Localized states in coupled Cahn–Hilliard equations

The classical Cahn-Hilliard equation corresponds to a gradient dynamics model that describes phase decomposition in a binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via a large-scale instability into drop, hole or labyrinthine concentration patterns of a typical structure length followed by a continuously ongoing coarsening process. Here we consider coupled Cahn-Hilliard dynamics for two concentration fields and show that nonvariational (or active, or nonreciprocal) coupling may induce a small-scale (Turing) instability. At the corresponding primary bifurcation a branch of periodically patterned steady states emerges. Furthermore, there exist localized states that consist of patterned patches coexisting with a homogeneous background. The branches of steady parity-symmetric and -asymmetric localized states form a slanted homoclinic snaking structure typical for systems with a conservation law.