Multiple front standing waves in the FitzHugh-Nagumo equations

TL;DR

This paper proves the existence of stable and unstable multiple front standing waves in the FitzHugh-Nagumo equations, using variational methods and Lyapunov-Schmidt reduction for systems with saddle-focus equilibria.

Contribution

It introduces a novel approach to construct and analyze multiple front standing waves in FitzHugh-Nagumo equations with saddle-focus equilibria, extending previous results.

Findings

Existence of stable multiple front standing waves.

Existence of unstable multiple front standing waves.

Application of variational characterization and Lyapunov-Schmidt reduction.

Abstract

There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite quadratic form in the velocity terms. When the system has two stable hyperbolic equilibria, there exist two stable standing fronts, which will be used in this paper as building blocks, to construct stable standing waves with multiple fronts in case the equilibria are of saddle-focus type. The idea to prove existence is somewhat close in spirit to [Buffoni-Sere, CPAM 49, 285-305]. However several differences are required in the argument: facing a strongly indefinite functional, we need to perform a nonlo-cal Lyapunov-Schmidt reduction; in order to justify the stability of multiple front standing waves, we rely on a more precise variational characterization of such critical points. Based on this approach, both stable and unstable standing waves are established.