Macroscopic dynamics of incoherent soliton ensembles: Soliton gas kinetics and direct numerical modelling

Abstract

We undertake a detailed comparison of the results of direct numerical simulations of the soliton gas dynamics for the Korteweg-de Vries equation with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. Two model problems are considered: i) the propagation of a “trial” soliton through a one-component “cold” soliton gas consisting of randomly distributed solitons of approximately the same amplitude; and ii) the collision of two cold soliton gases of different amplitudes (the soliton gas shock tube problem) leading to the formation of an expanding incoherent dispersive shock wave. In both cases excellent agreement is observed between the analytical predictions of the soliton gas kinetics and the direct numerical simulations. Our results confirm the relevance of the kinetic equation for solitons as a quantitatively accurate model for macroscopic non-equilibrium dynamics of incoherent soliton ensembles.

Transport fluctuations in integrable models out of equilibrium

“…We have provided explicit expressions for the first few cumulants. The results can be immediately applied, for instance, to the paradigmatic classical hard rod gas, the quantum Lieb-Liniger model, as well as other classical and quantum chains and field theories such as the sinh-Gordon and sine-Gordon model and the Heisenberg chain, and to soliton gases [80][81][82][83][84]. Importantly, we have explicitly verified the validity of the formalism by comparing the exact formulae for cumulants with Monte Carlo simulations in the classical hard rod gas.…”

“…This includes the Lieb-Liniger model [72,73] which has been shown to describe cold atomic gases in quasi-one-dimensional traps [74][75][76] (see e.g. the experiments [5,25] where the integrability of the Lieb-Liniger model played an important role, and the book [77] for a review), and many other quantum field theories, as well as integrable quantum chains, classical field theories, and classical gases such as the hard rod gas [78,79] and soliton gases [80][81][82][83][84].…”

Generalized hydrodynamics of the classical Toda system

“…That is, at fixed β i 's, the function g of −P given by (24) is the inverse of the function f of µ given by (25). Since g has a maximum as a function of P , its inverse has two branches, determined by sgn(ν).…”

“…We also define h 0 (p) = 1. From the knowledge of the scattering shift and of the conserved charges in scattering states, the TBA technology immediately gives a conjecture for the exact specific Landau potential f in (25) for any GGE, in terms of β i 's and g. The proposed expression, for classical particle systems, is…”

Generalized Gibbs Ensembles of the Classical Toda Chain

“…In the context of the Korteweg-de-Vries equation, Zakharov [9] proposed a hydrodynamic type theory. We refer to [10] for more details. For the relativistic sinh-Gordon model, also an integrable classical field theory, GHD is derived and studied in [11].…”