Curvature-driven front propagation through planar lattices in oblique directions

TL;DR

This paper studies the long-term evolution of solutions to the discrete Allen-Cahn equation on a 2D lattice, revealing how front-like initial conditions form stable planar traveling waves influenced by curvature in rational oblique directions.

Contribution

It generalizes previous results to rational oblique directions and links the wave behavior to discretized mean curvature flow, establishing nonlinear stability of traveling waves.

Findings

Traveling waves form in rational directions.

Wave behavior is linked to discretized mean curvature flow.

Traveling waves are nonlinearly stable under certain perturbations.

Abstract

In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift $\gamma_l(t)$ that depends on the coordinate $l$ transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of $\gamma$ can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.