Global bifurcation of traveling waves in discrete nonlinear Schr\"odinger equations

We consider discrete nonlinear Schr\"odinger equations of n sites with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Traveling waves appear as a symmetry-breaking from the standing waves for different amplitudes. The bifurcation is proved using the global Rabinowitz alternative in subspaces of symmetric functions. As examples, we present applications to the Schr\"odinger and Saturable lattices.