Slowly-migrating Transition Layers for the Discrete Allen-cahn and Cahn-hilliard Equations

It has recently been proposed that spatially discretized versions of the Allen-Cahn and Cahn-Hilliard equations for modeling phase transitions have certain theoretical and phenomenological advantages over their continuous counterparts. This paper deals with one-dimensional discretizations and examines the extent to which dynamical metastability, which manifests itself in the original partial differential equations in the form of solutions with slowly-moving transition layers, is also present for the discrete equations. It is shown that, in fact, there are transition-layer solutions that evolve at a speed bounded by C1ε(1 + C2/(nε)) −C3n+C4 for all n ≥ n0 and ε ≤ ε0, where 1/n is the spatial mesh size, ε is the interaction length, and n0 and ε0 are constants.