Geodesic
Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type
Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 279-288

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We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.
Keywords: compatible Hamiltonian operators - systems of hydrodynamic type - Lie derivative, integrable hierarchies, local Poisson brackets of hydrodynamic type, flat pencils of metrics. 
O. I. Mokhov. Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type. Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 279-288. http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a12/
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     title = {Compatible {Dubrovin{\textendash}Novikov} {Hamiltonian} {Operators,} {Lie} {Derivative,} and {Integrable} {Systems} of {Hydrodynamic} {Type}},
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