Discrete solitons and vortices in hexagonal and honeycomb lattices: Existence, stability, and dynamics

TL;DR

This paper systematically classifies and analyzes the existence, stability, and dynamics of discrete solitons and vortices in hexagonal and honeycomb lattice models, revealing stable configurations and rich nonlinear behaviors.

Contribution

It provides a comprehensive classification and stability analysis of discrete solitons and vortices in non-square lattices, including analytical and numerical methods.

Findings

Hexapole of alternating phases is stable.

Vortex of topological charge S=2 can be stable.

Vortex of topological charge S=1 may be stable in focusing nonlinearity.

Abstract

We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schroedinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and three-lattice-site contours in the form of both discrete multipole solitons and discrete vortices. Additionally to identifying the possible states, we analytically track their linear stability both qualitatively and quantitatively. We find that among the six-site configurations, the hexapole of alternating phases, as well as the vortex of topological charge S=2 have intervals of stability; among three-site states, only the vortex of topological charge S=1 may be stable in the case of focusing nonlinearity. These conclusions are confirmed both for hexagonal and for honeycomb lattices by means of detailed numerical bifurcation analysis of the stationary states from the anticontinuum limit, and by direct simulations to monitor the dynamical instabilities, when the latter arise. The dynamics reveal a wealth of nonlinear behavior resulting not only in single-site solitary wave forms, but also in robust multisite breathing structures.