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TMF, 2011, Volume 167, Number 1, Pages 3–22 (Mi tmf6623)  

This article is cited in 2 scientific papers (total in 2 papers)

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

O. I. Mokhovab

a M. V. Lomonosov Moscow State University, Moscow, Russia
b Landau Institute for Theoretical Physics, RAS, Chernogolovka, Moscow Oblast, Russia

Abstract: We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) $V^i_j(u)$ of this system of hydrodynamic type, the metrics $g^{ij}_1(u)$ and $g^{ij}_2(u)$, the affinor $v^i_j(u)=g_1^{is}(u)g_{2,sj}(u)$, and also the affinors $(w_{1,n})^i_j(u)$ and $(w_{2,n})^i_j(u)$ of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.

Keywords: bi-Hamiltonian system of hydrodynamic type, Riemann invariant, compatible metrics, diagonalizable affinor, bi-Hamiltonian structure, bi-Hamiltonian affinor, integrable system

DOI: https://doi.org/10.4213/tmf6623

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English version:
Theoretical and Mathematical Physics, 2011, 167:1, 403–420

Bibliographic databases:

Received: 14.10.2010

Citation: O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, TMF, 167:1 (2011), 3–22; Theoret. and Math. Phys., 167:1 (2011), 403–420

Citation in format AMSBIB
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\by O.~I.~Mokhov
\paper Compatible metrics and the~diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type
\jour TMF
\yr 2011
\vol 167
\issue 1
\pages 3--22
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\crossref{https://doi.org/10.4213/tmf6623}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2816136}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011TMP...167..403M}
\transl
\jour Theoret. and Math. Phys.
\yr 2011
\vol 167
\issue 1
\pages 403--420
\crossref{https://doi.org/10.1007/s11232-011-0032-z}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79956121870}


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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, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182  mathnet
    2. 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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