Geodesic
Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them
Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 1, pp. 60-73

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We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.
Mots-clés : compatible Poisson brackets, nonlocal Poisson brackets of hydrodynamic type.
Keywords: systems of hydrodynamic type, compatible metrics, integrable hierarchies, bi-Hamiltonian structures 
@article{TMF_2002_132_1_a2,
     author = {O. I. Mokhov},
     title = {Compatible {Nonlocal} {Poisson} {Brackets} of {Hydrodynamic} {Type} and {Integrable} {Hierarchies} {Related} to {Them}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {60--73},
     publisher = {mathdoc},
     volume = {132},
     number = {1},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_132_1_a2/}
}
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O. I. Mokhov. Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them. Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 1, pp. 60-73. http://geodesic.mathdoc.fr/item/TMF_2002_132_1_a2/