Snaking and isolas of localised states in bistable discrete lattices

We consider localised states in a discrete bistable Allen-Cahn equation This model equation combines bistability and local cell to-cell coupling in the simplest possible way The existence of stable localised states is made possible by pinning to the underlying lattice they do not exist in the equivalent continuum equation In particular we address the existence of isolas closed curves of solutions in the bifurcation diagram Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D