Modified Scattering for the Cubic Schrödinger Equation on Product Spaces and Applications

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R× Tᵈ . We prove modified scattering and construct modified wave operators for small initial and final data respectively ( 1 d 4 ). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d 2 . As a consequence, we obtain global strong solutions (for d 2 ) with infinitely growing high Sobolev norms Hˢ .