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。孤立子是一种特殊的波，在各种介质中传播时表现出显著的稳定性。例子包括众所周知的现象，如海啸波和光纤中的脉冲。1834年，斯科特·罗素（Scott Russell）首次对孤子进行了观测和科学描述。方程描述了罗素观察到的是在1895年得出的Korteweg和德弗里斯，但直到1967年时，这个方程，现在称为Korteweg-德弗里斯（KdV），是解决了封闭形式的加德纳，格林，克鲁斯卡尔，和三浦。他们的方法被认为是世纪科学的一项重大成就。它产生了孤立子理论，适用于广泛的物理重要的演化偏微分方程，从水波的流体力学（海洋中的流氓波）和非线性光学（光纤中的信息传播）到天体物理学，大气科学和基本粒子理论。该项目将开发新的方法，将理论扩展到物理和实际上重要的缓慢衰减波的情况，这仍然超出了目前的方法。该项目将有一个非常大的教育组成部分。研究者致力于继续他的研究经验，为大学生计划的非线性波动现象。该计划旨在识别和指导应用数学领域的年轻学者。他的目的是吸引一个多样化的（性别，种族，残疾）有才华的本科生群体进入该计划，以扩大在数学科学团体中代表性不足的参与。可积系统主要研究了从快速衰减或周期性初始数据开始的波的传播。在KdV的背景下，相应的解决方案有一个相对简单和很好理解的波结构的运行孤子伴随着辐射的衰减波，或周期性的波列及其调制。然而，任何偏离这些数据的情况都会遇到主要困难。主要的推力将放在理解的影响，较慢的衰变（甚至没有衰变）在空间正无穷大。物理动机包括建模流氓波，非线性波传播（伪）周期性介质中的振幅缓慢衰减，可积湍流，相干结构在嘈杂的介质中的传播。从数学的角度来看，这是一个未知的领域。在正无穷远处的慢衰变在IST的每一步都引起严重的并发症。主要的努力将放在理解如何使黎曼-希尔伯特问题的方法，一个现代强大的机器渐近分析，工作远远超出了经典问题的领域。为此，将需要发展长程电位的直接/反向散射理论。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，以及对噪声介质中更一般相干结构传播的研究，这些噪声介质出现在流体力学，电信，大气科学，非线性光学，等离子体，天体物理学，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.