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

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, coexisting 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 patterns 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 perpendicularly 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 nonvariational perturbation. The numerical algorithms presented in this paper can be adapted to general reaction-diffusion systems.

Automated summary

The study investigates hexagon fronts in the Swift-Hohenberg equation, revealing the phenomenon of homoclinic snaking where new hexagon cells are added to the core, leading to infinitely many coexisting localized states. In certain cases, defects in the cellular pattern may cause hexagon fronts to destabilize.