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

Abstract

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.

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