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学。尽管不同的物理环境，相似的波浪模式出现，它们的行为可以用相同的非线性偏微分方程来建模。在实际应用和实际测量中，非线性相互作用会导致异常复杂和明显随机性的波形。该项目将通过开发技术将这些波形建模为特殊孤立行波解（即孤子）的累积，从而进一步了解这些波形。当它们的性质被允许随机表现时，许多孤子的结果集合，被称为孤子气体，将被直接和统计地研究。其目的是精确描述并最终控制应用中观察到的非线性波动动力学，例如光纤传输。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特性，建立对孤子气体的严格分析描述，并利用这种描述来研究它们的统计特性；2)分析聚焦非线性薛定谔方程的小色散半经典极限，作为初始数据产生可积湍流和异常波的机制；3)研究可积偏微分方程谱奇异解的长时性。该研究将涉及使用和进一步开发复杂和渐近分析的工具，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，并进一步加深对异常波产生的理解。在数学上，对频谱奇异解和肥尾波的研究将导致反散射变换技术的扩展，以获得现有技术无法获得的更广泛的初始数据。该奖项反映了美国国家科学基金会的法定使命，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。

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.