Soliton gas at the crossroads of dispersive and generalised hydrodynamics

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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".