Localized states in coupled Cahn–Hilliard equations

• Published in: IMA journal of applied mathematics Vol. 86; no. 5; pp. 924 - 943
• Main Authors: Frohoff-Hülsmann, Tobias, Thiele, Uwe
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01.10.2021
• Subjects: non-reciprocal interaction; Cahn–Hilliard models; slanted homoclinic snaking; conservation laws; Turing instability; localized states
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxab026

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 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.