Travelling waves in a PDE-ODE coupled system with nonlinear diffusion

TL;DR

This paper provides an analytical framework for the existence and stability of travelling wave solutions in a coupled PDE-ODE system with nonlinear diffusion, supported by numerical simulations and biological relevance.

Contribution

It introduces a novel analytical approach to establish existence and stability of TW solutions in a degenerate PDE-ODE system with biological applications.

Findings

Existence of a unique travelling wave with a sharp front and diffusive tail.

Proven linear stability of the TW in two dimensions.

Numerical simulations confirm theoretical predictions.

Abstract

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the TWs.