Spectral Analysis for Periodic Solutions of the Cahn–Hilliard Equation on R

We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on R is linearized about a stationary periodic solution. Our analysis is particularly motivated by the study of spinodal decomposition, a phe-nomenon in which the rapid cooling (quenching) of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions of different crys-talline structure, separated by steep transition layers. In this context, a natural prob-lem regards the evolution of solutions initialized by small, random (in some sense) perturbations of the pre-quenching homogeneous state. Solutions initialized in this way appear to evolve transiently toward certain unstable periodic solutions, with the rate of evolution described by the spectrum associated with these periodic solutions. In the current paper, we use Evans function methods and a perturbation argument to locate the spectrum associated with such periodic solutions. We also briefly discuss a heuristic method due to J. S. Langer for relating our spectral information to coarsening rates