Integrable PDEs and Hankel operators

可积偏微分方程和 Hankel 算子

• 批准号: 1411560
• 负责人: Alexei Rybkin
• 金额: $ 21.3万
• 依托单位: University of Alaska Fairbanks Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411560&HistoricalAwards=false
• 关键词: Integrable; PDEs; Hankel; operators

This project is devoted to some fundamental issues of propagation of waves in various media. The principal methods of this work come from the soliton theory. Solitons are very special solitary waves ("bumps") of water that move with constant speed without any deterioration in their shape. The first soliton siting was described in 1834 by the Scottish naval architect John Scott Russell, who noticed this wave in a channel and pursued it on his horse for quite a while. The soliton theory was originated in the mid-1960s from the fundamental Gardner-Greene-Kruskal-Miura discovery of the inverse scattering transform (IST) for the Korteweg-de Vries (KdV) equation for shallow water waves (this equation happens to describe the channel phenomenon that Russell observed). Soon thereafter different versions of IST were found for many other physically important nonlinear evolution partial differential equations (PDEs) referred to as completely integrable systems. Being conceptually similar to the Fourier transform, the IST has yielded a tremendous amount of information about completely integrable systems, far beyond what standard PDE techniques may offer. Soliton theory is regarded as a major achievement of the 20th century science connecting different branches of pure mathematics and theoretical physics with numerous applications ranging from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Much of work in soliton theory has been done on the propagation of waves initiated by rapidly decaying or periodic initial data (the so-called classical data). The corresponding solutions have a relatively simple and well understood wave structure of running solitons accompanied by radiation of decaying waves, or periodic wave-trains and their modulations. However any deviation from classical data meets principal difficulties that are yet to be surmounted. The project will focus on soliton theory for initial profiles that are much broader than classical. We expect new types of solutions with much more complicated wave structure and far-reaching practical applications. It is expected that the results could be used for understanding rogue waves, soliton propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena (e.g. morning glory), or for the study of propagation of coherent structures in noisy media in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, etc. In the context of the KdV equation the principal investigator has reformulated the classical IST in terms of Hankel operators and Titchmarsh-Weyl m-functions that let one extend the IST to a surprisingly broad class of initial data. This was achieved by employing some subtle properties of the m-function and deep results from the theory of Hankel operators. The principal investigator plans to use powerful methods of Hankel operators to identify the broadest possible class of initial profiles for which a suitable analog of the IST exists. Another objective is asymptotic analysis of the underlying solutions. The well-known powerful machinery of the classical Riemann-Hilbert problem breaks down on such initial profiles in a number of serious ways. The main thrust will be put on understanding how to make the Riemann-Hilbert 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 Schrodinger operator, the cornerstone of quantum mechanics, and the theory of Hankel and Toeplitz operators, fundamental objects of operator theory. Uncovering connections between soliton theory and Hankel operators theory is of great independent interest and could potentially have a profound influence on both theories. The project will have a very large educational component. The principal investigator is committed to continuing his research experience for undergraduates program on nonlinear wave phenomena to identify and mentor young scholars in the field of applied mathematics. It is his intent to attract a diverse (gender, ethnicity, disability) group of talented undergraduates into the program to broaden the participation of underrepresented in the mathematical sciences groups.

这个项目致力于研究波在各种介质中传播的一些基本问题。本工作的主要方法来自孤立子理论。孤立子是水的非常特殊的孤立波（“颠簸”），以恒定的速度移动，形状没有任何恶化。苏格兰海军建筑师约翰·斯科特·罗素（John Scott Russell）在1834年首次描述了孤立子的位置，他注意到了海峡中的这种波，并骑着马追逐了很长一段时间。孤立子理论起源于20世纪60年代中期，Gardner-Greene-Kruskal-Miura发现了浅水波的Korteweg-de弗里斯（KdV）方程的逆散射变换（IST）（该方程恰好描述了罗素观察到的通道现象）。此后不久，不同版本的IST被发现为许多其他物理上重要的非线性演化偏微分方程（PDE）被称为完全可积系统。由于在概念上类似于傅立叶变换，IST已经产生了关于完全可积系统的大量信息，远远超出了标准PDE技术可能提供的信息。孤立子理论被认为是20世纪世纪科学的一项重大成就，它将纯数学和理论物理的不同分支与从流体力学和非线性光学到天体物理和基本粒子理论的众多应用联系起来。孤立子理论中的大部分工作都是针对由快速衰减或周期性初始数据（所谓的经典数据）发起的波的传播进行的。相应的解决方案有一个相对简单和容易理解的波结构的运行孤子伴随着辐射的衰减波，或周期性波列及其调制。然而，任何偏离经典数据的情况都会遇到尚待克服的主要困难。该项目将侧重于孤立子理论的初始配置文件，是比经典更广泛。我们期待着具有更复杂的波结构和深远的实际应用的新类型的解决方案。研究结果可为理解反常波、孤子在不同背景下的传播提供理论依据（包括噪音）、潮汐波、某些气象现象（例如牵牛花），或用于在诸如流体力学、电信、大气科学、非线性光学、等离子体、天体物理学等不同学科中的噪声介质中的相干结构的传播的研究，在KdV方程的背景下，主要研究者重新制定了经典的IST的汉克尔算子和Titchmarsh-Weyl m-函数，让一个扩展的IST到一个令人惊讶的广泛类的初始数据。这是通过使用m-函数的一些微妙性质和Hankel算子理论的深入结果来实现的。主要研究者计划使用Hankel算子的强大方法来确定最广泛的可能类别的初始配置文件，其中存在合适的IST类似物。另一个目标是渐近分析的基本解决方案。著名的强大的机械经典黎曼-希尔伯特问题打破了这样的初始配置文件在一些严重的方式。重点将放在理解如何使黎曼-希尔伯特问题的工作远远超出了经典问题的领域。预计结果将有助于各种应用。 伴随的数学问题对量子力学的基石薛定谔算子理论以及算子理论的基本对象汉克尔和托普利茨算子理论也非常重要。揭示孤立子理论和汉克尔算子理论之间的联系具有很大的独立意义，并可能对这两种理论产生深远的影响。 该项目将有一个非常大的教育组成部分。首席研究员致力于继续他的研究经验，为非线性波动现象的本科生计划，以确定和指导应用数学领域的年轻学者。这是他的意图，以吸引一个多元化的（性别，种族，残疾）有才华的本科生群体进入该计划，以扩大在数学科学团体代表性不足的参与。