Inverse scattering transform outside of classical conditions

经典条件之外的逆散射变换

• 批准号: 2307774
• 负责人: Alexei Rybkin
• 金额: $ 27.5万
• 依托单位: University of Alaska Fairbanks Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307774&HistoricalAwards=false
• 关键词: Inverse; scattering; transform; outside; classical

The investigator will study nonlinear partial differential equations that are used in applications, to model phenomena in hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, and astrophysics. The specific focus is on the so-called completely integrable systems, an area of applied mathematics also known as soliton theory. Much of what is currently known concerns rapidly decaying or periodic initial conditions, but physics and applications call for more general non-standard situations: the thrust of this project is on understanding the effect of slower decay (or no decay) initial data. The project will improve the modeling and predicting capabilities for rogue waves, integrable turbulence, propagation of coherent structures in noisy media, tidal waves, and meteorological phenomena (i.e., morning glory). The project will also offer research and mentoring opportunities at both undergraduate and graduate levels. Recruitment efforts will be mindful to broaden research participation in mathematical science, as well as target students intending to become high school mathematics teachers. The project studies in the context of the Korteweg-de Vries equation the effect that initial data with slower decay (or no decay) at spatial plus infinity have on the solutions of completely integrable systems, that is partial differential equations that can be solved and analyzed by means of the inverse scattering transform (IST). Approaching the case of slower decay at plus infinity presents severe mathematical challenges in the applications of the IST, and its study is a major unsolved open question. Predicting long-time behavior of the solutions is difficult, as the well-known powerful Riemann-Hilbert machinery breaks down on such initial profiles. The main effort of this project will be devoted to extending the Riemann-Hilbert method outside of the realm of classical situations. The investigator plans to use the methods of Hankel operators to tackle the arising issues and expects to discover new types of solutions, with more complicated wave structure, in contrast to the simplicity of existing solitons and radiation, or periodic (quasi-periodic) wave trains and their modulations. The results will have implications for applications and be relevant 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.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

研究人员将研究应用中使用的非线性偏微分方程，以对流体动力学、电信、大气科学、非线性光学、等离子体和天体物理学中的现象进行建模。具体重点是所谓的完全可积系统，这是应用数学的一个领域，也称为孤子理论。目前已知的大部分内容都涉及快速衰减或周期性初始条件，但物理学和应用需要更一般的非标准情况：该项目的主旨是了解较慢衰减（或无衰减）初始数据的影响。该项目将提高对异常波、可积湍流、噪声介质中相干结构的传播、潮汐波和气象现象（即牵牛花）的建模和预测能力。该项目还将提供本科生和研究生水平的研究和指导机会。招聘工作将注重扩大数学科学的研究参与，并瞄准有意成为高中数学教师的学生。该项目在 Korteweg-de Vries 方程的背景下研究了空间正无穷处衰减较慢（或无衰减）的初始数据对完全可积系统解的影响，即可以求解和分析的偏微分方程通过逆散射变换（IST）。在 IST 的应用中，接近正无穷大较慢衰减的情况提出了严峻的数学挑战，其研究是一个尚未解决的主要开放问题。预测解的长期行为是很困难的，因为众所周知的强大黎曼-希尔伯特机器在这种初始配置文件上会崩溃。该项目的主要工作将致力于将黎曼-希尔伯特方法扩展到经典情况领域之外。研究人员计划使用 Hankel 算子的方法来解决出现的问题，并期望发现新类型的解决方案，与现有孤子和辐射或周期（准周期）波列的简单性相比，波结构更复杂以及它们的调制。研究结果将对应用产生影响，并与量子力学基石薛定谔算子理论以及算子理论的基本对象汉克尔和托普利茨算子理论相关。该项目由 DMS 应用数学计划联合资助该奖项反映了 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。