KAM for the nonlinear Schrodinger equation

We consider the d-dimensional nonlinear Schrodinger equation under periodic boundary conditions: -i(u) over dot = -Delta u + V(x) *u + epsilon partial derivative F/partial derivative(u) over bar (x, u, (u) over bar), u = u(t, x), x is an element of Td where V(x) = Sigma(V) over cap (a)e(i < a, x >) is an analytic function with (V) over cap real, and F is a real analytic function in Ru, Ju 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 epsilon = 0 the equation is linear and has time-quasi-periodic solutions u(t, x) = Sigma(a is an element of A)<u(a)e(i(vertical bar a vertical bar 2+<(V)over cap>(a))t) e(i < a, x >), vertical bar(u) over cap (a)vertical bar > 0, where A is any finite subset of Z(d). We shall treat omega(a) = vertical bar a vertical bar(2) + (V) over cap (a), a is an element of A, as free parameters in some domain U subset of 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 vertical bar epsilon vertical bar is sufficiently small, then there is a large subset U' of U such that for all omega is an element of 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.