Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

Barbara Abraham-Shrauner

doi:10.13189/ms.2019.070102

Abstract

Exact traveling (solitary) wave solutions of nonlinear partial differential equations (NLPDEs) are analyzed for third-order nonlinear evolution equations. These equations have indeterminant homogenous balance and therefore cannot be solved by the Power Index Method (PIM). Some evolution equations are linearizable where solutions are transferred from those of a linear PDE. For other evolution equations transforming to a NLPDE which has a homogenous balance gives rise to possible solutions by the PIM. The solutions for evolution equations that are not linearizable are developed here.

Highlights

How can exact solutions be found for nonlinear partial differential equations (NLPDEs) for which homogeneous balance does not hold? This question is discussed for a particular class of NLPDEs, third-order nonlinear evolution equations
Homogeneous balance does not hold for linear partial differential equations (LPDEs) and it does not hold for some NLPDEs
Exact traveling wave solutions of third-order nonlinear evolution equations have been solved for the nonlinear partial differential equations that have indeterminant homogenous balance

Summary

Introduction

Determining exact, traveling wave solutions of some nonlinear partial differential equations (NLPDEs) has been codified recently where earlier references are noted [1,2]. The approach ties together various methods with new criteria in the PIM. A fundamental requirement of the PIM is that the homogeneous balance holds [3-6]. Homogeneous balance does not hold for linear partial differential equations (LPDEs) and it does not hold for some NLPDEs. Linearization of nonlinear ordinary differential equations (NLODEs) has a long history. Single NLODEs of various orders have been studied; past efforts are reviewed in a recent paper on the linearization of fifth-order NLODEs with emphasis on Lie symmetry analysis [7]. A type of hidden symmetry has been applied to Newtonian equations for a Hamiltonian system to reduce coupled ODEs to a single second-order linear ODE [8].

Objectives

Results

Conclusion