Breather and Soliton Gases for the Focusing Nonlinear Schrodinger Equation: Theoretical and Applied Aspects

最熟悉的非线性色散波的表现可能是破碎的海浪。非线性色散波在自然界中无处不在，出现在许多领域，从水波到光学、声学、凝聚态、宇宙学等。非线性色散波一直是应用数学和物理研究的热点。可积非线性Schrödinger （NLS）方程被普遍认为是波在非线性色散介质中传播的通用模型。这个项目的主要思想是用所谓的孤子或呼吸气体——众所周知的方程解的随机集合——来模拟自然现象中经常遇到的随机非线性波。最终，该项目旨在提高我们预测的能力，并在某些情况下，控制随机非线性波。这项研究采用了纯数学和应用数学的思想，这些思想通过与实验物理学家的合作得到了加强。该项目还将作为培养博士生的工具。本项目的主要目标可分为两大类：(a)发展聚焦NLS气体的严格光谱理论和构建其半经典极限实现；(b)呼吸子和孤子气体的光谱特性的统计特性及其在随机非线性波动问题中的应用。(a)部分的工作需要对描述气体光谱特性的积分微分方程进行严格的推导和分析。(b)部分中fNLS气体的重要统计特性（概率密度函数、功率谱、峰度等）的计算包含解析和数值两部分。预计获得的结果将导致与欧洲实验人员合作进行实验室实验。总的来说，该项目将对包括可积湍流在内的非线性波动理论的广泛领域产生影响。在光纤中，该项目的结果可能有助于研究并最终控制非线性光纤中噪声的演变。该项目还有望提高我们对随机非线性波（包括异常波）的一般认识，并改进其预测方法。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。

The most familiar manifestation of nonlinear dispersive waves is perhaps that of a breaking ocean wave. Nonlinear dispersive waves are ubiquitous in nature, appearing in many fields ranging from water waves to optics, acoustics, condensed matter, cosmology and beyond. Nonlinear dispersive waves have been an area of intensive research interest in applied mathematics and physics. The integrable Nonlinear Schrödinger (NLS) equation is generally accepted as the universal model for waves propagating in nonlinear dispersive media. The main idea of this project is to model random nonlinear waves, which are frequently encountered in natural phenomena, with the so-called soliton or breather gases – random ensembles of the well-known solutions of the equation. Ultimately, the project aims to enhance our ability to predict and, in some cases, to control random nonlinear waves. This research employs ideas of pure and applied mathematics that are enhanced by the collaboration with experimental physicists. The project will also serve as a vehicle for training PhD students.The main goals of this project can be divided in the two categories: (a) development of the rigorous spectral theory for the focusing NLS (fNLS) gases and construction of their semiclassical limit realizations; (b) statistical characterization of breather and soliton gases in terms of their spectral characteristics together with applications in random nonlinear wave problems. The work in part (a) requires rigorous derivation and analysis of integro-differential equations describing spectral characteristics of the gases. Calculation of important statistical characteristics (probability density function, power spectrum, kurtosis, etc.) of the fNLS gases in part (b) contains both analytical and numerical components. The obtained results are expected to lead to lab experiments in collaboration with European experimentalists. In general, the project will have an impact on a broad area of nonlinear wave theory, including integrable turbulence. In fiber optics, results of the project may help to examine, and ultimately to control the evolution of noise in nonlinear optical fibers. The project is also expected to advance our general knowledge of random nonlinear waves, including the rogue waves, and to improve methods of their prediction.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.