On Scattering for the Defocusing Quintic Nonlinear Schrödinger Equation on the Two-Dimensional Cylinder
Xing Cheng, Zihua Guo, Zehua Zhao
Abstract
In this article, we prove global well-posedness and scattering for the defocusing quintic nonlinear Schrödinger equation on the cylinder $\mathbb{R} \times \mathbb{T}$ in $H^1$. We establish an infinite vector-valued linear profile decomposition in $L^2_x h^\alpha$, $0 < \alpha \le 1$, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^d )$, $d\ge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques established by Dodson, to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration compactness/rigidity method. As a by-product of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrödinger system, we also prove the scattering conjecture for the two-discrete-component quintic resonant nonlinear Schrödinger system presented by Hani and Pausader in [Comm. Pure Appl. Math., 67 (2014), pp. 1466--1542].