AI 中文总结
本文在无衰减假设下，通过谱群作用构造三次非线性薛定谔方程的全局流，证明其在特定 Banach 空间上的全局适定性，并保持谱与概周期性质。
AI 中文摘要
对于每个正整数 $N$，我们在包含 $W^{2N+1,\infty}(\mathbb R)$ 的初始数据类上，为散焦和聚焦非线性薛定谔层级构造了全局谱群作用。对于 $N\ge2$，二次流给出全局经典三次 NLS 解。该构造将 Sato--Segal--Wilson 和 Kotani 框架推广到矩阵 Dirac 系统；一个齐次 tau 函数恒等式提供了聚焦情形所需的正值性。初始数据的包含不需要衰减、小性、周期性、算术条件或预设谱背景。作为主要应用，我们证明了在 $M_{\infty,1}^{5}(\mathbb R)$ 上对两种符号的全局适定性，具有有限时间界和对初始数据范数球的局部 Lipschitz 依赖性。过渡到该 Banach 空间流使用了近实轴 Weyl 渐近以及紧初始数据集上归一化 Toeplitz 算子的连续性，从而从五个有界导数获得一致空间界。在散焦情形中，该流可延拓到 $M_{\infty,1}^{5}(\mathbb R)+H^1(\mathbb R)$。进一步的结果包括定量逼近、空间 Bohr 概周期及其频率模的保持、空间平移包的共轭性，以及完整复 Dirac 谱和空间转移增长率的保持。
英文摘要
For each positive integer $N$, we construct global spectral group actions for the defocusing and focusing nonlinear Schrödinger hierarchies on initial-data classes containing $W^{2N+1,\infty}(\mathbb R)$. For $N\ge2$, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato--Segal--Wilson and Kotani frameworks to matrix Dirac systems; a homogeneous tau-function identity supplies the focusing positivity. No decay, smallness, periodicity, arithmetic condition, or prescribed spectral background is required for the initial-data inclusion. As a principal application, we prove global well-posedness on $M_{\infty,1}^{5}(\mathbb R)$ for both signs, with finite-time bounds and locally Lipschitz dependence on initial-data norm balls. The passage to this Banach-space flow uses near-real-axis Weyl asymptotics and continuity of normalized Toeplitz operators on compact initial-data sets to obtain uniform spatial bounds from five bounded derivatives. In the defocusing case, the flow extends to $M_{\infty,1}^{5}(\mathbb R)+H^1(\mathbb R)$. Further consequences include quantitative approximation, preservation of spatial Bohr almost periodicity and its frequency module, conjugacy of spatial translation hulls, and preservation of the full complex Dirac spectrum and spatial transfer growth rates.
