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Discrete Solitons and Vortices on Anisotropic Lattices

P. G. Kevrekidis·D. J. Frantzeskakis·R. Carretero-Gonzalez·B. A. Malomed·A. R. Bishop
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Abstract

We consider effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schr{ö}dinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation, which predicts that broad quasi-continuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons ("vortex crosses") feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called "super-symmetric" intersite-centered vortices ("vortex squares"), with the topological charge $S$ equal to the square's size $M$: we predict in an analytical form by means of the Lyapunov-Schmidt theory, and confirm by numerical results, that arbitrarily weak anisotropy results in dramatic changes in the stability and dynamics in comparison with the \emph{degenerate}, in this case, isotropic limit.

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P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, B. A. Malomed, A. R. Bishop. 2005-07-23. Discrete Solitons and Vortices on Anisotropic Lattices. https://doi.org/10.1103/physreve.72.046613

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