Collaborative Research: Spectra of Linear Differential Operators and Turbulence in Integrable Systems

合作研究：线性微分算子谱和可积系统中的湍流

基本信息

批准号：
2039071
负责人：
Sergey Dyachenko
金额：
$ 2.31万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-15 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2039071&HistoricalAwards=false
关键词：
Collaborative Research Spectra Linear Differential

项目摘要

Nonlinear evolution equations, also known as soliton equations, are used to model a wide variety of physical systems, such as ocean waves or fiber optic communications. Many such equations, for example the Korteweg-de Vries (KdV) equation for waves in shallow water, have the exciting feature that many of their solutions can be given exactly by an explicit formula. This allows one to develop a precise understanding of the underlying physical processes. However, the range of conditions for which exact solutions are currently known is restrictive, inhibiting the use of such solutions in real-world applications. This project aims to find larger families of exact solutions to soliton equations, and use these solutions to develop statistical theories of the corresponding physical systems. The main goal of the project is to construct and study new families of solutions of soliton equations such as KdV, Nonlinear Schrödinger, and Kadomtsev-Petviashvili. These solutions are obtained as limits of multisoliton solutions, and are bounded and non-decreasing at infinity. They are described by a Riemann-Hilbert problem and can be efficiently computed numerically. The first goal is a rigorous mathematical description of these new solutions. The PIs will investigate to what extent these solutions solve the initial value problem for KdV and related systems. They will study spectral properties of the associated linear operators and construct non-periodic one-dimensional ideal conductors. Finally, the PIs will develop a statistical theory of integrable turbulence for KdV and other soliton equations.

非线性进化方程（也称为固体方程）用于对多种物理系统（例如海浪或光纤通信）进行建模。许多这样的方程，例如浅水中波的korteweg-de Vries（KDV）方程，具有令人兴奋的特征，其许多解决方案可以通过明确的公式给出。这使人们能够对基本的物理过程进行精确的理解。但是，当前已知确切解决方案的条件范围是限制性的，从而抑制了在现实世界应用中使用此类溶液的使用。该项目旨在寻找较大的唯一方程式精确解决方案家族，并使用这些解决方案来开发相应的物理系统的统计理论。该项目的主要目的是构建和研究孤子方程解决方案的新家庭，例如KDV，NonlinearSchrödinger和Kadomtsev-Petviashvili。这些溶液作为多层溶液的限制获得，并在无穷大时被界定和不稳定。它们是由Riemann-Hilbert问题描述的，可以单独计算。第一个目标是对这些新解决方案的严格数学描述。 PI将在多大程度上研究这些解决方案解决KDV和相关系统的初始值问题。他们将研究相关线性操作员的光谱特性，并构建非周期性的一维理想导体。最后，PI将开发针对KDV和其他孤子方程的湍流的统计理论。