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

TL;DR

This paper conducts a detailed numerical bifurcation analysis of depinning fronts in the 2D Swift-Hohenberg equation, revealing how stripe orientation affects invasion dynamics and stability near the homoclinic snaking region.

Contribution

It introduces a boundary value problem approach for analyzing depinning fronts and uncovers the bifurcation structure and stability of various stripe invasion fronts in the Swift-Hohenberg equation.

Findings

Parallel depinning fronts select specific wavenumbers and speeds.

Almost planar invasion fronts bifurcate from parallel fronts near the snaking region.

Parallel stripe fronts can regain transverse stability above a critical speed.

Abstract

In this paper, we carry out 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 and 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 1D 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 localised patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts and reaction-diffusion systems.