Soliton gas in integrable dispersive hydrodynamics

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 Schrödinger (FNLS) equations. For the FNLS 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.

Rule 54: exactly solvable model of nonequilibrium statistical mechanics

“…There are many equivalent ways of finding this result: one can derive it from the definition of the dressed velocity [12,13] by using the TBA description put forward in [32]; one can obtain it by using the appropriate result for hard rods [56,58], or an appropriate soliton gas [57,[59][60][61]; or alternatively the dressed velocity can be identified in terms of expectation values in the GGE [31]. Additionally, the simplicity of the result allows us to obtain it through an elementary argument introduced in [31]: let us imagine that we are tracing a right-moving soliton that starts at time t = 0 at position x = 0 and at time t it arrives to the position x.…”

Form factors and generalized hydrodynamics for integrable systems

“…However, it is important to remark that actually the very first appearance of the collision rate conjecture dates back to half a century ago in the context of the classical soliton gas [73]. For the history of kinetic approaches to soliton gases and integrable turbulence, see the review [74] in the same volume. More recently the ansatz has also been applied to other classical Hamiltonian integrable models, foremost the classical Toda lattice [15,75].…”

Generalized hydrodynamics of the attractive non-linear Schrӧdinger equation

“…It is worth mentioning that far before the advent of GHD, the soliton picture (see reference [73] for a recent review) provides a hydrodynamic framework to study transport in classical integrable models bearing solitonic excitations. Indeed, the GHD equations describing inhomogeneous states, but with homogeneous and time-independent dynamics, reduce to the soliton gas picture when applicable.…”