Travelling wave solutions of equations in the Burgers Hierarchy

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.

Figures (4)

• Figure 1: Solutions of Burgers equation as a function of the travelling coordinate for three different values of the kinetic viscosity.
• Figure 2: Solutions of the first equation in Burgers hierarchy as a function of the travelling coordinate. The solid and dashed lines represent results for equations in absence and presence of the term $u_{3x}$.
• Figure 3: Solution of the first member of the Burgers hierarchy for $\xi$ in the interval $(-2,2)$. The solid and dashed curves carry the same meaning as those in Fig.2.
• Figure 4: Solutions of the fourth equation in Burgers hierarchy as a function of the travelling coordinate . The solid and dashed lines represent results for equations in absence and presence of the term $u_{5x}$.