Discrete Solitons and Vortices on Anisotropic Lattices

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.

10 pages + 7 figures

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• Discrete surface solitons in two dimensions
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• Semidiscrete vortex solitons
• Dark lattice solitons in one-dimensional waveguide arrays with defocusing saturable nonlinearities and alternating couplings
• Discrete vortices on spatially nonuniform two-dimensional electric networks
• Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links
• Self-trapped spatially localized states in combined linear-nonlinear periodic potentials
• Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices