KAM for the nonlinear Schrödinger equation

We consider the d-dimensional nonlinear Schrödinger equation under periodic boundary conditions: $-i\dot{u}=-\Delta u+V(x)\ast u+\varepsilon \frac{\partial F}{\partial \overline{u}}(x,u,\overline{u}),\quad u=u(t,x),x\in {\Bbb T}^{d}$ where $V(x)=\sum \hat{V}(a)e^{i\langle a,x\rangle}$ is an analytic function with V̂ real, and F is a real analytic function in Ru, Tu and x. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term V * u we have the potential term Vu.) For ε = 0 the equation is linear and has time—quasi-periodic solutions $u(t,x)=\sum_{a\in {\cal A}}\hat{u}(a)e^{i(|a|^{2}+\hat{V}(a))t}e^{i\langle a,x\rangle},\quad|\hat{u}(a)|>0,$ where A is any finite subset of Z d . We shall treat ω a = ǀaǀ2 + V̂ (a), a ∈ A, as free parameters in some domain U⊂R A . This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If ǀεǀ is sufficiently small, then there is a large subset U′ of U such that for all ω ∈ U′ the solution u persists as a time—quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients.