Quasi-periodic solutions in a nonlinear Schrödinger equation

Abstract

In this paper, one-dimensional (1D) nonlinear Schrödinger equation i u t − u x x + m u + | u | 4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1 , the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.