An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two dimensional nonlinear Schrödinger equationiut−Δu+|u|2u=0,t∈R,x∈T2 with periodic boundary conditions. We obtain for the equation a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.