  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.
