Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equationiut−uxx+mu+|u|4u=0iut−uxx+mu+|u|4u=0 with the periodic boundary condition is considered. It is proved that for each given constant potential m   and each prescribed integer N>1N>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.