Snakes in square, honeycomb and triangular lattices

R. Kusdiantara, H. Susanto

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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.

Original languageBritish English
Pages (from-to)5170-5190
Number of pages21
JournalNonlinearity
Volume32
Issue number12
DOIs
StatePublished - 12 Nov 2019

Keywords

  • bistable systems
  • discrete Allen-Cahn equation
  • homoclinic snaking
  • localised structures

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