Snakes in square, honeycomb and triangular lattices

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.

Localized patterns in planar bistable weakly coupled lattice systems

“…Using the cubic-quintic nonlinearity f (u, µ) = −µu + 2u 3 − u 5 , it was demonstrated in [35] using numerical continuation that the bifurcation curves of D 4 -symmetric square patches of (1.1) bear a striking resemblance to those of localized hexagon patches in the planar Swift-Hohenberg equation: compare Figure 1(ii) showing hexagon patches of the Swift-Hohenberg equation with Figure 2(iii) showing D 4 -symmetric off-site patterns of (1.1). Similar numerical results were obtained recently in [21] for equations similar to (1.1) posed on square, hexagonal, and triangular lattices 1 . An intuitive reasoning for the similarity between the continuous and discrete case is that domain-filling hexagons are the preferred planar state in the continuous Swift-Hohenberg PDE, and we can therefore think of hexagon patches as developing on an underlying hidden hexagonal lattice.…”

“…We note that the mechanisms that drive the fold bifurcations for planar lattices near the anti-continuum limit were also found independently in [21], where the leading-order interactions between neighboring sites were discussed; we will put their considerations on a rigorous footing here.…”

“…Since our analysis in the anti-continuum limit relied on Lyapunov-Schmidt reduction, these cases should be amenable to analysis as well, and it would be interesting to see whether the bifurcation diagrams are similar. Finally, other lattices could be explored: we refer to [5,21] for numerical computations of localized patterns on hexagonal lattices, to [21] for triangular lattices, and to [28] for a numerical study of snaking of localized patterns in a predator-prey model on Barabási-Albert networks, where the coupling operator is given by the graph laplacian.…”

Snaking bifurcations of localized patterns on ring lattices

“…Surprisingly, snaking was observed also on finite random graphs; see [26]. The formal analysis in [23] and the complementary rigorous results in [4] show that the shape of branches near folds seems to be determined by a small number of motifs where only a few nodes interact, whilst the states of all other nodes are essentially irrelevant. It might be this underlying generic structure that leads to the universality of snaking observed on random graphs in [26].…”

“…Many of these investigations were motivated by the observation of snaking patterns in experiments and models, including in ferrofluids [18,24,30], optical systems [9,15,17], and vegetation models [27], to name but a few. Similarly complex bifurcation structures of localized solutions have also been observed in spatially discrete systems posed on integer lattices in [9,10,22,23,26,28,32,34,35] and were explained in part by [3][4][5]. While the latter works have explained the bifurcation structure of localized solutions on "regular" graphs, such as the integer lattices, little is known about how graph structure and connection topology influence the connections of localized solutions in parameter space.…”