Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

Abstract

In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on “waveguide” manifolds, such as $\mathbb{R}^2\times \mathbb{T}$ in [X. Cheng et al., On Scattering for the Cubic Defocusing Nonlinear Schrödinger Equation on Waveguide $\mathbb{R}^2 \times \mathbb{T}$, preprint, arXiv:1705.00954 [math.AP], 2017]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in Cheng et al. closed as crucial ingredients for compactness of the critical elements.