Integrable turbulence and rogue waves: semi-classical nonlinear Schrödinger equation framework

湍流是最容易识别的，同时也是最耐人寻味的非线性运动形式之一，这是在日常现象中常见的，如狂风或湍急的河流。尽管湍流广泛存在，但湍流的数学描述仍然是现代科学中最具挑战性的问题之一。引起湍流运动的物理机制可能有很大的不同，但通常涉及某种耗散，例如粘性。目前的项目探索了一种非常不同的湍流，它不涉及任何耗散，但涉及随机非线性波的动力学和统计，这些随机非线性波由所谓的可积偏微分方程组(PDE)建模，如Korteweg-de Vries和非线性薛定谔(NLS)方程。这些方程是描述水波、光学介质、等离子体和超流体中广泛的非线性波动现象的通用数学模型。由于其丰富的数学结构和广泛的物理应用，可积偏微分方程组在过去的50多年里一直是非常热门的研究对象。最近，V.E.Zakharov提出了利用可积方程的随机(随机)解来模拟海洋和光学介质中的复杂非线性波动现象的想法，他创造了术语“可积湍流”。特别是，可积湍流框架可以帮助解释无赖波的形成和演化--这种罕见的大幅度事件出现在海洋表面，无法预测，可能会对船只和石油平台造成毁灭性的破坏。在光纤中也观察到了大量的自发波场起伏，这对高功率激光器和光通信系统有许多不利的影响。到目前为止，可积湍流中的分析结果很少，大多数发展都是数值的。该项目将通过在聚焦NLS方程的半经典极限的框架内构建第一个可积湍流分析模型来解决这一悬而未决的问题，NLS方程是非线性科学中的一个基本数学模型，适用于广泛的物理环境，包括水波、等离子体、非线性光纤和玻色-爱因斯坦凝聚体。特别是，NLS方程的所谓呼吸解具有强烈表明它们与海洋和光学介质中的无赖波有关的性质。在这个项目中，可积湍流和无赖波形成的数学描述将通过渐近方法来实现，该方法连接了色散偏微分方程组半经典分析中的两个主要技术：Whitham调制理论和Riemann-Hilbert问题分析。这一统一的方法是最近由国际和平研究所与A.Tovbis教授合作开发的，Tovbis教授也是当前项目的主要合作者之一。项目中要证明的基本数学假设之一是与发展的可积湍流的非线性谱的特殊热力学结构有关，然后将用于分析其动力学性质，特别是流氓波含量的确定。该项目的独特之处在于通过链接的PHD项目集成了撞击路径，涉及光纤实施的半经典NLS方法来处理可积湍流。国防科学和技术实验室批准资助的PHD项目的具体目标与开发分析和控制部分相干光通过光纤传输中的流氓波的实用方法有关。开发的方法将在里尔大学的PhLAM光学实验室进行实验验证。

Turbulence is one of the most recognisable, and at the same time, one of the most intriguing forms of nonlinear motion that is commonly observed in everyday phenomena such as wind blasts or fast flowing rivers. Despite its widespread occurrence, the mathematical description of turbulence remains one of the most challenging problems of modern science. Physical mechanisms giving rise to turbulent motion can be very different but typically they involve some sort of dissipation, e.g. viscosity.The current project explores a very different kind of turbulence that does not involve any dissipation but is concerned with dynamics and statistics of random nonlinear waves that are modelled by the so-called integrable partial differential equations (PDEs) such as the Korteweg - de Vries and nonlinear Schroedinger (NLS) equations. These equations are universal mathematical models for a broad spectrum of nonlinear wave phenomena in water waves, optical media, plasmas and superfluids. Owing to their rich mathematical structure and a wide range of physical applications, integrable PDEs have been the subject of incredibly intense research in the last 50 or so years.The idea of using random (stochastic) solutions to integrable equations for modelling complex nonlinear wave phenomena in the ocean and optical media has been recently put forward by V.E. Zakharov who has coined the term "integrable turbulence". In particular, the integrable turbulence framework can help to explain the formation and evolution of rogue waves - rare events of large amplitude that appear unpredictably on the ocean surface and can be devastating for ships and oil platforms. Rogue waves have also been observed in optical fibres as spontaneous field fluctuations of large amplitude with a number of undesirable implications for high power lasers and optical communications systems.To date, very few analytical results in integrable turbulence are available with the majority of the developments being numerical. The project will attack this outstanding issue by constructing the first analytical model of integrable turbulence in the framework of the semi-classical limit of the focusing NLS equation, which is a fundamental mathematical model in nonlinear science that applies to a wide range of physical contexts including water waves, plasmas, nonlinear optical fibres and Bose-Einstein condensates. In particular, the so-called breather solutions of the NLS equation have the properties that strongly suggest their links with rogue waves in the ocean and optical media. In the project, the mathematical description of integrable turbulence and the rogue wave formation will be achieved via the asymptotic approach bridging two major techniques in the semi-classical analysis of dispersive PDEs: the Whitham modulation theory and the Riemann-Hilbert problem analysis. This unified approach was recently developed by the PI in collaboration with Prof. A. Tovbis who is also one of the main collaborators in the current project. One of the fundamental mathematical hypotheses to be proved in the project is related to the special thermodynamic structure of the nonlinear spectrum of the developed integrable turbulence, which will then be used for the analysis of its kinetic properties and particularly, the determination of the rogue wave content.The unique feature of the project is the integrated pathway to impact via the linked PhD project concerned with the fibre optics implementation of the semi-classical NLS approach to integrable turbulence. The particular objectives of the PhD project, which is approved for funding by the Defence Science and Technology Laboratory, are related to the development of practical methods of analysis and control of the rogue wave formation in the partially coherent light propagation through optical fibres. The developed methods will be verified experimentally in the PhLAM optics laboratory at the University of Lille.