Discrete vector on-site vortices

We study discrete vortices in coupled discrete nonlinear Schrödinger 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.