Hexagon Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

TL;DR

This paper investigates the behavior of hexagon fronts outside the homoclinic snaking region in the Swift-Hohenberg equation, revealing how they invade the trivial state, destabilize, and lead to pattern selection and defects.

Contribution

It introduces a numerical path-following method to analyze hexagon front bifurcations and pattern compatibility in the Swift-Hohenberg equation with non-variational perturbations.

Findings

Hexagon fronts can destabilize and produce defects when stretched transversely.

Compatibility diagrams predict hexagon pattern selection on the plane.

Numerical algorithms are adaptable to general reaction-diffusion systems.

Abstract

Stationary fronts connecting the trivial state and a cellular (distorted) hexagonal pattern in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity are known to undergo a process of infinitely many folds as a parameter is varied, known as homoclinic snaking, where new hexagon cells are added to the core, leading to a region of infinitely-many, co-existing localized states. Outside the homoclinic snaking region, the hexagon fronts can invade the trivial state in a bursting fashion. In this paper, we use a far-field core decomposition to set up a numerical path-following routine to trace out the bifurcation diagrams of hexagon fronts for the two main orientations of cellular hexagon pattern with respect to the interface in the bistable region. We find for one orientation that the hexagon fronts can destabilize as the distorted hexagons are stretched in the transverse direction leading to defects occurring in the deposited cellular pattern. We then plot diagrams showing when the selected fronts for the two main orientations, aligned perpendicular to each other, are compatible leading to a hexagon wavenumber selection prediction for hexagon patches on the plane. Finally, we verify the compatibility criterion for hexagon patches in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity and a large non-variational perturbation. The numerical algorithms presented in this paper can be adapted to general reaction-diffusion systems.