Travelling wave solutions of equations in the Burgers Hierarchy

TL;DR

This paper introduces an algebraic method based on the Cole-Hopf transformation to construct travelling wave solutions for equations in the Burgers hierarchy, including higher-order equations where direct methods fail.

Contribution

The paper presents a simple algebraic approach for finding travelling wave solutions in the Burgers hierarchy, extending applicability to higher-order equations.

Findings

Method works well for higher Burgers equations

Diffusion term smooths the wave despite nonlinearity

All higher-order equations share identical solutions

Abstract

We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers hierarchy. Our method based on a judicious use of the well known Cole-Hopf transformation is found to work satisfactorily for higher Burgers equations for which the direct method of integration is inapplicable. For Burgers equation we clearly demonstrate how does the diffusion term in the equation counteract the nonlinearity to result in a smooth wave. We envisage a similar study for higher equations in the Buggers hierarchy and establish that (i) as opposed to the solution of the Burgers equation, the purely nonlinear terms of these equations support smooth solutions and more interestingly (ii) the complete solutions of all higher-order equations are identical.