P.G. Kevrekidis and D.E. Pelinovsky

Discrete vector on-site vortices

Proceedings of the Royal Society A 462, 2671-2694 (2006)

Abstract:
We study discrete vortices in coupled discrete nonlinear Schrodinger equations. We focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals. Stability of the single-component vortex cross in the anti-continuum limit of small coupling between lattice nodes is proved. In the vector case, we consider two coupled configurations of vortex crosses, namely the charge-one vortex in one component coupled in the other component to either the charge-one vortex (forming a double-charge vortex) or the charge-negative-one vortex (forming a, so-called, hidden-charge vortex). We show that both vortex configurations are stable in the anti-continuum limit if the parameter for the inter-component coupling is small and both of them are unstable when the coupling parameter is large. In the marginal case of the discrete two-dimensional Manakov system, the double-charge vortex is stable while the hidden-charge vortex is linearly unstable. Analytical predictions are corroborated with numerical observations that show good agreement near the anti-continuum limit but gradually deviate for larger couplings between the lattice nodes.

Keywords:
DISCRETE NONLINEAR SCHRODINGER EQUATIONS, DISCRETE SOLITONS, DISCRETE VORTICES, EXISTENCE AND STABILITY, EIGENVALUES OF LINEARIZED OPERATORS, PERTURBATION THEORY FOR EIGENVALUES, LYAPUNOV-SCHMIDT REDUCTIONS