Construction of exact solutions to the modified forms of DP and CH equations by analytical methods

Jalil Manafian Heris,Isa Zamanpour,Jalal Jalali,Mohammad Asadpour,Reza Shahabi

doi:10.19139/131

Abstract

In this work, we establish the exact solutions to the modified forms of Degasperis–Procesi (DP) and Camassa–Holm (CH) equations. The generalized (G’/G)-expansion and generalized tanh-coth methods were used to construct solitary wave solutions of nonlinear evolution equations. The generalized (G’/G)-expansion method presents a wider applicability for handling nonlinear wave equations. It is shown that the (G’/G)-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Highlights

The investigation of the traveling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena
Both mathematicians and physicists have devoted considerable effort to the study of exact and numerical solutions of the nonlinear ordinary or partial differential equations corresponding to the nonlinear problems
We use of an effective method, (G’/G)-expansion method, for constructing a range of exact solutions for the nonlinear partial differential equations, first proposed by Wang et al [19]

Summary

Introduction

The investigation of the traveling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. CH equation and some its generalized forms have been studied by many authors, for instance, Wazwaz [40] have obtained new solitary wave solutions to the modified forms of DP and CH equations He [41] derived exact travelling wave solutions of a generalized CH equation using the integral bifurcation method. In [47] bifurcations of travelling wave solutions for a variant of CH equation investigated by He. In [48] Rui et al, applied the integral bifurcation method and its application for solving a family of third-order dispersive PDEs. Liu and Qian [49] have derived peakons and their bifurcation for the generalized CH equation. Some references are given at the end of this paper

Basic ideas of two methods

The modified Degasperis–Procesi equation

The modified Camassa–Holm equation

Conclusion