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

TL;DR

This paper studies the long-term behavior of solutions to the discrete Allen-Cahn equation on a 2D lattice, showing convergence to shifted planar fronts and stability under certain perturbations, with connections to mean curvature flow.

Contribution

It introduces an asymptotic phase function for the discrete setting and links the front evolution to a discretized mean curvature flow with directional drift, extending continuous case results.

Findings

Solutions converge to shifted planar fronts over time.

The phase evolution approximates a discretized mean curvature flow.

Horizontal planar waves are nonlinearly stable under asymptotically periodic perturbations.

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.