SLOWLY-MIGRATING TRANSITION LAYERS FOR THE DISCRETE ALLEN-CAHN AND CAHN-HILLIARD EQUATIONS

It has recently been proposed that spatially discretized versions of t he Allen-Cahn and Cahn-Hilliard equations for modelling phase transiti ons have certain theoretical and phenomenological advantages over thei r continuous counterparts. This paper deals with one-dimensional discr etizations and examines the extent to which dynamical metastability, w hich manifests itself in the original partial differential equations i n the form of solutions with slowly-moving transition layers, is also present for the discrete equations. It is shown that, in fact, there a re transition-layer solutions that evolve at a speed bounded by C-1 ep silon(1 + C-2/(n epsilon))(-C3n+C4) for all n greater than or equal to n(0) and epsilon less than or equal to epsilon(0), where 1/n is the s patial mesh size, epsilon is the interaction length, and n(0) and epsi lon(0) are constants.