Modified scattering for the cubic Schr\"odinger equation on product spaces and applications

We consider the cubic nonlinear Schr\"odinger 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 ($1\leq d\leq 4)$. 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 $d\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\mathbb{R}\times \mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobolev norms $H^s$.