Co-existence of planar and non-planar traveling waves in a sharp interface model

Abstract

Traveling waves modeled with reaction-diffusion differential equations have been studied for decades. Less common are waves for sharp interface models, i.e., sets (or characteristic functions) that move with steady velocities. Our focus belongs to the latter category: the waves are critical points of a geometric variational functional which comes as the Γ-limit of the FitzHugh-Nagumo equations. We demonstrate that planar traveling fronts can become unstable when subject to 2D perturbation. With the same physical parameters the co-existence of 2 planar waves and 1 non-planar wave, each with its distinct speed, is established; this may be the first time a result of this kind is obtained for sharp interface models. As a by-product conclusions on traveling waves of the original FitzHugh-Nagumo equations can be drawn.