Discrete Solitons and Vortices in Anisotropic Hexagonal and Honeycomb Lattices

TL;DR

This paper investigates how anisotropy affects discrete solitons and vortices in hexagonal and honeycomb lattices modeled by the discrete nonlinear Schrödinger equation, revealing bifurcations and stability changes.

Contribution

It provides analytical and numerical analysis of anisotropy-induced bifurcations and stability properties of discrete vortices in hexagonal and honeycomb lattices.

Findings

Discrete vortices undergo destabilizing bifurcations with anisotropy.

Instabilities lead to localized waveforms or breathing modes.

Stability is confirmed through spectral analysis and numerical simulations.

Abstract

In the present work, we consider the self-focusing discrete nonlinear Schrodinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that important classes of solutions, such as the so-called discrete vortices, undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. We quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation of these instabilities to be the spontaneous rearrangement of the solution, for larger values of the coupling, into localized waveforms typically centered over fewer sites than the original unstable structure. For weak coupling, the instability appears to result in a robust breathing of the relevant waveforms.