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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：
1716822
负责人：
Sergey Dyachenko
金额：
$ 12.76万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716822&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.

非线性演化方程，也称为孤子方程，用于模拟各种物理系统，例如海浪或光纤通信。许多这样的方程，例如浅水中波浪的 Korteweg-de Vries (KdV) 方程，具有令人兴奋的特征，即它们的许多解可以通过显式公式精确给出。这使得人们能够准确地理解潜在的物理过程。然而，目前已知的精确解决方案的条件范围是有限的，限制了此类解决方案在实际应用中的使用。该项目旨在找到更多的孤子方程精确解，并使用这些解来发展相应物理系统的统计理论。该项目的主要目标是构建和研究新的孤子方程组解，例如 KdV、非线性薛定谔和 Kadomtsev-Petviashvili。这些解是作为多孤子解的极限而获得的，并且是有界的且在无穷远处不递减。它们由黎曼-希尔伯特问题描述，并且可以有效地进行数值计算。第一个目标是对这些新解决方案进行严格的数学描述。 PI 将调查这些解决方案在多大程度上解决了 KdV 及相关系统的初始值问题。他们将研究相关线性算子的光谱特性并构造非周期一维理想导体。最后，PI 将开发 KdV 和其他孤子方程的可积湍流统计理论。