A KAM algorithm for two-dimensional nonlinear Schrödinger equations with spatial variable

Abstract

We consider the two-dimensional nonlinear Schrödinger equationiut−△u+|u|2pu+H(x,u,u¯)=0,t∈R,x∈T2 with periodic boundary conditions, where the nonlinearity H(x,u,u¯)=∑m=1∞αm(x)|u|2p+2mu is a real analytic function in a neighborhood of the origin. We obtain, through a KAM algorithm, a Whitney smooth family of small–amplitude quasi–periodic solutions.