Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation

Lijun Zhang,Chaudry Masood Khalique

doi:10.1186/s13662-015-0510-y

Lijun Zhang, Chaudry Masood Khalique

Open Access

https://doi.org/10.1186/s13662-015-0510-y

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Journal: Advances in Difference Equations	Publication Date: Jun 26, 2015
Citations: 7	License type: CC BY 4.0

Abstract

In this paper we derive new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which we introduce here for the first time. Firstly, we reduce the associated fourth-order nonlinear ordinary differential equation (ODE) into a solvable first-order nonlinear ODE to obtain new exact traveling wave solutions, including the solitary wave and periodic solutions. Furthermore, using the new method we derive the quasi-periodic wave solutions of this equation by assuming that the solutions of the corresponding higher-order ODE are the sum of the solutions of two solvable first-order nonlinear ODEs. This new method can be used to investigate the exact traveling wave solutions and quasi-periodic wave solutions of a general class of higher-order wave equations.

Highlights

In this paper we study the KdV-Sawada-Kotera-Ramani equation [ – ]ut + a u + uxx x + b u + uuxx + uxxxx x =, ( . )which was used to theoretically study the resonances of solitons in a one-dimensional space by Hirota and Ito [ ]
Equation ( . ) is reduced to the KdV equation when b = and to the Sawada-Kotera equation when a = ; it is a linear combination of the KdV equation and the Sawada-Kotera equation
4 Conclusion and discussion In this paper, we studied the exact traveling wave solutions to the KdV-Sawada-KoteraRamani equation ( . ) via the sub-equation in the form = a y + a y + a y + a

Summary

Introduction

In this paper we study the KdV-Sawada-Kotera-Ramani equation [ – ]. Ut + a u + uxx x + b u + uuxx + uxxxx x = , ). which was used to theoretically study the resonances of solitons in a one-dimensional space by Hirota and Ito [ ]. The existence of conservation laws for this equation was proved by Konno [ ]. Some traveling wave solutions were derived in [ ] by the (G /G)-expansion method. In [ ], the traveling wave solutions of ) were studied by using the generalized auxiliary equation method. Too many undetermined coefficients were involved in this method and some conditions on these coefficients were ignored, and some wrong results were given in [ ], which can be checked by Maple

Objectives

Conclusion