H Igher - O Rder V Ortices in N Onlinear D Ynamical L Attices

In this paper, we investigate localized discrete states with a non-zero topological charge (discrete vortices) in a prototypical model of dynamical lattice systems, based on the twoand three-dimensional (2D and 3D) discrete nonlinear Schrödinger (DNLS) equation, with both attractive and repulsive on-site cubic nonlinearity. Systems of two nonlinearly coupled DNLS equations are considered too. We report new results concerning the existence and, especially, stability of the vortices with higher values of the topological charge S (S = 2; 3; 4). Quasi-vortices, i.e., stable solutions of the quadrupole and octupole type, which replace unstable vortices with S = 2 and 4, respectively, are also found. The vortices of the gap-soliton type, which are found in the defocusing (repulsive) model, are quite different, as concerns the stability, from their counterparts in the focusing (attractive) models. In the two-component system, stable compound vortices of the type (S1; S2) = (1; 1) are found, the stability area being larger for the (+S; S) species. In the 3D case, besides finding stable vortices with S = 1 and 3, a novel possibility is reported, viz., a stable two-component complex with mutually orthogonal vortices in the components. Applications of the results to nonlinear optics and Bose-Einstein condensates are briefly discussed. URL: http://nlds.sdsu.edu 138 P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis et. al