Localized states in coupled Cahn–Hilliard equations

The classical Cahn–Hilliard (CH) 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 the coupled CH dynamics of two concentration fields and show that non-reciprocal (or active or non-variational) 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 parity-asymmetric localized states form a slanted homoclinic snaking structure typical for systems with a conservation law. In contrast to snaking structures in systems with gradient dynamics, here, Hopf instabilities occur at a sufficiently large activity, which results in oscillating and travelling localized patterns.