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P.G. Kevrekidis and D.E. Pelinovsky

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Discrete vector on-site vortices

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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