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

Abstract

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger 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 Schrödinger equation in $L^2({\mathbb{R}}^2)$. Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method.