Well-Posedness for Integrable Dispersive Partial Differential Equations

KDV方程是在19世纪90年代引入的，用来解释对浅水航道表面孤立波的实验观测。这些波在保持其轮廓的同时，会传播很远的距离。更令人惊讶的是，这样的波发生了粒子状的相互作用；这促使研究人员引入了“孤子”一词来描述它们。这个项目的目标是加深我们对其他表现出相同现象学的方程的理解。该项目试图克服的一个特别挑战是，现有方法不太适合研究在空间中没有很好定位的波。该项目为研究生提供了研究训练的机会，所要研究的问题是非线性色散偏微分方程组、完全可积系统和概率论的交集。这些问题是公认的利益问题，之所以选择这些问题，既是因为它们抵制以前的技术，也是因为首席研究员相信，她帮助开发的新技术最终使它们变得容易获得。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，周期和潮汐完全可积系统的适定性，为Landau-Lifshitz模型构建Gibbs动力学，以及了解散焦和聚焦完全可积系统解的长期行为。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.