Integrable PDEs and Hankel operators

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。这项工作的主要方法来自孤子理论。孤子是一种非常特殊的水的孤立波(“颠簸”)，它们以恒定的速度移动，形状没有任何恶化。苏格兰海军建筑师约翰·斯科特·拉塞尔于1834年描述了第一个孤立波的位置，他在一条海峡中注意到了这一波，并骑在马上追逐了很长一段时间。孤子理论起源于20世纪60年代中期Gardner-Greene-Kruskal-Miura发现的浅水波Korteweg-de Vries(KdV)方程的逆散射变换(IST)(该方程恰好描述了Russell观测到的信道现象)。此后不久，许多其他物理上重要的非线性发展偏微分方程(PDE)被称为完全可积系统，发现了不同版本的IST。在概念上类似于傅里叶变换，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技术所能提供的。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的不同分支，有着从流体力学、非线性光学到天体物理和基本粒子理论的众多应用。孤子理论中的许多工作都是关于由快速衰减或周期性的初始数据(即所谓的经典数据)引起的波的传播。相应的解具有相对简单和容易理解的运行孤子伴随着衰减波的辐射或周期波列及其调制的波结构。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。该项目将专注于孤子理论的初始轮廓，这些理论比经典的孤子理论要广泛得多。我们期待着具有更复杂的波结构和更深远的实际应用的新型解决方案。预计这些结果可用于理解流氓波、孤子在不同背景下的传播(包括噪声)、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，这些学科涉及流体力学、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。在KdV方程的背景下，首席研究人员根据Hankel算子和Titchmarsh-Weyl m函数重新定义了经典的IST，使人们能够将IST扩展到令人惊讶的广泛的初始数据类别。这是通过利用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。首席研究人员计划使用汉克尔算子的强大方法来确定存在合适的IST模拟的最广泛的初始轮廓类别。另一个目标是对潜在解决方案进行渐近分析。众所周知，经典黎曼-希尔伯特问题的强大机制在这样的初始轮廓上以许多严肃的方式分解。主要的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领域。预计研究结果将对各种应用具有一定的指导意义。伴随而来的数学问题对于量子力学的基石薛定谔算符理论以及算符理论的基本对象汉克尔和托普利茨算符理论也是非常重要的。揭示孤子理论和Hankel算符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。这位首席研究员致力于继续他的本科生非线性波动现象项目的研究经验，以识别和指导应用数学领域的年轻学者。他的意图是吸引不同的(性别、种族、残疾)有才华的本科生加入该计划，以扩大在数学科学小组中代表性不足的人的参与。

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.