Well-Posedness for Integrable Dispersive Partial Differential Equations

Korteweg-de弗里斯方程在19世纪90年代被引入，用来解释浅水渠道表面孤立波的实验观测。这些波传播很远的距离，同时保持其轮廓。更令人震惊的是，这种波经历了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。这个项目的目标是加深我们对其他表现出相同现象的方程的理解。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培训的机会。要研究的问题位于非线性色散偏微分方程，完全可积系统和概率论的交叉点。这些问题都是公认的兴趣，并选择了，因为他们已经抵制以前的技术，因为首席研究员相信，她帮助开发的新技术，使他们最终访问。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，Landau-Lifshitz模型的Gibbs动力学构造，并了解长期以来，该奖项反映了NSF的法定使命，并被认为是值得的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，

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.