Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation

TL;DR

This paper proves the existence of special travelling wave solutions in a non-local Korteweg-de Vries-Burgers equation, revealing non-classical shocks in shallow water models through dynamical systems techniques.

Contribution

It establishes the rigorous existence of undercompressive travelling waves in a non-local PDE with fractional derivatives, extending shock analysis in shallow water flows.

Findings

Existence of non-classical shock solutions.

Travelling waves can be characterized as delayed integro-differential equations.

Methodology involves dynamical systems and shooting arguments.

Abstract

We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equation with a non-local diffusion term and a concave-convex flux. 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 a fractional type derivative with order between $1$ and $2$. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of travelling waves that, formally, in the limit of vanishing diffusion and dispersion would give rise to non-classical shocks, that is, shocks that violate the Lax entropy condition. The proof is based on arguments that are typical in dynamical systems. The nature of the non-local operator makes this possible, since the resulting travelling wave equation can be seen as a delayed integro-differential equation. Thus, linearisation around critical points, continuity with respect to parameters and a shooting argument, are the main steps that we have proved and adapted for solving this problem.