MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS

Abstract

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$ . We prove modified scattering and construct modified wave operators for small initial and final data respectively ( $1\leqslant d\leqslant 4$ ). 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 $d\geqslant 2$ . As a consequence, we obtain global strong solutions (for $d\geqslant 2$ ) with infinitely growing high Sobolev norms $H^{s}$ .