Orientation-dependent pinning and homoclinic snaking on a planar lattice

Abstract

We study homoclinic snaking of one-dimensional, localised states on two-dimensional, bistable lattices via the method of exponential asymptotics. Within a narrow region of parameter space, fronts connecting the two stable states are pinned to the underlying lattice. Localised solutions are formed by matching two such stationary fronts back-to-back; depending on the orientation relative to the lattice, the solution branch may `snake' back and forth within the pinning region via successive saddle-node bifurcations. Standard continuum approximations in the weakly nonlinear limit (equivalently, the limit of small mesh size) do not exhibit this behaviour, due to the resultant leading-order reaction-diffusion equation lacking a periodic spatial structure. By including exponentially small effects hidden beyond all algebraic orders in the asymptotic expansion, we find that exponentially small but exponentially growing terms are switched on via error function smoothing near Stokes lines. Eliminating these otherwise unbounded beyond-all-orders terms selects the origin (modulo the mesh size) of the front, and matching two fronts together yields a set of equations describing the snaking bifurcation diagram. This is possible only within an exponentially small region of parameter space---the pinning region. Moreover, by considering fronts orientated at an arbitrary angle $\psi$ to the $x$-axis, we show that the width of the pinning region is non-zero only if $\tan\psi$ is rational or infinite. This is the first time a formula explicitly relating the orientation of a front to the width of its pinning region has been derived. The asymptotic results are compared with numerical calculations, with good agreement.