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Differential Geometry of Hydrodynamic Vlasov Equations

John Gibbons·Andrea Raimondo
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Abstract

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 $A^n =\int_{-\infty}^\infty p^n f dp.$ 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 (arXiv:nlin.SI/0505013) for the integrability of these hydrodynamic chains are also sufficient.

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John Gibbons, Andrea Raimondo. 2007-03-07. Differential Geometry of Hydrodynamic Vlasov Equations. https://doi.org/10.1016/j.geomphys.2007.03.002

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