Revista Matemática Iberoamericana


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Published online first: 2020-01-10
DOI: 10.4171/rmi/1155

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

Xing Cheng, Zihua Guo[1], Kailong Yang[2] and Lifeng Zhao[3]

(1) Monash University, Clayton, Australia
(2) University of Sciences and Technology, Beijing, China
(3) University of Scinece and Technology of China, Hefei, China

In this article, we will show the scattering of the cubic defocusing nonlinear Schrödinger equation on the 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 cubic resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method.

Keywords: Nonlinear Schrödinger equation, scattering, profile decomposition, waveguide

Cheng Xing, Guo Zihua, Yang Kailong, Zhao Lifeng: On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$. Rev. Mat. Iberoam. Electronically published on January 10, 2020. doi: 10.4171/rmi/1155 (to appear in print)