Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice

Abstract

In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze the large-time behaviour of solutions that start as bounded perturbations to the well-known planar front solution that travels in the horizontal direction. In particular, we construct an asymptotic phase function $\gamma_j(t)$ and show that for each vertical coordinate $j$ the corresponding horizontal slice of the solution converges to the planar front shifted by $\gamma_j(t)$. We exploit the comparison principle to show that the evolution of these phase variables can be approximated by an appropriate discretization of the mean curvature flow with a direction-dependent drift term. This generalizes the results obtained in [Matano & Nara, 2011] for the spatially continuous setting. Finally, we prove that the horizontal planar wave is nonlinearly stable with respect to perturbations that are asymptotically periodic in the vertical direction.