Well-Posedness for Integrable Dispersive Partial Differential Equations

可积色散偏微分方程的适定性

• 批准号: 2054194
• 负责人: Monica Visan
• 金额: $ 29.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054194&HistoricalAwards=false
• 关键词: Well; Posedness; Integrable; Dispersive; Partial

The Korteweg-de Vries equation was introduced in the 1890s to explain the experimental observation of solitary waves on the surface of shallow channels of water. These waves travel large distances while maintaining their profile. Still more astounding is the fact that such waves undergo particle-like interactions; this prompted researchers to introduce the term "soliton" to describe them. The goal of this project is to deepen our understanding of other equations that exhibit the same phenomenology. One particular challenge that the project seeks to overcome is the fact that existing methods are poorly suited to the study of waves that are not well localized in space. The project provides research training opportunities for graduate students.The problems to be studied lie at the intersection of nonlinear dispersive PDE, completely integrable systems, and probability theory. These problems are of well-established interest and chosen both because they have resisted previous technology and because the principal investigator believes that the new techniques she helped develop make them finally accessible. Among the topics that will be investigated as part of the project are large-data sharp well-posedness for the derivative nonlinear Schrodinger equation, well-posedness for periodic and tidal completely integrable systems, constructing Gibbs dynamics for the Landau-Lifshitz model, and understanding the long-time behavior of solutions to both defocusing and focusing completely integrable systems.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.

Korteweg-de Vries 方程于 1890 年代引入，用于解释浅水通道表面孤立波的实验观察。这些波在保持其轮廓的同时传播很远的距离。更令人震惊的是，这种波会发生类似粒子的相互作用。这促使研究人员引入“孤子”一词来描述它们。该项目的目标是加深我们对表现出相同现象的其他方程的理解。该项目试图克服的一个特殊挑战是现有方法不太适合研究空间中不太局域化的波。该项目为研究生提供研究训练机会。要研究的问题是非线性色散偏微分方程、完全可积系统和概率论的交叉点。这些问题引起了广泛的兴趣，之所以选择这些问题，既是因为它们抵制了以前的技术，也因为首席研究员相信她帮助开发的新技术使它们最终变得可行。作为该项目的一部分，将研究的主题包括导数非线性薛定谔方程的大数据尖锐适定性、周期性和潮汐完全可积系统的适定性、为朗道-利夫什茨模型构建吉布斯动力学以及理解散焦和聚焦完全可积系统的解决方案的长期行为。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。