The fractional Allen–Cahn equation with the sextic potential

Abstract

We extend the classical Allen–Cahn (AC) equation to the fractional Allen–Cahn equation (FAC) with triple-well potential. By replacing the spatial Laplacian and double-well potential with fractional Laplacian and triple-well potential, we observe different dynamics. This study leads us to understand different properties of the FAC equation. We seek the existence, boundedness, and unique solvability of numerical solutions for the FAC equation with triple-well potential. In addition, the inclusion principle for the Allen–Cahn equation is considered. These different properties make us enhance the applicability of the phase-field method to the mathematical modeling in materials science. In computation, the spectral decomposition for the fractional operator allows us to develop a numerical method for the fractional Laplacian problem. For the periodic and discrete Laplacian matrix and vector multiplication, circulant submatices are formed in more than two-dimensional case. Even if the fast Fourier transform (FFT) can be utilized in this modeling, we construct the inverse of the doubly-block-circulant matrix for the solution of the fractional Allen–Cahn equation. In doing so, it helps to straightforwardly understand the numerical treatment, and exploit the properties of the discrete Fourier transforms.