Well-posedness and Long-time Behavior of Dispersive Integrable Systems

可积系统长期以来一直是研究哈密顿偏微分方程的向导。它们是真实物理系统的有效模型，包括在光学和多体量子力学中。在完全可积系统的背景下，孤子和多孤子首次被发现。从那以后，这些结构在应用科学中得到了许多应用：例如，在光纤中，孤子被用于远距离数字信号的传输，而在生物学中，它们被用来描述神经系统中的信号传播和蛋白质中的低频集体运动。该项目旨在研究长期存在的和新引入的可积模型。具体来说，我们试图找到初始状态的最小条件，在此条件下可以构建全局实时动力学，研究特殊结构（如孤子和多孤子）的稳定性，并阐明一般解的长时间行为。该项目为研究生提供了重要的研究训练机会，他们融入了项目的主要目标。该项目研究了新引入的连续统Calogero-Moser方程的以下具体问题：(1)尺度不变空间中的大数据适定性；(2)散焦模型和聚焦方程中质量小于基态孤子的初始数据的散射；(3)聚焦情况下爆炸阈值的确定。进一步的目标包括Benjamin-Ono方程在最优适定空间中多孤子解的轨道稳定性和渐近稳定性，该方程大解的孤子分量的色散衰减，以及Landau-Lifshitz模型的Gibbs动力学的构建。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Integrable systems have long served as guides in the study of Hamiltonian partial differential equations. They arise as effective models of real physical systems, including in optics and many-body quantum mechanics. It is in the setting of completely integrable systems that solitons and multisolitons were first discovered. These structures have since found numerous applications in the applied sciences: for example, in fiber optics, solitons have been employed in the transmission of digital signals over long distances, while in biology, they are used to describe signal propagation in the nervous system and low-frequency collective motion in proteins. This project seeks to investigate both longstanding and newly introduced integrable models. Specifically, we seek to find the minimal conditions on the initial state under which one can construct global-in-time dynamics, investigate the (in)stability of special structures (such as solitons and multisolitons), and elucidate the long-time behavior of general solutions. The project provides significant research training opportunities for graduate students, who are integrated into the main objectives of the project. The project investigates the following specific questions for the newly introduced continuum Calogero-Moser equations: (1) large data well-posedness in the scaling-invariant space, (2) scattering for both the defocusing model and the focusing equation for initial data with mass less than that of the ground state soliton, and (3) the determination of the blowup threshold in the focusing case. Further objectives include orbital and asymptotic stability of multisoliton solutions to the Benjamin-Ono equation in optimal well-posedness spaces, dispersive decay away from the soliton component for large solutions to this equation, and the construction of Gibbs dynamics for the Landau-Lifshitz model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.