A KAM Theorem for Two-dimensional Nonlinear Schrödinger Equations

Abstract

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schrödinger equation $$i{u}_t-△u+|u{|}^{2}u+\frac{\partial f(x,u,\bar{u})}{\partial \bar{u}}=0,t\in \mathbb{R},x\in {\mathbb{T}}^2$$ with periodic boundary conditions, where the nonlinearity $f(x,u,\bar{u})=\sum _{j,l,j+l\ge 6}{a}_{jl}(x)u^j{\bar{u}}^l$, $a_{jl}=a_{lj}$ is a real analytic function in a neighborhood of the origin. We obtain for the equation a Whitney smooth family of small--amplitude quasi--periodic solutions which are partially hyperbolic.