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

色散可积系统的适定性和长期行为

• 批准号: 2348018
• 负责人: Monica Visan
• 金额: $ 38.87万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348018&HistoricalAwards=false
• 关键词: Well; posedness; Long; time; Behavior

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.

长期以来，整合系统一直是对哈密顿局部差分方程的研究指南。它们是实际物理系统的有效模型，包括光学和多体量子力学。正是在完全可集成的系统的情况下，首先发现了孤子和多层。此后，这些结构在应用科学中发现了许多应用：例如，在光纤中，孤子已用于长距离传输数字信号，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低频集体运动。该项目旨在调查长期和新引入的集成模型。具体而言，我们试图在初始状态下找到最小的条件，在该状态下，人们可以构建时间全球动力学，研究特殊结构（例如孤子和多层）的（在）稳定性（在）稳定性上，并阐明一般解决方案的长期行为。该项目为研究生提供了重要的研究培训机会，这些研究生纳入了项目的主要目标。 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.进一步的目标包括多层解决方案对本杰明·索尼（Benjamin-Ono）方程的轨道和渐近稳定性，在最佳的良好空间中方程，从孤子组件中脱落了该方程式的大型解决方案的分散性衰减，以及该方程式的构建，以及通过landau-lifshitz Models的构建gibbs的构建。优点和更广泛的影响审查标准。