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

非线性波相互作用描述了在整个科学和工程中观察到的现象，从水波的运动到光纤传输，再到与恒星和聚变反应堆相关的等离子体物理。尽管物理环境不同，但出现了相似的波型，它们的行为可以用相同的非线性偏微分方程组来建模。在应用和实际测量中，非线性相互作用导致了波型的极端复杂性和明显的随机性。这个项目将通过开发技术来将这些波形建模为特殊孤立行波解的累积，即孤子，从而进一步加深对这些波形的理解。由此产生的许多孤子的系综，称为孤子气体，将被直接和统计地研究，当它们的性质被允许随机行为时。其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。该项目有三个主要的数学目标：1)通过表征孤子气体的光谱性质，建立一个严格的解析描述，并利用这个描述来研究它们的统计性质；2)分析聚焦非线性薛定谔方程的小色散半经典极限，作为从初始数据产生可积湍流和流浪波的机制；3)研究可积偏微分方程谱奇异解的长时间行为。这项研究将涉及使用和进一步开发复数和渐近分析的工具，重点是逆散射变换方法。聚焦非线性薛定谔方程的双标度极限有可能在破波时引入一类新的普适性，并进一步加深对无赖波产生的理解。从数学上讲，关于频谱奇异解和厚尾波的工作将导致逆散射变换技术扩展到现有技术无法获得的更广泛类别的初始数据。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。

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.