Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity

Abstract

We study the existence and stability of localized states in the discrete nonlinear Schrödinger equation (DNLS) on two-dimensional non-square lattices. The model includes both the nearestneighbor and long-range interactions. For the fundamental strongly localized soliton, the results depend on the coordination number, i.e., on the particular type of the lattice. The long-range interactions additionally destabilize the discrete soliton, or make it more stable, if the sign of the interaction is, respectively, the same as or opposite to the sign of the short-range interaction. We also explore more complicated solutions, such as twisted localized modes (TLM's) and solutions carrying multiple topological charge (vortices) that are specific to the triangular and honeycomb lattices. In the cases when such vortices are unstable, direct simulations demonstrate that they turn into zero-vorticity fundamental solitons.

The Discrete Nonlinear Schrödinger Equation — 20 Years On

“…Typically they become unstable as the coupling is increased; the mechanisms of these instabilities were described in some detail in 76,63 . Let us finally also mention a recent study 62 exploring numerically different types of breathers (including vortex-breathers) and their stability in triangular and hexagonal DNLS-lattices.…”

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

“…In recent years there has been a considerable interest in the study of solitons in lattice-type systems. Such solitons have been observed in optics using waveguide arrays, photo-refractive materials, photonic crystal fibers, etc., in both one-dimensional and multidimensional lattices, mostly periodic sinusoidal square lattices [1,2,3,4,5,6,7,8] or single waveguide potentials [9,10,11], but also in discontinuous lattices (surface solitons) [12], radially-symmetric Bessel lattices [13], lattices with triangular or hexagonal symmetry [14,15], lattices with defects [16,17,18,19,20,21,22], with quasicrystal structures [16,23,24,25,26,27,28] or with random potentials [29,30]. Solitons have also been observed in the context of Bose-Einstein Condensates (BEC) [31,32], where lattices have been induced using a variety of techniques.…”

Influence of geometry of waveguide arrays to get discrete solitons

“…Up to now, there have been published many works, both theoretical and experimental, about the beam self-trapping and propagation in Waveguide Arrays (WGA) of different geometries [4][5][6][7][8][9][10][11][12][13][14] . Therefore, the studies conducted by Kevrekidis, et al 15 …”