic solutions to NDDEs is depicted. Some new solitary wave solutions and periodic solutions of Hybrid lattice, discrete mKdV lattice, and modified Volterra lattice have been obtained.

The Improved Method

Conclusions