Quasi-periodic solutions for 1D Schrödinger equation with the nonlinearity |u|2pu|u|2pu

In this paper, one-dimensional (1D) nonlinear Schrödinger equationiut−uxx+|u|2pu=0,p∈N, with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial normal form and scaling skills.