Travelling-wave spatially periodic forcing of asymmetric binary mixtures

Highlights: • We study the Cahn–Hilliard equation with a forcing term. • This is a model for phase separation with external heating. • We model the forcing term/external heating as a travelling wave. • Then the solutions of the equation have a well-defined mathematical structure. • We use numerical and analytical methods to characterize this structure. -- Abstract: We study travelling-wave spatially periodic solutions of a forced Cahn–Hilliard equation. This is a model for phase separation of a binary mixture, subject to external forcing. We look at arbitrary values of the mean mixture concentration, corresponding to asymmetric mixtures (previous studies have only considered the symmetric case). We characterize in depth one particular solution which consists of an oscillation around the mean concentration level, using a range of techniques, both numerical and analytical. We determine the stability of this solution to small-amplitude perturbations. Next, we use methods developed elsewhere in the context of shallow-water waves to uncover a (possibly infinite) family of multiple-spike solutions for the concentration profile, which linear stability analysis demonstrates to be unstable. Throughout the work, we perform thorough parametric studies to outline for which parameter values the different solution types occur.