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

用于聚焦非线性薛定谔方程的呼吸气体和孤子气体：理论和应用方面

• 批准号: 2009647
• 负责人: Alexander Tovbis
• 金额: $ 24.2万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009647&HistoricalAwards=false
• 关键词: Breather; Soliton; Gases; Focusing; Nonlinear

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.

非线性色散波最常见的表现形式可能是破碎的海浪。非线性色散波在自然界中普遍存在，出现在从水波到光学、声学、凝聚态物质、宇宙学等诸多领域。非线性色散波一直是应用数学和物理领域的研究热点。可积非线性薛定谔(NLS)方程被公认为波在非线性色散介质中传播的普遍模型。这个项目的主要思想是用所谓的孤子或呼吸气体来模拟自然现象中经常遇到的随机非线性波-方程众所周知的解的随机集合。归根结底，该项目旨在提高我们预测并在某些情况下控制随机非线性波的能力。这项研究使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。这个项目的主要目标可以分为两类：(A)发展聚焦NLS(FNLS)气体的严格光谱理论并构造它们的半经典极限实现；(B)呼吸和孤子气体的光谱特性的统计表征以及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征的积分-微分方程式进行严格的推导和分析。重要统计特性(概率密度函数、功率谱、峰度等)的计算(B)部分中的fNLS气体中既有分析成分，也有数值成分。预计所获得的结果将导致与欧洲实验者合作进行实验室实验。总体而言，该项目将对非线性波动理论的广泛领域产生影响，包括可积湍流。在光纤方面，该项目的结果可能有助于检查并最终控制非线性光纤中的噪声演化。该项目还有望促进我们对随机非线性波(包括无赖波)的一般知识，并改进其预测方法。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。

项目成果

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

• DOI: 10.1088/1361-6544/accfdf
• 发表时间: 2022-10
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Marco Bertola;Robert Jenkins;A. Tovbis

Elliptic finite-band potentials of a non-self-adjoint Dirac operator

• DOI: 10.1016/j.aim.2023.109188
• 发表时间: 2022-10
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: G. Biondini;Xu‐Dan Luo;Jeffrey Oregero;A. Tovbis

Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering

通过非线性谱工程预测和操纵水动力流氓波

• DOI: 10.1103/physrevfluids.7.054401
• 发表时间: 2022
• 期刊: Physical Review Fluids
• 影响因子: 2.7
• 作者: Tikan, Alexey;Bonnefoy, Felicien;Roberti, Giacomo;El, Gennady;Tovbis, Alexander;Ducrozet, Guillaume;Cazaubiel, Annette;Prabhudesai, Gaurav;Michel, Guillaume;Copie, Francois
• 通讯作者: Copie, Francois

On the spectrum of the periodic focusing Zakharov–Shabat operator

• DOI: 10.4171/jst/432
• 发表时间: 2020-10
• 期刊: Journal of Spectral Theory
• 影响因子: 1
• 作者: G. Biondini;Jeffrey Oregero;A. Tovbis

Local Emergence of Peregrine Solitons: Experiments and Theory

• DOI: 10.3389/fphy.2020.599435
• 发表时间: 2020-11
• 作者: A. Tikan;S. Randoux;G. El;A. Tovbis;F. Copie;P. Suret