Integrable Partial Differential Equations Beyond Standard Assumptions on Initial Data

超出初始数据标准假设的可积偏微分方程

基本信息

批准号：
1716975
负责人：
Alexei Rybkin
金额：
$ 23.96万
依托单位：
University of Alaska Fairbanks Campus
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：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716975&HistoricalAwards=false
关键词：
Integrable Partial Differential Equations Beyond

项目摘要

This project is concerned with some fundamental problems of soliton theory which deals with nonlinear wave propagation in various media. Solitons are very special solitary waves that move with constant speed without changing their shape; the most prominent example is a tsunami wave. The first scientific description of a soliton was given in 1834 by Scott Russell. The equation describing what Russel had observed was derived in 1895 by Korteweg and de Vries (KdV), but it was not until 1967 when the KdV equation was solved in closed form. The method of solution, the inverse scattering transform (IST), is regarded as a major achievement of the 20th century mathematics. It gave rise to soliton theory that is dealing with broad classes of physically important differential equations which can be solved by a suitable IST (such equations are also called completely integrable systems). The range of applications is enormous: from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Integrable systems have been primarily studied in the connection with propagation of waves initiated by rapidly decaying or periodic initial data (the "classical" data). The corresponding solutions have a relatively simple wave structure of traveling solitons accompanied by radiation of decaying waves, or periodic wave-trains and their modulations. However, any deviation from classical data leads to fundamental difficulties; it is the main focus of the project to overcome them. Entirely new types of solutions, with much more complicated wave structure and far-reaching practical applications, are expected to arise. The results could be used for understanding rogue waves, soliton propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena (i.e. morning glory), or the study of propagation of coherent structures in noisy media (or in a general wave setting), in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma physics, astrophysics, etc. The project will have a very large educational component. The principal investigator (PI) will continue his research experience for undergraduates program to identify and mentor young scholars in the field of applied mathematics. In the KdV setting the PI has reformulated the IST in terms of Hankel operators and Weyl m-functions. It lets one extend the IST to a surprisingly broad class of initial data. The PI plans to continue using these powerful tools to identify the broadest possible class of initial data for which a suitable analog of the IST exists. Another objective is asymptotic analysis of the underlying solutions. The most powerful approach is based on the Riemann-Hilbert (RH) problem which also breaks down on such initial data in a number of serious ways. The main thrust will be put on understanding how to make the RH problem work far outside of the realm of classical problems. The results are expected to be instrumental for various applications. The accompanying mathematical problems are also very important to the theory of the Schrödinger operator, and the theory of Hankel and Toeplitz operators, fundamental objects of operator theory. Uncovering connections between soliton theory and Hankel operators is of great independent interest and could potentially have a profound influence on both theories.

该项目涉及孤子理论的一些基本问题，孤子理论涉及各种介质中的非线性波传播，孤子是一种非常特殊的孤波，它们以恒定的速度移动而不改变其形状；最突出的例子是海啸波。斯科特·拉塞尔 (Scott Russell) 于 1834 年给出了描述拉塞尔观察到的现象的方程，该方程由 Korteweg 和 de Vries (KdV) 于 1895 年导出，但直到 1895 年才得到。 1967 年，KdV 方程以封闭形式求解，逆散射变换 (IST) 被认为是 20 世纪数学的一项重大成就，它催生了涉及广泛物理领域的孤子理论。可以通过合适的 IST 求解的重要微分方程（此类方程也称为完全可积系统），其应用范围非常广泛：从流体动力学和非线性光学到天体物理学和基本粒子。理论。可积系统主要研究了由快速衰减或周期性初始数据（“经典”数据）引发的波传播，相应的解决方案具有相对简单的行进孤子波结构，并伴随着衰减波的辐射。或周期性波列及其调制，但是，任何与经典数据的偏差都会导致根本性的困难；克服它们是该项目的主要焦点，具有更复杂的波结构和。预计这些结果将产生深远的实际应用，可用于理解异常波、不同背景下的孤子传播（包括噪声）、潮汐波、某些气象现象（即牵牛花）或相干传播的研究。噪声介质（或一般波环境）中的结构，涉及流体动力学、电信、大气科学、非线性光学、等离子体物理学、天体物理学等不同学科。该项目将首席研究员 (PI) 将继续本科生项目的研究经验，以识别和指导应用数学领域的年轻学者。 Weyl m 函数使人们能够将 IST 扩展到令人惊讶的广泛类别的初始数据。 PI 计划继续使用这些强大的工具来识别最广泛的初始数据类别，其中存在 IST 的合适模拟。是渐近的最强大的方法是基于 Riemann-Hilbert (RH) 问题，该问题也以多种严肃的方式分解此类初始数据。其结果远远超出了经典问题的范围，预计对各种应用都有帮助，伴随的数学问题对于薛定谔算子的理论以及算子的基本对象汉克尔和托普利茨算子的理论也非常重要。发现联系。孤子理论和汉克尔算子之间的关系具有很大的独立意义，并且可能对这两种理论产生深远的影响。