Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

TL;DR

This paper proves the existence and nonlinear stability of traveling pulse solutions in a discrete FitzHugh-Nagumo model with infinite-range interactions, bridging the gap between discrete and continuum systems.

Contribution

It introduces a novel approach to analyze spectral properties without exponential dichotomies, transferring Fredholm properties from PDEs to discrete models.

Findings

Existence of stable pulse solutions near the continuum limit

Development of a new method for spectral analysis of mixed-type MFDEs

Extension of Fredholm property transfer from PDEs to discrete equations

Abstract

We establish the existence and nonlinear stability of travelling pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions close to the continuum limit. For the verification of the spectral properties, we need to study a functional differential equation of mixed type (MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and phase spaces, by building on a technique developed by Bates, Chen and Chmaj for the discrete Nagumo equation. This allows us to transfer several crucial Fredholm properties from the PDE setting to our discrete setting.