AI 中文总结
本文对包含中间型NLS与Calogero–Moser导数NLS的一类方程，基于Ozawa和Tsutsumi的双线性Strichartz估计，证明了$L^2$下的小数据整体适定性及特定条件下的散射结果，且不依赖方程可积性。
AI 中文摘要
本文以统一方式研究一类非线性薛定谔方程（NLS），该类包含中间型NLS（INLS）与Calogero–Moser导数NLS（CM-DNLS）两个重要例子。主要结果分为两部分：第一，对一大类方程证明$L^2(\boldsymbol{R})$下的小数据整体适定性，包括所有参数选择下的聚焦型、散焦型CM-DNLS及INLS；第二，在额外假设下证明$L^2(\boldsymbol{R})$下的小数据散射，该结果覆盖特定参数选择下的聚焦型、散焦型CM-DNLS及INLS。问题的表述（含解的概念）与证明均关键依赖含粗糙时变势的薛定谔方程线性理论，因允许的势粗糙到标准Duhamel公式可能无意义，该理论本身也具独立研究价值。本文采用微扰方法，基于Ozawa与Tsutsumi于1998年证明的双线性Strichartz估计构建，论证不依赖可积性。
英文摘要
In this paper, we study a class of nonlinear Schrödinger equations (NLS) in a unified way. This class includes two important examples: the intermediate NLS (INLS) and the Calogero--Moser derivative NLS (CM-DNLS). Our main results are twofold. First, we prove small-data global well-posedness in $L^2(\mathbb{R})$ for a broad class of equations. This includes both focusing and defocusing CM-DNLS and the INLS for arbitrary choices of its parameters. Second, we prove small-data scattering in $L^2(\mathbb{R})$ under an additional assumption. This result covers both focusing and defocusing CM-DNLS and the INLS for specific choices of its parameters. Both the formulation of the problem, including the notion of solution, and the proofs rely crucially on a linear theory for Schrödinger equations with rough time-dependent potentials. This theory is also of independent interest, since we allow potentials so rough that the standard Duhamel formulation may not make sense. Our approach is perturbative and is built on the bilinear Strichartz estimate proved by Ozawa and Tsutsumi in 1998. In particular, our argument does not rely on integrability.
Comments32 pages
