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TMF, 2012, Volume 171, Number 1, Pages 26–32 (Mi tmf6912)  

This article is cited in 7 scientific papers (total in 7 papers)

Bi-Hamiltonian ordinary differential equations with matrix variables

A. V. Odesskiia, V. N. Rubtsovbc, V. V. Sokolovd

a Brock University, St. Catharines, Canada
b Institute for Theoretical and Experimental Physics, Moscow, Russia
c LAREMA, CNRS, Université d'Angers, Angers, France
d Landau Institute for Theoretical Physics, RAS, Moscow, Russia

Abstract: We consider a special class of Poisson brackets related to systems of ordinary differential equations with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets, and find the corresponding hierarchy of integrable models, which generalizes the two-component Manakov matrix system to the case of an arbitrary number of matrices.

Keywords: integrable ordinary differential equation with matrix unknowns, bi-Hamiltonian formalism, Manakov model

DOI: https://doi.org/10.4213/tmf6912

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English version:
Theoretical and Mathematical Physics, 2012, 171:1, 442–447

Bibliographic databases:

Received: 07.05.2011

Citation: A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Bi-Hamiltonian ordinary differential equations with matrix variables”, TMF, 171:1 (2012), 26–32; Theoret. and Math. Phys., 171:1 (2012), 442–447

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. V. V. Sokolov, “Classification of constant solutions of the associative Yang–Baxter equation on $\operatorname{Mat}_3$”, Theoret. and Math. Phys., 176:3 (2013), 1156–1162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. Odesskii, V. Rubtsov, V. Sokolov, “Double Poisson brackets on free associative algebras”, Noncommutative Birational Geometry, Representations and Combinatorics, Contemporary Mathematics, 592, eds. A. Berenstein, V. Retakh, Amer. Math. Soc., Providence, RI, 2013, 225–239  crossref  mathscinet  zmath  isi
    3. A. I. Zobnin, “Anti-Frobenius algebras and associative Yang–Baxter equation”, Matem. modelirovanie, 26:11 (2014), 51–56  mathnet  mathscinet  elib
    4. A. Odesskii, V. Rubtsov, V. Sokolov, “Parameter-dependent associative Yang–Baxter equations and Poisson brackets”, Int. J. Geom. Methods Mod. Phys., 11:9 (2014), 1460036  crossref  mathscinet  zmath  isi
    5. S. Arthamonov, “Noncommutative inverse scattering method for the Kontsevich system”, Lett. Math. Phys., 105:9 (2015), 1223–1251  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. S. Arthamonov, “Modified double Poisson brackets”, J. Algebra, 492 (2017), 212–233  crossref  mathscinet  zmath  isi
    7. M. N. Hounkonnou, G. D. Houndedji, “Solutions of associative Yang–Baxter equation and $D$-equation in low dimensions and associated Frobenius algebras and Connes cocycles”, J. Algebra. Appl., 17:1 (2018), 1850010, 26 pp.  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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