Integrable Partial Differential Equations Beyond Standard Assumptions on Initial Data

本项目涉及孤子理论的一些基本问题，涉及非线性波在各种介质中的传播。孤波是一种非常特殊的孤波，它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·罗素于1834年首次对孤子进行了科学描述。描述罗素所观察到的东西的方程是由Korteweg和de Vries(KdV)在1895年推导出来的，但直到1967年KdV方程才以封闭的形式求解。逆散射变换(IST)被认为是20世纪数学的一项重要成果。它导致了孤子理论的产生，它处理了一类广泛的物理上重要的微分方程，可以用合适的IST来求解(这种方程也被称为完全可积系统)。它的应用范围是巨大的：从流体力学和非线性光学到天体物理学和基本粒子理论。可积系统主要研究由快速衰减或周期性的初始数据(“经典”数据)引发的波的传播。相应的解具有相对简单的行进孤子伴随衰减波辐射的波结构，或周期波列及其调制。然而，任何与经典数据的偏差都会导致根本的困难；克服这些困难是该项目的主要重点。预计将出现具有复杂得多的波浪结构和深远的实际应用的全新类型的解决方案。这些结果可用于理解流氓波、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)，或研究相干结构在噪声介质中的传播(或在一般波环境中)，涉及不同的学科，如流体力学、电信、大气科学、非线性光学、等离子体物理、天体物理等。首席研究员(PI)将继续他的本科生项目的研究经验，以确定和指导应用数学领域的年轻学者。在KdV设置中，PI根据Hankel算子和Weyl m-函数重新制定了IST。它让人可以将IST扩展到令人惊讶的广泛的初始数据类别。PI计划继续使用这些强大的工具来确定存在合适的IST模拟的最广泛的初始数据类别。另一个目标是对潜在解决方案进行渐近分析。最强大的方法是基于Riemann-Hilbert(RH)问题，该问题也以许多严肃的方式分解这些初始数据。主要的重点将放在理解如何使相对论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薛定谔算子理论和Hankel和Toeplitz算子理论也是非常重要的，它们是算子理论的基本对象。揭示孤子理论和Hankel算符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。

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.