Quasi-periodic solutions of nonlinear Schrödinger equations on ${\mathbb{T}}^d$

Abstract

We present recent existence results of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on T_d_, $d\ge 1$ , 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 $C^{\infty }$ then the solutions are in $C^{\infty }$ . The proofs are based on an improved Nash–Moser iterative scheme and a new multiscale inductive analysis for the inverse linearized operators.