Localized states in coupled Cahn-Hilliard equations

TL;DR

This paper investigates how nonreciprocal coupling in coupled Cahn-Hilliard equations can lead to small-scale Turing instabilities, resulting in localized and oscillating patterns, expanding understanding of pattern formation in active systems.

Contribution

It introduces the emergence of localized states and oscillatory patterns in coupled Cahn-Hilliard equations with nonreciprocal coupling, revealing new bifurcation structures and dynamics.

Findings

Nonreciprocal coupling induces Turing instabilities.

Localized patterned states form homoclinic snaking structures.

Hopf instabilities lead to oscillating and traveling localized patterns.

Abstract

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 nonreciprocal (or active, or nonvariational) 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 sufficiently large activity which result in oscillating and traveling localized patterns.