Collisionless Boltzmann equations and integrable moment equations

Gibbons, John

doi:10.1016/0167-2789(81)90036-1

Cited by 108 publications

(156 citation statements)

References 9 publications

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“…In the case of Benney's equations, however, N is infinite. Identical moment equations can alternatively be derived from a Vlasov equation [8], [16] ∂f ∂t 2 + p ∂f ∂x − ∂A 0 ∂x ∂f ∂p = 0.…”

Section: The Benney Moment Equationsmentioning

confidence: 99%

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Hyperelliptic reduction of the Benney moment equations

Baldwin¹,

Gibbons²

2003

J. Phys. A: Math. Gen.

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Abstract. We consider N -parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263 ) to correspond to N −parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605 ) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157).

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Section: The Benney Moment Equationsmentioning

confidence: 99%

“…We consider N -parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett.…”

mentioning

confidence: 99%

Hyperelliptic reduction of the Benney moment equations

Baldwin¹,

Gibbons²

2003

J. Phys. A: Math. Gen.

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“…As was shown in [17] by one of the present authors, if we restrict the bracket (13) to functionals depending on the moments alone:…”

Section: Vlasov Formalism For Hamiltonian Hydrodynamic Chainsmentioning

confidence: 91%

“…In this section we define the Vlasov equations, and we recall ( [17]) how a special class of these equations can be related with hydrodynamic chains which are Hamiltonian with respect to the Kupershmidt-Manin bracket. Moreover, we show how to construct all the differential geometric objects related with such chains in the Vlasov picture.…”

Section: Vlasov Formalism For Hamiltonian Hydrodynamic Chainsmentioning

confidence: 99%

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Differential geometry of hydrodynamic Vlasov equations

Gibbons

Raimondo

2007

Journal of Geometry and Physics

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We consider hydrodynamic chains in (1 + 1) dimensions which are Hamiltonian with respect to the Kupershmidt-Manin Poisson bracket. These systems can be derived from single (2+1) equations, here called hydrodynamic Vlasov equations, under the map A n = ∞ −∞ p n f dp. For these equations an analogue of the Dubrovin-Novikov Hamiltonian structure is constructed. The Vlasov formalism allows us to describe objects like the Haantjes tensor for such a chain in a much more compact and computable way. We prove that the necessary conditions found by Ferapontov and Marshall in [1] for the integrability of these hydrodynamic chains are also sufficient.

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On the Integrability of the Averaged KdV and Benney Equations

Tsarëv

1994

Singular Limits of Dispersive Waves

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Collisionless Boltzmann equations and integrable moment equations

Cited by 108 publications

References 9 publications

Hyperelliptic reduction of the Benney moment equations

Hyperelliptic reduction of the Benney moment equations

Differential geometry of hydrodynamic Vlasov equations

On the Integrability of the Averaged KdV and Benney Equations

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