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Soliton gas at the crossroads of dispersive and generalised hydrodynamics

孤子气体处于色散和广义流体动力学的十字路口

基本信息

批准号：
EP/W032759/1
负责人：
Gennady El
金额：
$ 10.26万
依托单位：
Northumbria University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW032759%2F1
关键词：
Soliton gas crossroads dispersive generalised

项目摘要

Long wavelength, hydrodynamic theories abound in physics, from fluids to optics, condensed matter to quantum mechanics, and beyond. The familiar occurrences of hydrodynamic motion in fluids like shock waves and turbulence involve a complex interplay between the large-scale, nonlinear fluid motion and the small-scale, "microscopic", dissipative processes. In media where the microscopic dynamics are dominated by conservative, dispersive, processes the shock waves and turbulent motions of a spectacularly different nature are described by dispersive hydrodynamic theories. Ubiquitous nonlinear waves in dispersive media include localised solitons, exhibiting particle-like properties, and expanding, oscillatory dispersive shock waves. When the dispersive hydrodynamics are described by one of the completely integrable nonlinear partial differential equations that support an infinite number of conserved quantities, an intriguing turbulent wave motion, called soliton gas, becomes possible. Soliton gas can be viewed as an infinite random ensemble of interacting solitons, a "soliton soup'', displaying a nontrivial large-scale, hydrodynamic behaviour, ultimately determined by the properties of elementary two-soliton nonlinear interactions. More generally, the emergence at large scales of a rich, sometimes counter-intuitive, phenomenology from otherwise simple microscopic interactions in complex systems is at the forefront of contemporary mathematical and theoretical physics. Indeed, recent theoretical and experimental research has shown that soliton gas dynamics is instrumental in the understanding of a number of fundamental physical phenomena such as spontaneous modulation instability and the formation of rogue waves. It has been realised recently that the equations describing soliton gases in dispersive hydrodynamics are strikingly similar to those arising in the description of quantum many-body systems. The emerging hydrodynamics of quantum systems, called generalised hydrodynamics (GHD), has proven extremely successful in the description of far from equilibrium behaviour of quantum gases but also turns out to be revealing for classical many-body systems. The remarkable parallels between the ideas of GHD and the spectral theory of soliton gases in integrable dispersive hydrodynamics open a number of potentially transformative perspectives for both areas. While these parallels have already been recognised at both ends, a formal relation between those two theories is lacking. The project aims to establish the precise mathematical relation between the theories of soliton gases in dispersive hydrodynamics and GHD. We shall then explore the implications of this relation, putting a particular emphasis on the applications to dispersive shock and rogue waves and their potential counterparts in GHD. The GHD tools will be used to investigate the statistics and thermodynamics of dispersive hydrodynamic soliton gases and, more generally, to gain further insight into the notion of ``integrable turbulence".

长波长的流体力学理论在物理学中比比皆是，从流体到光学，从凝聚态到量子力学，等等。流体中常见的流体动力学运动，如冲击波和湍流，涉及大尺度非线性流体运动和小尺度“微观”耗散过程之间的复杂相互作用。在介质中，微观动力学是由保守的，分散的，过程的冲击波和湍流运动的一个非常不同的性质是由分散流体动力学理论描述。色散介质中普遍存在的非线性波包括局域孤子，表现出类似粒子的特性，以及膨胀的振荡色散冲击波。当色散流体动力学的一个完全可积的非线性偏微分方程，支持无限数量的守恒量，一个有趣的湍流波运动，称为孤子气体，成为可能。孤立子气体可以被看作是相互作用的孤立子的无限随机系综，一个“孤立子汤”，显示出非平凡的大尺度，流体动力学行为，最终由基本的两个孤立子非线性相互作用的性质决定。更一般地说，从复杂系统中简单的微观相互作用中，在大尺度上出现丰富的，有时是反直觉的现象学，是当代数学和理论物理学的前沿。事实上，最近的理论和实验研究表明，孤子气体动力学是有助于理解一些基本的物理现象，如自发调制不稳定性和流氓波的形成。最近人们认识到，色散流体力学中描述孤立子气体的方程与描述量子多体系统中出现的方程惊人地相似。新兴的量子系统流体力学，称为广义流体力学（GHD），已被证明在描述量子气体远离平衡行为方面非常成功，但也被证明是经典多体系统的启示。GHD的想法和孤子气体的光谱理论在可积色散流体力学显着的相似之处，为这两个领域开辟了一些潜在的变革前景。虽然这些相似之处已经在两端得到承认，但这两种理论之间缺乏正式的关系。该项目旨在建立色散流体力学中孤立子气体理论与GHD之间的精确数学关系。然后，我们将探讨这种关系的影响，把一个特别强调的应用程序，色散冲击波和流氓波和他们的潜在对手在GHD。GHD工具将用于研究色散流体力学孤立子气体的统计和热力学，更一般地说，将进一步深入了解"可积湍流”的概念。