Metastable Speeds in the Fractional Allen-Cahn Equation

TL;DR

This paper investigates how the speed and width of interfaces in the fractional Allen-Cahn equation depend on the fractional Laplacian exponent, combining asymptotic analysis with numerical simulations to understand metastable dynamics.

Contribution

It derives asymptotic formulas for interface speed and collision time in the fractional Allen-Cahn equation and validates them through numerical simulations across different fractional exponents.

Findings

Interface speed varies with the fractional Laplacian exponent.

Asymptotic formulas accurately predict interface behavior for large intervals.

Numerical results confirm the dependence of interface dynamics on al and scaling parameters.

Abstract

We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-\Delta)^{\alpha/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $\alpha$ of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of $\alpha$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for $\alpha\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.