  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.
