Variational properties of space-periodic standing waves of nonlinear Schr{\"o}dinger equations with general nonlinearities

Abstract

Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles. This involves minimizing energy while keeping mass and momentum constant, as well as minimizing the action over the Nehari manifold. These variational approaches are considered both in the periodic and anti-periodic settings, and for focusing and defocusing nonlinearities. In appendix, we study the existence properties of periodic solutions of the triple power nonlinearity.