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

It is well-known that stationary localized patterns involving a periodic stripe core can undergo a process that is known as "homoclinic snaking" where patterns are added to the stripe core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes place usually occupies a small region in the bistability region between the stripes and quiescent state. Outside the homoclinic snaking region, the localized patterns invade or retreat where stripes are either added or removed from the core forming depinning fronts. It remains an open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper, we carry out a numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique, or perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed, whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the one-dimensional homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localized patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts in reaction-diffusion systems.