Analysis of Regular and Random Soliton Gases in Integrable Dispersive Partial Differential Equations.

可积色散偏微分方程中规则和随机孤子气体的分析。

• 批准号: 2307142
• 负责人: Robert Jenkins
• 金额: $ 20.95万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Standard 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=2307142&HistoricalAwards=false
• 关键词: Analysis; Regular; Random; Soliton; Gases

Nonlinear wave interactions describe phenomena observed throughout science and engineering, from the motion of water waves to fiber optic transmissions to the physics of plasmas relevant to star and fusion reactors. Despite the disparate physical contexts, similar wave patterns emerge, and their behavior can be modeled by the same nonlinear partial differential equations. In applications and real-world measurements, the nonlinear interactions lead to waves patterns of exceeding complexity and apparent randomness. This project will further the understanding of these waveforms by developing techniques to model them as accumulations of special isolated traveling wave solutions, i.e., solitons. The resulting ensembles of many solitons, known as soliton gases, will be studied directly and statistically, when their properties are allowed to behave randomly. The aim is to precisely describe, and ultimately control, the nonlinear wave dynamics observed in applications, for example in fiber optic transmission. The project will provide research training opportunities for undergraduate and graduate students. The project has three main mathematical goals: 1) to build a rigorous analytical description of soliton gases, via characterization of their spectral properties, and use this description to study their statistical properties; 2) to analyze the small dispersion semiclassical limit of the focusing nonlinear Schrodinger equation, as a mechanism for generating integrable turbulence and rogue waves from initial data; 3) to study the long-time behavior of spectrally singular solutions of integrable partial differential equations. The study will involve using and further developing tools from complex and asymptotic analysis, with an emphasis in the Inverse Scattering Transform method. The double scaling limit of the focusing nonlinear Schrodinger equation has the potential to introduce a new class of universality at wave breaking and further the understanding of rogue wave generation. Mathematically, the work on spectrally singular solutions and fat-tailed waves would lead to an extension of Inverse Scattering Transform techniques to broader classes of initial data inaccessible by existing techniques.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.

非线性波相互作用描述了整个科学和工程中观察到的现象，从水波到纤维传播到光纤传播再到与恒星和融合反应器相关的等离子体物理学。尽管物理背景不同，但类似的波模式出现，并且它们的行为可以通过相同的非线性偏微分方程来建模。在应用和现实世界测量中，非线性相互作用导致波浪模式超出复杂性和明显的随机性。该项目将通过开发技术将它们建模为特殊孤立的波动波解决方案（即孤子群）的积累来进一步了解这些波形。当允许其特性随机行为时，将直接和统计地研究许多孤子的合奏，称为孤子气体。目的是精确地描述并最终控制在应用中观察到的非线性波动力学，例如在光纤传输中。 该项目将为本科生和研究生提供研究培训机会。该项目具有三个主要的数学目标：1）通过表征其光谱特性来构建对孤子气体的严格分析描述，并使用此描述来研究其统计特性； 2）分析焦点非线性schrodinger方程的小分散度半经典极限，作为从初始数据产生可集成的湍流和流氓波的机制； 3）研究可集成偏微分方程的光谱奇异解的长期行为。这项研究将涉及使用复杂和渐近分析的工具并进一步开发工具，重点是反向散射变换方法。聚焦非线性Schrodinger方程的双缩放限制具有在波浪破裂时引入新的普遍性，并进一步了解流氓波产生。从数学上讲，在频谱奇异的解决方案和脂肪波浪方面的工作将导致反向散射转换技术扩展到现有技术无法访问的更广泛的初始数据类别。该奖项反映了NSF的法定任务，并通过评估该基金会的知识绩效和广泛的影响来评估NSF的法定任务。