Global well-posedness and scattering for the defocusing cubic Schr\"odinger equation on waveguide $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$

TL;DR

This paper proves large data scattering for the defocusing cubic nonlinear Schrödinger equation on a waveguide, using global Strichartz estimates, resonant system approximation, and profile decomposition, assuming scattering for the 2D resonant system.

Contribution

It establishes large data scattering on waveguides by linking it to the scattering of the 2D resonant system, introducing new analytical techniques.

Findings

Proves large data scattering under certain assumptions.

Develops global Strichartz estimates for the waveguide setting.

Connects the problem to the 2D cubic resonant system.

Abstract

We consider the problem of large data scattering for the defocusing cubic nonlinear Schr\"odinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$. This equation is critical both at the level of energy and mass. The key ingredients contain a global-in-time Stricharz estimate, resonant system approximation and profile decomposition. Assuming the large data scattering for the 2d cubic resonant system, we can prove the large data scattering for this problem.