Quasi-periodic solutions of Nonlinear Schrodinger on T-d

We present recent existence results of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on Td , finitely di¤erentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. The proofs are based on an improved Nash–Moser iterative scheme and a new multiscale inductive analysis for the inverse linearized operators