D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis

Persistence and stability of discrete vortices in nonlinear Schrodinger lattices

Physica D 212, 20-53 (2005)

Abstract:
We study discrete vortices in the anti-continuum limit of the discrete two-dimensional nonlinear Schr\"{o}dinger (NLS) equations. The discrete vortices in the anti-continuum limit represent a finite set of excited nodes on a closed discrete contour with a non-zero topological charge. Using the Lyapunov--Schmidt reductions, we find sufficient conditions for continuation and termination of the discrete vortices for a small coupling constant in the discrete NLS lattice. An example of a closed discrete contour is considered that includes the vortex cell (also known as the off-site vortex). We classify the symmetric and asymmetric discrete vortices that bifurcate from the anti-continuum limit. We predict analytically and confirm numerically the number of unstable eigenvalues associated with various families of such discrete vortices.

Keywords:
DISCRETE NONLINEAR SCHRODINGER EQUATIONS, DISCRETE SOLITONS, DISCRETE VORTICES, EXISTENCE AND STABILITY, EIGENVALUES OF LINEARIZED OPERATORS, PERTURBATION THEORY FOR EIGENVALUES, LYAPUNOV-SCHMIDT REDUCTIONS, INVERSE FUNCTION THEOREM