Snakes in square, honeycomb, and triangular lattices

TL;DR

This paper investigates localized solutions and bifurcation structures in the two-dimensional discrete Allen-Cahn equation across square, honeycomb, and triangular lattices, revealing how lattice geometry influences solution complexity.

Contribution

It introduces an active-cell approximation method to analyze bifurcations and stability, highlighting the impact of lattice interfaces on snaking diagram complexity.

Findings

Snaking structures depend on the number of lattice interfaces.

Active-cell approximation accurately predicts bifurcation points.

Lattice geometry significantly influences solution bifurcation complexity.

Abstract

We present a study of time-independent solutions of the two-dimensional discrete Allen-Cahn equation with cubic and quintic nonlinearity. Three different types of lattices are considered, i.e., square, honeycomb, and triangular lattices. The equation admits uniform and localised states. We can obtain localised solutions by combining two different states of uniform solutions, which can develop a snaking structure in the bifurcation diagrams. We find that the complexity and width of the snaking diagrams depend on the number of "patch interfaces" admitted by the lattice systems. We introduce an active-cell approximation to analyse the saddle-node bifurcation and stabilities of the corresponding solutions along the snaking curves. Numerical simulations show that the active-cell approximation gives good agreement for all of the lattice types when the coupling is weak. We also consider planar fronts that support our hypothesis on the relation between the complexity of a bifurcation diagram and the number of interface of its corresponding solutions.