On invariant tori of full dimension for 1D periodic NLS

The multiplier M=(Vn) in (0.3) is to be considered as a parameter and (0.1) a parameter-dependent equation. The role of this parameter is essential to ensure appropriate nonresonance properties of the (modulated) frequencies along the iteration. In the absence of exterior parameters, these conditions need to be realized from amplitude-frequency modulation and suitable restriction of the action-variables. This problem is harder. Indeed, a fast decay of the action-variables (enhancing convergence of the process) allows less frequency modulation and worse small divisors (cf. [Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000) 1-35]).