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.

在1890年代引入了Korteweg-de Vries方程，以解释对浅水河流表面上孤立波的实验观察。这些波浪在维持其轮廓的同时，它们的距离很大。更令人惊讶的是，这种波发生了类似粒子的相互作用。这促使研究人员介绍了“ Soliton”一词来描述它们。该项目的目的是加深我们对表现出相同现象学的其他方程式的理解。该项目试图克服的一个特别的挑战是，现有方法不适合研究在太空中局部局限的波浪的研究。该项目为研究生提供了研究培训机会。在非线性色散PDE，完全可集成的系统和概率理论的交集中，研究的问题在于。这些问题具有良好的兴趣，并且之所以选择这两个问题是因为它们已经抵制了以前的技术，并且因为主要研究人员认为她帮助开发的新技术使它们最终易于访问。在该项目的一部分中将进行调查的主题中，有大型数据是针对衍生性非线性schrodinger方程式的较大尖锐的体积，适合定期和潮汐完全集成的系统，为Landau-Lifshitz模型构建Gibbs动力学，并理解了对decocusision和Cospote neprys neprys the nss ness的长期行为，并理解了dectory nesss的范围。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。