On scattering for the cubic defocusing nonlinear Schr\"odinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$

TL;DR

This paper proves global well-posedness and scattering for the cubic defocusing nonlinear Schrödinger equation on a waveguide structure, extending understanding of dispersive PDEs on mixed Euclidean and periodic domains.

Contribution

It introduces a linear profile decomposition in $H^1$ for the waveguide setting and employs a vector-valued resonant system to establish scattering results.

Findings

Proved global well-posedness in $H^1$ for the equation.

Established scattering in $H^1$ for the waveguide domain.

Developed a new profile decomposition technique for mixed domains.

Abstract

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schr\"odinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{ 1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schr\"odinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schr\"odinger system to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method.