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 TMF, 2002, Volume 132, Number 1, Pages 60–73 (Mi tmf346)

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: 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.

Keywords: compatible Poisson brackets, systems of hydrodynamic type, compatible metrics, integrable hierarchies, bi-Hamiltonian structures, nonlocal Poisson brackets of hydrodynamic type

DOI: https://doi.org/10.4213/tmf346

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English version:
Theoretical and Mathematical Physics, 2002, 132:1, 942–954

Bibliographic databases:

Citation: O. I. Mokhov, “Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them”, TMF, 132:1 (2002), 60–73; Theoret. and Math. Phys., 132:1 (2002), 942–954

Citation in format AMSBIB
\Bibitem{Mok02} \by O.~I.~Mokhov \paper Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them \jour TMF \yr 2002 \vol 132 \issue 1 \pages 60--73 \mathnet{http://mi.mathnet.ru/tmf346} \crossref{https://doi.org/10.4213/tmf346} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1956677} \zmath{https://zbmath.org/?q=an:1067.37099} \elib{http://elibrary.ru/item.asp?id=13401722} \transl \jour Theoret. and Math. Phys. \yr 2002 \vol 132 \issue 1 \pages 942--954 \crossref{https://doi.org/10.1023/A:1019659324655} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000177713500003} 

• http://mi.mathnet.ru/eng/tmf346
• https://doi.org/10.4213/tmf346
• http://mi.mathnet.ru/eng/tmf/v132/i1/p60

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249
2. Pavlov, MV, “Hydrodynamic chains and the classification of their Poisson brackets”, Journal of Mathematical Physics, 47:12 (2006), 123514
3. Pavlov, MV, “Algebro-geometric approach in the theory of integrable hydrodynamic type systems”, Communications in Mathematical Physics, 272:2 (2007), 469
4. Artur Sergyeyev, “Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility”, SIGMA, 3 (2007), 062, 14 pp.
5. O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420
6. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
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