Snaking and isolas of localised states in bistable discrete lattices

TL;DR

This paper investigates localized states and isolas in a discrete bistable Allen-Cahn model, revealing how lattice pinning enables stable localized solutions absent in the continuum, with distinct bifurcation structures in 1D and 2D.

Contribution

It demonstrates the existence of stable localized states and isolas in a discrete bistable lattice model, highlighting differences from continuum equations and explaining bifurcation features across dimensions.

Findings

Localized states are stabilized by lattice pinning.

Isolas appear in bifurcation diagrams under certain boundary conditions.

Differences in bifurcation structures between 1D and 2D are explained.

Abstract

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.