Persistence and stability of discrete vortices in nonlinear Schrödinger lattices

We study discrete vortices in the two-dimensional nonlinear Schrödinger lattice with small coupling between lattice nodes. The discrete vortices in the anti-continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non-zero charge. Using the Lyapunov-Schmidt reductions, we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes. An example of a square discrete contour is considered that includes the vortex cell (also known as the off-site vortex). We classify families of 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 such discrete vortices.