Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system

TL;DR

This paper proves the nonlinear stability of pattern-forming fronts in the FitzHugh-Nagumo system, providing a rigorous foundation for pattern selection laws in reaction-diffusion systems.

Contribution

It establishes the first stability results for critical pattern-selecting fronts, especially pulled fronts, in reaction-diffusion models like FitzHugh-Nagumo.

Findings

Proved nonlinear stability of pattern-forming fronts in FitzHugh-Nagumo system.

Developed tools for analyzing interaction of marginal stability modes.

Provided a framework for understanding wave number selection in growth processes.

Abstract

We establish sharp nonlinear stability results for fronts that describe the creation of a periodic pattern through the invasion of an unstable state. The fronts we consider are critical, in the sense that they are expected to mediate pattern selection from compactly supported or steep initial data. We focus on pulled fronts, that is, on fronts whose propagation speed is determined by the linearization about the unstable state in the leading edge, only. We present our analysis in the specific setting of the FitzHugh-Nagumo system, where pattern-forming uniformly translating fronts have recently been constructed rigorously, but our methods can be used to establish nonlinear stability of pulled pattern-forming fronts in general reaction-diffusion systems. This is the first stability result of critical pattern-selecting fronts and provides a rigorous foundation for heuristic, universal wave number selection laws in growth processes based on a marginal stability conjecture. The main technical challenge is to describe the interaction between two separate modes of marginal stability, one associated with the spreading process in the leading edge, and one associated with the pattern in the wake. We develop tools based on far-field/core decompositions to characterize, and eventually control, the interaction between these two different types of diffusive modes. Linear decay rates are insufficient to close a nonlinear stability argument and we therefore need a sharper description of the relaxation in the wake of the front using a phase modulation ansatz. We control regularity in the resulting quasilinear equation for the modulated perturbation using nonlinear damping estimates.