标题： 非线性薛定谔方程在波导流形上的临界散射 显示英文标题

标题： Critical scattering for the nonlinear Schrödinger equation on waveguide manifolds

摘要： 我们研究了非线性薛定谔方程（NLS）在波导流形上临界空间中的小数据散射问题。 我们的工作主要受到Kwak和Kwon最近发表的论文\cite{KwakKwon}的启发，该论文建立了具有可能非代数非线性的周期NLS的局部适定性。 尽管我们在解决问题时采用了类似于\cite{KwakKwon}的框架，但有两个主要障碍阻止了其直接适应波导设置。 首先，Hani和Pausader引入的临界乘积空间中NLS的经典Strichartz估计具有有限端点，因此无法应用于高维波导。 其次，\cite{KwakKwon}中使用的关键分数阶论证依赖于Strichartz的一个广为人知的分数阶导数公式，该公式只能推广到Hilbert空间值，并且因此与我们的模型设置不兼容。 为了克服这些困难，我们利用Tzvetkov和Visciglia建立的各向异性Strichartz估计，发展了\cite{KwakKwon}框架的一种各向异性推广，这些估计允许几乎不受限制的端点。 我们也通过在Besov空间内采用新的插值技术解决了由模型的向量值和各向异性特性引起的若干新挑战。 作为进一步的新颖之处，我们基于经典的不动点论证提供了主要结果的新证明，不同于\cite{KwakKwon}使用的逼近方法。 最终，我们解决了波导流形上任意质量超临界非线性NLS在临界空间中的小数据散射问题。 显示英文摘要

摘要： We study the small data scattering problem in critical spaces for the nonlinear Schr\"odinger equation (NLS) on waveguide manifolds. Our work is primarily inspired by the recent paper of Kwak and Kwon \cite{KwakKwon} that established the local well-posedness of the periodic NLS with possibly non-algebraic nonlinearity. While we adopt a framework similar to \cite{KwakKwon} for our problem, two main obstacles prevent its direct adaptation to the waveguide setting. First, the classical Strichartz estimates for NLS in critical product spaces, introduced by Hani and Pausader, possess limited endpoints and are thus inapplicable to high-dimensional waveguides. Second, the crucial fractional arguments used in \cite{KwakKwon} rely on a well-known fractional derivative formula due to Strichartz, which admits only a Hilbert space-valued extension and is therefore incompatible with our model setting. To overcome these difficulties, we develop an anisotropic generalization of the framework in \cite{KwakKwon} using the anisotropic Strichartz estimates established by Tzvetkov and Visciglia, which allow for nearly unlimited endpoints. We also resolve several new challenges arising from the vector-valued and anisotropic nature of the model by employing novel interpolation techniques within Besov spaces. As a further novelty, we provide a new proof of the main result based on classical fixed point arguments, differing from the approximation methods used in \cite{KwakKwon}. Consequently, we settle the small data scattering problem in critical spaces for the NLS with arbitrary mass-supercritical nonlinearity on waveguide manifolds.