Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

Pavlov, Maxim V.; Taranov, V. B.; El, Gennady

doi:10.1007/s11232-012-0064-z

Cited by 12 publications

(39 citation statements)

References 9 publications

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“…The hydrodynamic reductions obtained by (7) for arbitrary N have been thoroughly analysed in [17,18]. Here we shall be mostly looking at the case of a two-component gas yielding the simplest nontrivial results that can be verified numerically.…”

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confidence: 99%

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

Carbone¹,

Dutykh²,

El³

2016

EPL

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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.

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confidence: 99%

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

Carbone¹,

Dutykh²,

El³

2016

EPL

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“…, η n ). Systems of this form have appeared recently as delta-functional reductions of the kinetic equation for soliton gas [15], see below, as well as hydrodynamic reductions of linearly degenerate dispersionless integrable systems in multidimensions, see Section 5.2. In Section 2 we derive Jordan block analogues of equations for commuting flows (4) and integrability conditions (5).…”

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confidence: 99%

“…was derived in [5] as thermodynamic limit of the KdV Whitham equations and generalised in [6,7] to the NLS case. It was demonstrated in [15] that under a delta-functional ansatz,…”

Section: Introductionmentioning

confidence: 99%

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Kinetic equation for soliton gas: integrable reductions

Ferapontov,

Pavlov

2021

Preprint

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Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2 × 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev's theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.MSC: 35Q51, 35Q83, 37K10.

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“…This provides a partial answer to the question posed in Refs. [28,35]: in what circumstances is the kinetic equation described by Eqs. (1.1) and (1.4) integrable?…”

Section: Introductionmentioning

confidence: 99%

On classical integrability of the hydrodynamics of quantum integrable systems

Bulchandani¹

2017

J. Phys. A: Math. Theor.

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Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain "Bethe-Boltzmann" kinetic equation, which has the form of an integro-differential conservation law in (1 + 1)D. The purpose of the present work is to investigate the sense in which the Bethe-Boltzmann equation defines an "integrable kinetic equation". To this end, we study a class of N dimensional systems of evolution equations that arise naturally as finite-dimensional approximations to the Bethe-Boltzmann equation. We obtain non-local Poisson brackets and Hamiltonian densities for these equations and derive an infinite family of first integrals, parameterized by N functional degrees of freedom. We find that the conserved charges arising from quantum integrability map to Casimir invariants of the hydrodynamic bracket and their group velocities map to Hamiltonian flows. Some results from the finite-dimensional setting extend to the underlying integro-differential equation, providing evidence for its integrability in the hydrodynamic sense.

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Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

Cited by 12 publications

References 9 publications

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

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

Kinetic equation for soliton gas: integrable reductions

On classical integrability of the hydrodynamics of quantum integrable systems

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