Invariant tori of full dimension for a nonlinear Schrödinger equation

In this paper, we consider the one-dimensional nonlinear Schrödinger equationiut−uxx+mu+f(|u|2)u=0iut−uxx+mu+f(|u|2)u=0 with periodic boundary conditions or Dirichlet boundary conditions, where f   is a real analytic function in some neighborhood of the origin satisfying f(0)=0f(0)=0, f′(0)≠0f′(0)≠0. We prove that for each given constant potential m, when the frequencies, as a function of the amplitudes, can be regarded as the independent parameters, the equation admits a Whitney smooth family of small-amplitude, time almost-periodic solutions with all frequencies. The proof is based on a Birkhoff normal form reduction and an improved version of the KAM theorem.