# Resting and Traveling Localized States in an Active Phase-Field-Crystal Model

## TL;DR

This paper investigates the formation and dynamics of localized crystalline states in an active phase-field-crystal model, revealing how activity influences bifurcation structures and the transition from resting to traveling states.

## Contribution

It introduces a detailed bifurcation analysis of active PFC models, highlighting how activity modifies localized state behavior and motion onset mechanisms.

## Key findings

- Active PFC model exhibits resting and traveling localized states.
- Activity alters the bifurcation structure, including homoclinic snaking.
- Motion onset occurs via drift bifurcations, with an analytical criterion derived.

## Abstract

The conserved Swift-Hohenberg equation (or Phase-Field-Crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles Menzel and L\"owen [Phys. Rev. Lett. 110, 055702 (2013)] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.