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