Travelling waves for spatially discrete systems of FitzHugh-Nagumo type with periodic coefficients

TL;DR

This paper proves the existence and stability of travelling wave solutions in spatially discrete FitzHugh-Nagumo systems with periodic coefficients, using advanced spectral analysis techniques.

Contribution

It introduces a generalized spectral convergence method for analyzing multi-component MFDEs in periodic lattice systems, extending previous scalar approaches.

Findings

Existence of travelling wave solutions in periodic FitzHugh-Nagumo systems.

Nonlinear stability of these travelling waves.

Generalization of spectral convergence techniques to multi-component MFDEs.

Abstract

We establish the existence and nonlinear stability of travelling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh-Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable travelling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multi-component functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies.