Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation

TL;DR

This paper analyzes traveling wave solutions in a modified Allen-Cahn model incorporating relaxation effects, proving their nonlinear stability through spectral analysis and supporting findings with numerical simulations of wave speed and dynamics.

Contribution

It provides the first proof of nonlinear stability for traveling fronts in the Allen-Cahn model with relaxation, combining spectral analysis with numerical validation.

Findings

Traveling waves exist and are stable in the relaxed Allen-Cahn model.

Wave speed differs from the classical parabolic case, as shown numerically.

Large perturbations exhibit complex dynamics explored through simulations.

Abstract

A modification of the parabolic Allen-Cahn equation, determined by the substitution of Fick's diffusion law with a relaxation relation of Cattaneo-Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.