A Brief Description Of The Proof Of Quasi-periodic Solutions With Sobolev Regularity Of NLS On T~d With A Multiplicative Potential

In this thesis,we mainly discuss that Berti and Bolle use the improved method of Nash-Moser iterative scheme to prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential onTd,d?1,merely 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 C~? then the solutions are C~?.The proofs are based on an improved Nash-Moser iterative scheme,which assumes the weakest tame estimates for the inverse linearized operators("Green functions")along scales of Sobolev spaces.The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis.The main novelty of Berti and Bolle concerns the measure and "complexity" estimates.