Reducible KAM tori for two-dimensional nonlinear Schrödinger equations with explicit dependence on the spatial variable

Abstract

We study the two-dimensional nonlinear Schrödinger equationiut−△u+|u|2u+∂f(x,u,u¯)∂u¯=0,t∈R,x∈T2 with periodic boundary conditions. The nonlinearity f(x,u,u¯)=∑j,l,j+l≥6ajl(x)uju¯l, ajl=alj is a real analytic function in a neighborhood of the origin. We obtain, through an infinite dimensional KAM theorem, a Whitney smooth family of small–amplitude quasi–periodic solutions.