Nonexistence of small, smooth, time-periodic, spatially periodic solutions for nonlinear Schrödinger equations

Abstract

We study the question of nonexistence of small spatially periodic, timeperiodic solutions for cubic nonlinear Schrödinger equations. We prove that for almost any value in a bounded set of possible temporal periods, there is an amplitude threshold, below which any initial value is not the initial value for a time-periodic solution. The proof requires a certain level of Sobolev regularity on solutions. The methods used are not based on any special structure of the nonlinear Schrödinger equation, and can be applied more generally.