Differential Geometry of Hydrodynamic Vlasov Equations

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 $A^n={\int }_{-\infty }^{\infty }{p}^{n}fdp.$ 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.

arXiv comment24 pages

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Cited by in corpus (5)

• On Classical Integrability of the Hydrodynamics of Quantum Integrable Systems
• Hamiltonian Formalism of Two-Dimensional Vlasov Kinetic Equation
• Integrable hydrodynamic chains associated with Dorfman Poisson brackets
• Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation
• Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions