KAM for quasi-linear forced hamiltonian NLS

In this paper we prove the existence of quasi-periodic, small-amplitude, solutions for quasi-linear Hamiltonian perturbations of the non-linear Schroedinger equation on the torus in presence of a quasi-periodic forcing. In particular we prove that such solutions are linearly stable. The proof is based on a Nash-Moser implicit function theorem and on a reducibility result on the linearized operator in a neighborhood of zero. The proof of the reducibility relies on changes of coordinates such as diffeomorphisms of the torus, pseudo-differential operators and a KAM-reducibility arguments. Due to the multiplicity of the eigenvalues we obtain a block-diagonalization.