Quasi-periodic solutions of nonlinear Schrodinger equations on Td

We present recent existence results of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on T-d, d >= 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are in C-infinity then the solutions are in C-infinity. The proofs are based on an improved Nash-Moser iterative scheme and a new multiscale inductive analysis for the inverse linearized operators.