Localized states in passive and active phase-field-crystal models

TL;DR

This paper investigates localized states in passive and active phase-field-crystal models, analyzing their bifurcation behavior, snaking phenomena, and the effects of coupling and activity on localized structures.

Contribution

It provides a comprehensive review of localized states in both passive and active PFC models, including new insights into bifurcation behaviors and coupling effects.

Findings

Localized states exhibit characteristic snaking in passive PFC models.

Active PFC models show parity-breaking drift bifurcations.

Coupling can break gradient dynamics, affecting localized state behavior.

Abstract

The passive conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) corresponds to a gradient dynamics for a single order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we first briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC equations described by common gradient dynamics, (ii) two coupled passive PFC where the coupling breaks the gradient dynamics structure, and (iii) a passive PFC coupled to an active PFC.