Soliton gas at the crossroads of dispersive and generalised hydrodynamics

从流体到光学，从凝聚态物质到量子力学等等，长波、流体动力学理论在物理学中比比皆是。流体中常见的流体运动，如激波和湍流，涉及到大尺度、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。在微观动力学由保守的色散过程主导的介质中，用色散流体动力学理论描述了性质截然不同的激波和湍流运动。色散介质中普遍存在的非线性波包括具有粒子性质的局域孤子，以及扩展的、振荡的色散冲击波。当色散流体动力学用支持无穷多个守恒量的完全可积的非线性偏微分方程组描述时，一种称为孤子气体的有趣的湍流波动成为可能。孤子气体可以看作是相互作用的孤子的无限随机集合，表现出非平凡的大尺度流体力学行为，最终由基本的双孤子非线性相互作用的性质决定。更广泛地说，从复杂系统中其他简单的微观相互作用中大规模出现丰富的、有时与直觉相反的现象学，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，描述色散流体力学中的孤子气体的方程与描述量子多体系统的方程惊人地相似。量子系统的新兴流体动力学，称为广义流体动力学(GHD)，已被证明在描述远离平衡的量子气体行为方面极其成功，但也被证明对经典的多体系统具有揭示作用。在可积色散流体力学中，GHD的思想和孤子气体的光谱理论之间的惊人相似之处为这两个领域打开了许多潜在的变革性的视角。尽管两端都已经认识到了这些相似之处，但这两种理论之间缺乏正式的联系。该项目旨在建立色散流体力学中的孤子气体理论和GHD之间的精确数学关系。然后，我们将探索这种关系的含义，特别强调对弥散激波和流氓波及其在GHD中的潜在对应波的应用。GHD工具将被用来研究色散流体动力孤子气体的统计和热力学，更广泛地说，将进一步深入了解“可积湍流”的概念。

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".