Symmetry, winding number and topological charge of vortex solitons in   discrete-symmetry media
  We determine the functional behavior near the discrete rotational symmetry axis of discrete vortices of the nonlinear Schr\"odinger equation. We show that these solutions present a central phase singularity whose charge is restricted by symmetry arguments. Consequently, we demonstrate that the existence of high-charged discrete vortices is related to the presence of other off-axis phase singularities, whose positions and charges are also restricted by symmetry arguments. To illustrate our theoretical results, we offer two numerical examples of high-charged discrete vortices in photonic crystal fibers showing hexagonal discrete rotational invariance. 
Miguel-\'Angel Garc\'ia-March, Albert Ferrando, Mario Zacar\'es,   Sarira Sahu, Daniel E. Ceballos-Herrera