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

TL;DR

This paper develops a modified scattering theory for a wave guide nonlinear Schrödinger equation, demonstrating the existence of global solutions with unbounded Sobolev trajectories, advancing understanding of long-term dynamics in such systems.

Contribution

It introduces a novel modified scattering framework for the wave guide NLS and proves the existence of solutions with unbounded Sobolev norms, extending previous results.

Findings

Established a modified scattering theory for (WS)

Proved existence of global solutions unbounded in Sobolev spaces

Connected scattering results with unbounded Sobolev trajectories

Abstract

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 $s\textgreater{}\frac 12$.