Discrete vortex solitons.

Localized states in the discrete two-dimensional (2D) nonlinear Schrödinger equation is found: vortex solitons with an integer vorticity S. While Hamiltonian lattices do not conserve angular momentum or the topological invariant related to it, we demonstrate that the soliton's vorticity may be conserved as a dynamical invariant. Linear stability analysis and direct simulations concur in showing that fundamental vortex solitons, with S=1, are stable if the intersite coupling C is smaller than some critical value C((1))(cr). At C>C((1))(cr), an instability sets in through a quartet of complex eigenvalues appearing in the linearized equations. Direct simulations reveal that an unstable vortex soliton with S=1 first splits into two usual solitons with S=0 (in accordance with the prediction of the linear analysis), but then an instability-induced spontaneous symmetry breaking takes place: one of the secondary solitons with S=0 decays into radiation, while the other one survives. We demonstrate that the usual (S=0) 2D solitons in the model become unstable, at C>C((0))(cr) approximately 2.46C((1))(cr), in a different way, via a pair of imaginary eigenvalues omega which bifurcate into instability through omega=0. Except for the lower-energy S=1 solitons that are centered on a site, we also construct ones which are centered between lattice sites which, however, have higher energy than the former. Vortex solitons with S=2 are found too, but they are always unstable. Solitons with S=1 and S=0 can form stable bound states.