Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice

We predict new generic types of vorticity-carrying soliton complexes in a class of physical systems including an attractive Bose-Einstein condensate in a square optical lattice (OL) and photonic lattices in photorefractive media. The patterns include ring-shaped higher-order vortex solitons and supervortices. Stability diagrams for these patterns, based on direct simulations, are presented. The vortex ring solitons are stable if the phase difference ∆φ between adjacent solitons in the ring is larger than π/2, while the supervortices are stable in the opposite case, ∆φ < π/2. A qualitative explanation to the stability is given. Recent experimental observations of two-dimensional (2D) solitons [1] and localized vortices [2] in photonic lattices in self-focusing photorefractive crystals (PRCs), and prediction of similar structures in the 2D Gross-Pitaevskii equation (GPE) for a self-attractive Bose-Einstein condensate (BEC) loaded into the square optical lattice (OL) [3, 4], have drawn interest to solitons of this type, including vorticity-carrying ones. Vortex solitons of the gap type have also been predicted in the repulsive GPE with the square OL [9, 10], including higher-order ones, with the vorticity S = 2 [10]. However, the results which were reported thus far for the self-focusing media, both saturable and cubic, pertain solely to the vortices with S = 1 (including strongly deformed ones [11]). In this work, we present stable vortex solitons with higher values of S and more complex vortex patterns, which suggests a possibility to create them in the experiment. We report the results for both BEC and PRC models, which attests to their generic character. It is relevant to mention that, in discrete models, which may be considered as a limiting case corresponding to a very strong OL, with