Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation

We consider the following wave guide nonlinear Schrödinger equation, WS(i∂t+∂xx-|Dy|)U=|U|2U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (i\partial _t+\partial _{xx}-\vert D_y\vert )U=\vert U\vert ^2U\ \end{aligned}$$\end{document}on the spatial cylinder Rx×Ty\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}_x\times {\mathbb T}_y$$\end{document}. We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. The proof is an adaptation of the method of Hani et al. (Modified scattering for the cubic Schrödinger equation on product spaces and applications, 2015. arXiv:1311.2275v3). Combining this scattering theory with a recent result by Gérard and Grellier (On the growth of Sobolev norms for the cubic Szegő equation, 2015), we infer existence of global solutions to (WS) which are unbounded in the space Lx2Hys(R×T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2_xH^s_y({\mathbb R}\times {\mathbb T})$$\end{document} for every s>12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>\frac{1}{2}$$\end{document}.