A non-local KdV-Burgers equation: Numerical study of travelling waves

Abstract

We present numerical simulations that support our previous study of travelling wave solutions of a Korteweg-de Vries-Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is of the fractional derivative type with order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. In this paper we give numerical evidence of stability of non-monotone travelling wave. We also confirm the existence of travelling waves that are everywhere monotone except over a bounded interval where they exhibit oscillations.