Pure cross-diffusion models: Implications for traveling wave solutions

TL;DR

This paper investigates traveling wave solutions in pure cross-diffusion systems, revealing conditions for various wave types and demonstrating that wave trains are a common phenomenon, aiding the modeling of systems with directional movement.

Contribution

It provides a qualitative phase plane analysis for pure cross-diffusion systems, establishing conditions for wave solutions and highlighting the prevalence of wave trains.

Findings

Wave trains are a generic phenomenon in pure cross-diffusion systems.

Conditions for existence of different wave solutions are formulated.

The results assist in modeling systems with directional movement.

Abstract

An analysis of traveling wave solutions of pure cross-diffusion systems, i.e., systems that lack reaction and self-diffusion terms, is presented. Using the qualitative theory of phase plane analysis the conditions for existence of different types of wave solutions are formulated. In particular, it is shown that family of wave trains is a generic phenomenon in pure cross-diffusion systems. The results can be used for construction and analysis of different mathematical models describing systems with directional movement.