Global bifurcation of travelling waves in discrete nonlinear Schrödinger equations

Abstract

The discrete nonlinear Schrödinger equations of n sites are studied with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Travelling 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. Applications to the Schrödinger and Saturable lattices are presented.

Keywords:

• NLS-like equations
• periodic solutions
• symmetries
• equivariant bifurcation
• degree theory

AMS Subject Classifications:

• 34C25
• 37G40
• 47H11
• 35Q55

Acknowledgements

C. García is grateful to P. Panayotaros, M. Tejada-Wriedt and the referee and editor for their useful comments which greatly improved the presentation of this manuscript.

Notes

No potential conflict of interest was reported by the author.