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 ($1\le d\le 4)$. 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 $d\ge 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\mathbb{R}\times {\mathbb{T}}^d$ (for any $d\ge 2$) with infinitely growing high Sobolev norms $H^s$.

47 pages. Minor corrections and several typos fixed