The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation

Medhat M Helal,Emad A Mohamed,Mohammad L Mekky

doi:10.4236/am.2012.31002

Abstract

In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional disper-sive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Rungekutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

Highlights

Exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) have been of a major concern for both mathematicians and physicists
The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Runge-kutta scheme
The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions

Summary

Introduction

Exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) have been of a major concern for both mathematicians and physicists. Long wave in shallow water is a subject of broad interests and has a long colorful history It has a rich variety of phenomenological manifestation, especially the existence of waves permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail. These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion. It has been noted that validity of theoretical models critically epends on the domain of underlying key parameters which characterize the specific motions to be modelled

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