We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schr\"{ o}dinger equation $$iu_t-\triangle u +|u|^2u+\frac{ \partial{ f(x,u,\bar u)}}{ \partial{ \bar u}}=0, \quad t\in\Bbb R, x\in\Bbb T^2$$ with periodic boundary conditions, where the nonlinearity $\displaystyle f(x,u,\bar u)=\sum_{ j,l,j+l\geq6}a_{ jl}(x)u^j\bar u^l$, $a_{ jl}=a_{ lj}$ 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.
Categories: math.DS