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.

中文翻译：

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关于波导$ \ mathbb R ^ 2 \ times \ mathbb T $上三次离焦非线性Schrödinger方程的散射

在本文中，我们将展示三次离焦非线性Schrödinger方程在$ H ^ 1 $中在波导$ \ mathbb {R} ^ 2 \ times \ mathbb {T} $上的散射。我们首先在$ H ^ {1}（\ mathbb {R} ^ 2 \ times \ mathbb {T}）$中建立线性轮廓分解，该分解是由质量关键的Schrödinger方程在$ L ^ 2（ \ mathbb {R} ^ 2）$。然后，通过使用三次共振非线性Schrödinger系统的解来近似非线性轮廓，我们可以使用浓度-紧致度/刚度方法证明$ H ^ 1 $中的散射。