Calogero--Moser导数非线性薛定谔方程在临界空间中的孤子分解猜想
Soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space
- China University of Mining and Technology（中国矿业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明Calogero--Moser导数非线性薛定谔方程在临界空间中的孤子分解猜想，通过模算子框架和刚性准则统一描述全局与爆破渐近，实现最优正则性。
AI中文摘要：
我们证明了Calogero--Moser导数非线性薛定谔方程在临界空间$L^2_+(\mathbb R)$中的孤子分解猜想。对于一类包含广泛谱Dini--矩类的自然初始数据，全局流--Lax容许解分解为有限多个显式调制孤子加上色散辐射，而有限时间爆破解则分解为量子化的零载波$R$-气泡和一个强收敛的端点余项。这项工作将孤子分解推广到最优临界正则性。此前的结果要求加权的$H^{1,1}$初始数据，并且在全局情形下还要求解在所有时间内保持在$H^{1,1}$中，这相当于额外的空间衰减假设。我们论证中的一个关键要素是新的模算子理论框架，它分离离散和连续谱通道，并通过畸变傅里叶变换识别辐射轮廓。第二个要素是为离散孤子轮廓发展了若干独立的刚性准则，这避免了逆散射并处理嵌入特征值。我们的框架还给出了全局和有限时间渐近的统一描述，具有精确的质量分配和质量缺陷量子化。
英文摘要：
We prove the soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space $L^2_+(\mathbb R)$. For a natural class of initial data including a broad spectral Dini---moment class, global flow--Lax admissible solutions decompose into finitely many explicit modulated solitons plus a dispersive radiation, while finite-time blow-up solutions resolve into quantized zero-carrier $R$-bubbles and a strongly convergent endpoint remainder. This work extends soliton resolution to the optimal critical regularity. Previous results had required weighted $H^{1,1}$ initial data, and in the global case the solution was additionally required to remain in $H^{1,1}$ for all time, which amounts to an extra spatial decay assumption. One key ingredient in our argument is a new modular operator-theoretic framework that separates the discrete and continuous spectral channels and identifies the radiation profile through the distorted Fourier transform. A second ingredient is the development of several independent rigidity criteria for the discrete soliton profiles, which avoids inverse scattering and handles embedded eigenvalues. Our framework also yields a unified description of both global and finite-time asymptotics, with exact mass partition and mass-defect quantization.
