Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schr\"odinger equation

We consider the following wave guide nonlinear Schr\"odinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T} \_y$. We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szeg\H{o} equation. The proof is an adaptation of the method of Hani--Pausader--Tzvetkov--Visciglia. Combining this scattering theory with a recent result by G\'erard--Grellier, we infer existence of global solutions to (WS) which are unbounded in the space $L^2\_xH^s\_y(\mathbb{R} \times \mathbb{T} )$ for every $s\textgreater{}\frac 12$.