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.

该项目致力于研究波在各种介质中传播的一些基本问题。这项工作的主要方法来自于孤子理论。孤子是一种非常特殊的孤立的水波（“波”），它以恒定的速度移动，而不会改变其形状。 1834 年，苏格兰造船工程师约翰·斯科特·拉塞尔 (John Scott Russell) 描述了第一个孤子定位，他在海峡中注意到了这种波，并骑着马追踪了它很长一段时间。孤子理论起源于 20 世纪 60 年代中期，源于 Gardner-Greene-Kruskal-Miura 对浅水波 Korteweg-de Vries (KdV) 方程的逆散射变换 (IST) 的基本发现（该方程恰好描述了罗素观察到的通道现象）。此后不久，针对许多其他物理上重要的非线性演化偏微分方程（PDE）（称为完全可积系统）发现了不同版本的 IST。 IST 在概念上类似于傅里叶变换，它产生了大量有关完全可积系统的信息，远远超出了标准 PDE 技术所能提供的信息。孤子理论被认为是 20 世纪科学的一项重大成就，它连接了纯数学和理论物理学的不同分支，具有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的众多应用。孤子理论的大部分工作都是关于由快速衰减或周期性初始数据（所谓的经典数据）引发的波的传播。相应的解决方案具有相对简单且易于理解的运行孤子的波结构，伴随着衰变波的辐射，或周期性波列及其调制。然而，任何与经典数据的偏差都会遇到尚未克服的主要困难。该项目将重点关注比经典更广泛的初始轮廓的孤子理论。我们期待新型解决方案具有更复杂的波浪结构和深远的实际应用。预计该结果可用于理解异常波、不同背景（包括噪声）下的孤子传播、潮汐波、某些气象现象（例如牵牛花），或用于研究相干结构在噪声介质中的传播。流体动力学、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。在 KdV 方程的背景下，首席研究员用 Hankel 算子重新表述了经典 IST Titchmarsh-Weyl m 函数可让人们将 IST 扩展到令人惊讶的广泛类别的初始数据。这是通过利用 m 函数的一些微妙属性和 Hankel 算子理论的深层结果来实现的。首席研究员计划使用 Hankel 算子的强大方法来识别最广泛的初始剖面类别，其中存在 IST 的合适模拟。另一个目标是对底层解决方案进行渐近分析。众所周知的经典黎曼-希尔伯特问题的强大机制在许多严重的方面在这种初始轮廓上崩溃了。主要重点将放在理解如何使黎曼-希尔伯特问题远远超出经典问题的范围。预计结果将对各种应用有所帮助。 随之而来的数学问题对于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，并且可能对这两种理论产生深远的影响。 该项目将具有非常大的教育成分。首席研究员致力于继续他在非线性波现象本科生项目中的研究经验，以识别和指导应用数学领域的年轻学者。他的目的是吸引多元化（性别、种族、残疾）的才华横溢的本科生加入该计划，以扩大数学科学群体中代表性不足的人的参与。