Nonlinear Schrödinger lattices II: Persistence and stability of discrete vortices

We study discrete vortices in the anti-continuum limit of the discrete two-dimensional nonlinear Schrödinger (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 each family of the discrete vortices.