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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
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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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http://mi.mathnet.ru/eng/tmf6912https://doi.org/10.4213/tmf6912 http://mi.mathnet.ru/eng/tmf/v171/i1/p26
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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
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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
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A. I. Zobnin, “Anti-Frobenius algebras and associative Yang–Baxter equation”, Matem. modelirovanie, 26:11 (2014), 51–56
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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
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S. Arthamonov, “Noncommutative inverse scattering method for the Kontsevich system”, Lett. Math. Phys., 105:9 (2015), 1223–1251
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S. Arthamonov, “Modified double Poisson brackets”, J. Algebra, 492 (2017), 212–233
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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.
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