Metastable Speeds in the Fractional Allen-Cahn Equation

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.