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

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:

Received: 05.02.2002

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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    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 Lamй Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. 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
    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. 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
    6. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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