Integrable PDEs beyond standard assumptions on initial data

本项目致力于研究孤子理论的一些基本问题。孤子是一种特殊类型的波，在通过各种介质时表现出显著的稳定性。例如，光纤中的海啸波和脉冲等众所周知的现象。斯科特·罗素于1834年首次对孤子进行了观测和科学描述。描述罗素所观察到的东西的方程是由Korteweg和de Vries在1895年推导出来的，但直到1967年，这个方程，现在被称为Korteweg-de Vries(KdV)，才被Gardner，Greene，Kruskal和Miura以封闭的形式求解。他们的方法被认为是20世纪科学的重大成就。它产生了孤子理论，适用于广泛类别的物理上重要的演化偏微分方程组，范围从水波的流体力学(海洋中的无赖波)和非线性光学(光纤中的信息传播)到天体物理、大气科学和基本粒子理论。这个项目将开发新的方法，将该理论扩展到物理上和实际上重要的缓慢衰减波的情况，这些情况仍然超出了当前方法的范围。该项目将有一个非常大的教育组成部分。这位研究人员致力于继续他在本科生项目中关于非线性波动现象的研究经验。该计划旨在确定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。可积系统主要研究与快速衰减或周期性初始数据引发的波的传播有关的问题。在KdV背景下，相应的解具有相对简单和易于理解的运行孤子伴随衰减波的辐射的波结构，或周期波列及其调制。然而，任何与这些数据的偏差都会遇到主要的困难。主要的推力将放在理解空间加无限大的较慢衰变(甚至没有衰变)的影响上。物理动机包括模拟流浪波、非线性波在具有缓慢衰减幅度的(伪)周期介质中的传播、可积湍流以及相干结构在噪声介质中的传播。从数学的角度来看，这是一个未知的领域。正无穷大的衰变速度较慢，在IST的每一步都会导致严重的并发症。主要的努力将放在理解如何使Riemann-Hilbert问题的方法--一种现代强大的渐近分析机器--远远超出经典问题的领域。为此，需要发展长程势的正/逆散射理论。研究人员希望找到具有深远实际应用的新型解决方案，包括但不限于对流氓波的理解，不同背景下的孤子传播，以及在噪声介质中出现的更一般相干结构的传播研究，这些学科出现在流体力学、电信、大气科学、非线性光学、等离子体、天体物理学和其他自然存在可积系统的领域。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to the study of some fundamental problems of soliton theory. A soliton is a special type of wave that shows a remarkable stability when traveling through various media. Examples include such well-known phenomena as tsunami waves and pulses in optical fibers. The first observation and scientific description of a soliton was given by Scott Russell in 1834. The equation describing what Russel had observed was derived in 1895 by Korteweg and de Vries but it was not until 1967 when this equation, now called Korteweg-de Vries (KdV), was solved in closed form by Gardner, Greene, Kruskal, and Miura. Their method is regarded as a major achievement of the 20th century science. It gave rise to soliton theory, applicable to broad classes of physically important evolution partial differential equations, ranging from hydrodynamics of water waves (rogue waves in the ocean) and nonlinear optics (propagation of information in optical fibers) to astrophysics, atmospheric sciences, and elementary particle theory. This project will develop novel approaches to extend the theory to the physically and practically important cases of slowly decaying waves, which are still beyond the reach of the current methods. The project will have a very large educational component. The investigator is committed to continuing his research experience for undergraduates program on nonlinear wave phenomena. This program is designed 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.Integrable systems have been primarily studied in the connection with propagation of waves initiated from rapidly decaying or periodic initial data. In the KdV context, 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 such data meets principal difficulties. The main thrust will be put on understanding of the effect of slower decay (or even no decay) at spatial plus infinity. Physical motivations include modeling rogue waves, nonlinear wave propagation in (pseudo) periodic media with slowly decaying amplitude, integrable turbulence, and propagation of coherent structures in noisy media. From the mathematical viewpoint, it is an uncharted territory. Slower decay at plus infinity causes serious complications at every step of the IST. The main effort will be put on understanding how to make the method of the Riemann-Hilbert problem, a modern powerful machinery of asymptotic analysis, work far outside of the realm of classical problems. To this end, developing direct/inverse scattering theory for long-range potentials will be required. The investigator expects to find new types of solutions with far-reaching practical applications, which include, but not limited to, the understanding of rogue waves, soliton propagation on different backgrounds, and the study of propagation of more general coherent structures in noisy media appearing in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, and other areas where integrable systems naturally arise.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.