On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D iut + u + Mu ± εu|u|4 = 0, (0.1) where M is a random Fourier multiplier defined by Mu(n) = Vn ˆu(n) (0.2) and (Vn)n∈Z are independently chosen in [−1, 1]. The quintic nonlinearity in (0.1) is unimportant and may be replaced by u|u|p−2, p ∈ 2Z, p 4. We give a proof of the following fact.