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

TL;DR

This paper investigates the bifurcation of traveling waves from standing waves in discrete nonlinear Schrödinger equations with periodic boundary conditions, using global bifurcation theory and symmetry analysis.

Contribution

It proves the existence of traveling wave bifurcations from standing waves in discrete nonlinear Schrödinger equations using the Rabinowitz alternative.

Findings

Traveling waves bifurcate from standing waves at different amplitudes.

The bifurcation analysis applies to Schrödinger and Saturable lattice models.

Global bifurcation results are established in symmetric subspaces.

Abstract

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.