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.

24 pages

References in corpus (4)

• Classification of integrable hydrodynamic chains and generating functions of conservation laws
• The Hamiltonian approach in classification and integrability of hydrodynamic chains
• Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains
• Transformations of integrable hydrodynamic chains and their hydrodynamic reductions