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

• 批准号: 1715323
• 负责人: Vladimir Zakharov
• 金额: $ 14.24万
• 依托单位: University of Arizona
• 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 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1715323&HistoricalAwards=false
• 关键词: Collaborative; Research; Spectra; Linear; Differential

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，也被称为孤子方程，被用来模拟各种各样的物理系统，如海浪或光纤通信。许多这样的方程，例如浅水波的Korteweg-de弗里斯（KdV）方程，具有令人兴奋的特征，即它们的许多解可以由显式公式精确地给出。这使人们能够对潜在的物理过程有一个精确的理解。然而，目前已知的精确解的条件范围是限制性的，抑制了这种解在现实世界应用中的使用。本项目的主要目的是寻找孤子方程的更大的精确解族，并利用这些解发展相应物理系统的统计理论。本项目的主要目标是构造和研究新的孤子方程解族，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得到的，并且在无穷远处是有界且非减的。它们被描述为一个黎曼-希尔伯特问题，可以有效地计算数值。第一个目标是对这些新的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和相关系统的初值问题。他们将研究相关线性算子的谱特性，并构造非周期一维理想导体。最后，PI将发展KdV和其他孤子方程的可积湍流的统计理论。

项目成果

Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface

• DOI: 10.1017/jfm.2019.219
• 发表时间: 2018-09
• 期刊: Journal of Fluid Mechanics
• 影响因子: 3.7
• 作者: A. Dyachenko;P. Lushnikov;Vladimir E Zakharov

On symmetric primitive potentials

关于对称本原势

• DOI: 10.1093/integr/xyz006
• 发表时间: 2019
• 期刊: Journal of Integrable Systems
• 作者: Nabelek, Patrik;Zakharov, Dmitry;Zakharov, Vladimir
• 通讯作者: Zakharov, Vladimir

Non-periodic One-gap Potentials in Quantum Mechanics

量子力学中的非周期单能隙势

• 发表时间: 2018
• 期刊: Geometric Methods in Physics. XXXV Workshop
• 作者: Zakharov, D;Zakharov, V
• 通讯作者: Zakharov, V

Algebro-geometric finite gap solutions to the Korteweg–de Vries equation as primitive solutions

Kortewegâde Vries 方程的代数几何有限间隙解作为原始解

• DOI: 10.1016/j.physd.2020.132709
• 发表时间: 2020
• 期刊: Physica D: Nonlinear Phenomena
• 作者: Nabelek, Patrik V.

Generalized Primitive Potentials

广义原势

• DOI: 10.1134/s1064562420020258
• 发表时间: 2020
• 期刊: Doklady Mathematics
• 影响因子: 0.6
• 作者: Zakharov, V. E.;Zakharov, D. V.
• 通讯作者: Zakharov, D. V.