Propagation failure for traveling fronts of the Nagumo equation on a lattice

Abstract

In this work, we study the phenomenon of propagation failure for fronts in a bistable reaction–diffusion equation on a lattice in one and two dimensions. Using asymptotic methods and modulation theory, we approximate the anchoring region in the parameter space defined by the coupling coefficient and the bistability parameter. For the one-dimensional case, modulation theory yields a periodic, time-dependent velocity of the wavefront, governed by a Peierls–Nabarro potential. We provide a simple explanation for a numerical observation made in previous work by Mallet-Paret, Hoffman and Mallet-Paret (2010), regarding the fact that a stationary vertical wavefront begins to advance when its direction is perturbed. We also present numerical evidence demonstrating the accuracy of our approximations.