Integrable Partial Differential Equations Beyond Standard Assumptions on Initial Data

本项目研究孤立子理论中的一些基本问题，这些问题涉及各种介质中的非线性波传播。孤立波是一种非常特殊的孤立波，它以恒定的速度运动而不改变形状;最突出的例子是海啸波。1834年，斯科特·罗素（Scott Russell）首次对孤子进行了科学描述。方程描述了什么罗素观察到的是在1895年由Korteweg和德弗里斯（KdV），但它不是直到1967年时，KdV方程是解决封闭形式。逆散射变换（IST）是20世纪世纪数学的一项重要成就。它产生了孤立子理论，该理论处理广泛的物理重要的微分方程，这些方程可以通过合适的IST来求解（这些方程也称为完全可积系统）。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。可积系统主要是在研究由快速衰减或周期性初始数据（“经典”数据）引发的波的传播。相应的解具有一个相对简单的行波孤子的波结构，伴随着衰减波的辐射，或周期波列及其调制。然而，任何偏离经典数据的情况都会导致根本性的困难;克服这些困难是该项目的主要重点。预计将出现具有更复杂的波结构和更广泛的实际应用的新类型的解决方案。这一结果可用于理解不同背景下的流氓波、孤子传播等（包括噪音）、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（或在一般的波设置），在流体力学，电信，大气科学，非线性光学，等离子体物理学，天体物理学，该项目将有一个非常大的教育组成部分。首席研究员（PI）将继续他的本科生计划的研究经验，以确定和指导应用数学领域的年轻学者。在KdV设置PI已重新制定了IST的汉克尔运营商和Weyl m-功能。它允许将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识别最广泛的初始数据类别，其中存在合适的IST模拟。另一个目标是渐近分析的基本解决方案。最强大的方法是基于Riemann-Hilbert（RH）问题，该问题也以许多严重的方式分解了这些初始数据。主要的推力将放在理解如何使RH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。伴随的数学问题也是非常重要的理论薛定谔运营商，理论的汉克尔和Toeplitz运营商，基本对象的运营商理论。揭示孤立子理论和汉克尔算子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.