A KAM algorithm for the resonant non-linear Schrödinger equation

Abstract We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in [26] , [27] and a generalization of the quasi-Toplitz functions defined in [31] .