Selection of pushed pattern-forming fronts in the FitzHugh-Nagumo system

TL;DR

This paper proves the nonlinear stability and wave number selection of pushed pattern-forming fronts in the FitzHugh-Nagumo system, confirming the marginal stability conjecture and providing a framework applicable to general dissipative PDE models.

Contribution

It establishes the first proof of the marginal stability conjecture for pattern-forming fronts and introduces a novel analysis method for the interaction of localized modes and diffusive modes.

Findings

Pushed fronts attract localized initial data and determine invasion speeds.

Confirmation of universal wave number selection laws in pattern formation.

Development of a far-field/core decomposition technique for stability analysis.

Abstract

We establish nonlinear stability of fronts that describe the creation of a periodic pattern through the invasion of an unstable state. Our results concern pushed fronts, that is, fronts whose propagation is driven by a localized mode at the front interface. We prove that these pushed pattern-forming fronts attract initial data supported on a half-line, and therefore determine both propagation speeds and selected wave numbers for invasion from localized initial conditions. This provides to our knowledge the first proof of the marginal stability conjecture for pattern-forming fronts, thereby confirming universal wave number selection laws widely used in the physics literature. We present our analysis in the specific setting of the FitzHugh-Nagumo system, but our methods can be applied to general dissipative PDE models which exhibit pattern formation. The main technical challenge is to control the interaction between the localized mode driving the propagation and outgoing diffusive modes in the wake of the front. Through a subtle far-field/core decomposition of the linearized evolution, we resolve this interaction and describe the nonlinear response of the front to perturbations as a dynamically driven phase mixing problem for the pattern in the wake. The methods we develop are generally useful in any setting involving the interaction of localized modes and outward diffusive transport, such as in the nonlinear stability of undercompressive viscous shock waves or source defects.