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

Abstract

We consider the following wave guide nonlinear Schrödinger 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 Szegő equation. The proof is an adaptation of the method of Hani--Pausader--Tzvetkov--Visciglia. Combining this scattering theory with a recent result by Gérard--Grellier, we infer existence of global solutions to (WS) which are unbounded in the space $L^2\_x{H}^s\_y(\mathbb{R}\times \mathbb{T})$ for every $s\textgreater{}\frac 12$.

arXiv admin note: text overlap with arXiv:1311.2275, arXiv:1408.6213 by other authors