Breathing discrete nonlinear Schrödinger vortices

Abstract

Breathing discrete vortices are obtained as numerically exact and generally quasiperiodic, localized solutions to the discrete nonlinear Schrödinger equation with cubic (Kerr) on-site nonlinearity, on a two-dimensional square lattice with nearest-neighbor couplings. We identify and analyze three different types of solutions characterized by circulating currents and time-periodically oscillating intensity distributions, two of which have been discussed in earlier works while the third being, to our knowledge, presented here for the first time. (i) A vortex-breather, constructed from the anticontinuous limit as a superposition of a single-site breather and a discrete vortex surrounding it, where the breather and vortex are oscillating at different frequencies. (ii) A charge-flipping vortex, constructed from an anticontinuous solution with an even number of sites on a closed loop, with alternating sites oscillating at different frequencies. (iii) A breathing vortex, constructed by continuation of a non-resonating linear internal eigenmode of a stationary discrete vortex. We illustrate by examples, using numerical Floquet analysis for solutions obtained from a Newton scheme, that linearly stable solutions exist from all three categories, at sufficiently strong discreteness.