Differential Geometry of Hydrodynamic Vlasov Equations

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 = ∫ ∞ − ∞ pn 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 [1] for the integrability of these hydrodynamic chains are also sufficient. 1 Systems of hydrodynamic type Systems of hydrodynamic type are quasilinear first order PDE of the form u i t = vi j (u)ujx, i, j = 1...N, (1) where (x, t) are the independent and (u 1,..., u N) the dependent variables. Here and below sums over repeated indices are assumed. A Hamiltonian 1 formalism for systems of this type was introduced in [2] by Dubrovin and Novikov, who defined a Poisson bracket of the form δIα