Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

TL;DR

This paper investigates localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity, revealing their existence in the phase diagram and their relation to homoclinic snaking, with implications for models of phase transitions.

Contribution

It provides a detailed numerical analysis of localized structures in a conserved phase field crystal model, connecting their behavior to homoclinic snaking phenomena.

Findings

Localized states exist in the coexistence region of phases.

Localized states may lie outside the binodal.

Results relate to standard homoclinic snaking patterns.

Abstract

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.