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Teoreticheskaya i Matematicheskaya Fizika, 2002, Volume 132, Number 1, Pages 60–73
DOI: https://doi.org/10.4213/tmf346 (Mi tmf346) 		 
This article is cited in 6 scientific papers (total in 6 papers)

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
Full-text PDF (249 kB) Citations (6)
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DOI: https://doi.org/10.4213/tmf346
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.
Received: 05.02.2002
English version:
Theoretical and Mathematical Physics, 2002, Volume 132, Issue 1, Pages 942–954
DOI: https://doi.org/10.1023/A:1019659324655
Bibliographic databases:
Language: Russian
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
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\paper Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them
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\transl
\jour Theoret. and Math. Phys.
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\vol 132
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\pages 942--954
\crossref{https://doi.org/10.1023/A:1019659324655}
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Linking options:
  • https://www.mathnet.ru/eng/tmf346
  • https://doi.org/10.4213/tmf346
  • https://www.mathnet.ru/eng/tmf/v132/i1/p60
    • This publication is cited in the following 6 articles:
    1. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    3. Pavlov, MV, “Algebro-geometric approach in the theory of integrable hydrodynamic type systems”, Communications in Mathematical Physics, 272:2 (2007), 469  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Artur Sergyeyev, “Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility”, SIGMA, 3 (2007), 062, 14 pp.  mathnet  crossref  mathscinet  zmath
    5. Pavlov, MV, “Hydrodynamic chains and the classification of their Poisson brackets”, Journal of Mathematical Physics, 47:12 (2006), 123514  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamй Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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    References:57
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