Similarity Reductions and New Exact Solutions for B-Family Equations

G M Moatimid,M.H.M Moussa,Rehab M El-Shiekh,A A El-Satar

doi:10.5923/j.ajms.20120203.05

Abstract

In this paper, Lie-Group method is applied to the b-family equations which includes two important nonlinear partial differential equations Camassa--Holm (CH) equation and the Degasperis--Procesi (DP) equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub-algebras of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations (ODEs) for the b-family equations. The generalized He's Exp-Function method is used to drive exact solutions for the reduced nonlinear ODEs, some figures are given to show the properties of the solutions.

Highlights

IntroductionEq (1) represent the competition, or balance, in fluid convection between nonlinear transport and amplification due to b-dimensional stretching[2,3]
In this paper we consider the following b-family of equations[1]ut − uxxt + (b +1)u= ux buxuxx + uuxxx (1.1)where b is a dimensionless constant
(1) was included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations[4]

Summary

Introduction

Eq (1) represent the competition, or balance, in fluid convection between nonlinear transport and amplification due to b-dimensional stretching[2,3]. (1) was included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations[4]. Degasperis and Procesi[5] showed that the family of equations (1) cannot be integrable unless b=2 or b=3 by using the method of asymptotic integrability. The CH and the DP equations are bi-Hamiltonian and have an associated isospectral problem, they are both formally integrable[6,7,8,9]. Both equations admit peaked solitary wave solutions and present similarities they are truly different[10,11,12,13]

Solution of the Problem

Similarity Reduction and Exact Solutions

Conclusions