Remark on the scattering theory of the nonlinear Schr\"odinger equation on the cylinders

TL;DR

This paper investigates the scattering behavior of the nonlinear Schrödinger equation on cylindrical domains, establishing non-existence of scattering states in the long range case and decay/scattering results for small data in the short range case.

Contribution

It provides new insights into the scattering theory of nonlinear Schrödinger equations on cylinders, distinguishing between long and short range cases.

Findings

No linear scattering states in the long range case.

Decay and scattering for small data in the short range case.

Results extend understanding of nonlinear Schrödinger dynamics on cylindrical geometries.

Abstract

In this article, we consider the nonlinear Schr\"odinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schr\"odinger equation on $\mathbb{R}^d \times \mathbb{T}$. In the short range case, we show the decay and scattering of solutions of the nonlinear Schr\"odinger equation on $\mathbb{R}^d \times \mathbb{T}$ for small data.