Index theory for traveling waves in reaction diffusion systems with skew gradient structure

TL;DR

This paper extends a geometric stability analysis framework to traveling front solutions in reaction-diffusion systems with skew-gradient structure, using Maslov index to determine spectral stability and unstable eigenvalues.

Contribution

It introduces a Maslov index approach for traveling front solutions, filling a gap in stability analysis for reaction-diffusion systems with skew-gradient structure.

Findings

Maslov index is well-defined for traveling front solutions.

The index provides the exact number of unstable eigenvalues.

Application to FitzHugh-Nagumo equation demonstrates practical utility.

Abstract

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation.