Snaking and isolas of localized states in bistable discrete lattices

TL;DR

This paper investigates localized states in a bistable discrete lattice model, revealing the existence of stable isolated solution curves called isolas, which are influenced by boundary conditions and dimensionality.

Contribution

It demonstrates the existence of stable localized states and isolas in a discrete bistable Allen-Cahn model, highlighting differences from the continuum case and effects of boundary conditions.

Findings

Isolas appear in 1D with non-periodic boundaries.

Isolas seem to occur generically in 2D.

Bifurcation features in 1D help explain 2D behaviors.

Abstract

We consider localized 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 localized 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.