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

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.

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• Spatiotemporal vortex solitons in hexagonal arrays of waveguides
• Non-Nearest-Neighbor Interactions in Nonlinear Dynamical Lattices
• Discrete Breathers in Hexagonal Dusty Plasma Lattices
• Existence and stability of multisite breathers in honeycomb and hexagonal lattices
• Geometric stabilization of extended S=2 vortices in two-dimensional photonic lattices: theoretical analysis, numerical computation and experimental results
• Stability of non-time-reversible phonobreathers
• Discrete Solitons and Vortices in Anisotropic Hexagonal and Honeycomb Lattices
• Wavepacket spreading dynamics under a non-instantaneous nonlinearity: Self-trapping, defocusing and focusing