Families of traveling impulses and fronts in some models with cross-diffusion

TL;DR

This paper analyzes traveling wave solutions in PDE systems with cross-diffusion, especially in chemotaxis models, identifying conditions for various wave types and their mathematical properties.

Contribution

It introduces a comprehensive phase plane analysis for traveling waves in Keller-Segel type models without boundary restrictions, revealing new solution families and singular point implications.

Findings

Existence conditions for front-impulse, impulse-front, and front-front waves.

Identification of impulse-impulse solutions in a simple model.

Link between singular points and free-boundary fronts.

Abstract

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse-impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.