Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

M V Pavlov,V B Taranov,G A El

doi:10.1007/s11232-012-0064-z

Abstract

We derive generalized multiflow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type beyond the conventional semi-Hamiltonian framework.

Highlights

The generalised soliton-gas kinetic equation represents an integro-differential system [2] ft +x = 0, (1) ∞s(η) = S(η) + G(η, μ)f (μ)[s(μ) − s(η)]dμ. (2) ηHere f (η) ≡ f (η, x, t) is the distribution function and s(η) ≡ s(η, x, t) is the associated transport velocity
S(η) = 4η2, η−μ G(η, μ) = log η+μ was derived in [1] as an infinite-genus thermodynamic limit of the Whitham modulation equations associated with the KdV equation, φt − 6φφx + φxxx = 0, and was shown to describe macroscopic dynamics of a soliton gas, a disordered infinite-soliton ensemble of finite density [4]
We show that the corresponding N -flow non-isospectral hydrodynamic reductions have the form of 2N -component hydrodynamic type systems uit =x, ηti = viηxi, i = 1, 2, . . . , N, (9)

Summary

Introduction

The generalised soliton-gas kinetic equation represents an integro-differential system [2]. M=1 where the ‘spectral’ components ηN > ηN−1 > · · · > η1 > 0 are arbitrary numbers These ‘isospectral’ cold-gas reductions were shown to have the form of systems of hydrodynamic conservation laws uit = (uivi)x , i = 1, . Having in mind that this system is obtained as an exact reduction of an integrable system (at least for S(η), G(η, μ) defined by (3) — the KdV case), one can expect that the multi-flow reductions (9) will be integrable by some modification of the generalised hodograph method [9] This could lead to an extension of the conventional notion of an integrable system of hydrodynamic type. We are going to investigate this problem in detail in future publications

Evolution equations

Closure relations

The structure of generalised multi-flow hydrodynamic reductions

Conclusion