Inhomogeneous frozen states in the Swift-Hohenberg equation: hexagonal patterns vs. localized structures

TL;DR

This paper investigates the behavior of interfaces and localized structures in the Swift-Hohenberg model, revealing frozen states with pinned interfaces and metastable localized structures in a bistable pattern regime.

Contribution

It provides a detailed analysis of interface dynamics and localized structures in the Swift-Hohenberg equation, highlighting the conditions for frozen states and metastability.

Findings

Interfaces exhibit crystal-like properties such as faceting and grooving.

Frozen states consist of polygonal domains with pinned interfaces.

Localized structures can be metastable or shrink, depending on parameters.

Abstract

We revisit the Swift-Hohenberg model for two-dimensional hexagonal patterns in the bistability region where hexagons coexist with the uniform quiescent state. We both analyze the law of motion of planar interfaces (separating hexagons and uniform regions), and the stability of localized structures. Interfaces exhibit properties analogous to that of interfaces in crystals, such as faceting, grooving and activated growth or " melting". In the nonlinear regime, some spatially disordered heterogeneous configurations do not evolve in time. Frozen states are essentially composed of extended polygonal domains of hexagons with pinned interfaces, that may coexist with isolated localized structures randomly distributed in the quiescent background. Localized structures become metastable at the pinning/depinning transition of interfaces. In some region of the parameter space, localized structures shrink meanwhile interfaces are still pinned. The region where localized structures have an infinite life-time is relatively limited.