Soliton gas in integrable dispersive hydrodynamics

TL;DR

This paper reviews the spectral theory and kinetic modeling of soliton gases in integrable dispersive hydrodynamics, introducing new concepts like soliton condensates and breather gases, and comparing theoretical predictions with numerical simulations.

Contribution

It develops a detailed thermodynamic spectral theory for soliton gases, including novel concepts such as soliton condensates and breather gases, and connects kinetic equations with physical phenomena.

Findings

Spectral theory provides a framework for soliton gas dynamics.

Introduction of soliton condensate and breather gas concepts.

Comparison of kinetic theory with numerical simulations confirms validity.

Abstract

We review spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then a more detailed theory is presented in which soliton gas dynamics are modelled by a thermodynamic type limit of modulated finite-gap spectral solutions of the Korteweg-de Vries and the focusing nonlinear Schr\"odinger equations. For the focusing nonlinear Schr\"odinger equation the notions of soliton condensate and breather gas are introduced that are related to the phenomena of spontaneous modulational instability and the rogue wave formation. Integrability properties of the kinetic equation for soliton gas are discussed and some physically relevant solutions are presented and compared with direct numerical simulations of dispersive hydrodynamic systems.