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Theoretical and Mathematical Physics

, Volume 171, Issue 1, pp 442–447 | Cite as

Bi-Hamiltonian ordinary differential equations with matrix variables

Article

Abstact

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 

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References

  1. 1.
    S. V. Manakov, Funct. Anal. Appl., 10, No. 4, 328–329 (1976).MathSciNetCrossRefGoogle Scholar
  2. 2.
    F. Magri, J. Math. Phys., 19, 1156–1162 (1978).MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    F. Magri, P. Casati, G. Falqui, and M. Pedroni, “Eight lectures on integrable systems,” in: Integrability of Nonlinear Systems (Lect. Notes Phys., Vol. 495, Y. Kosmann-Schwarzbach, K. M. Tamizhmani, and B. Grammaticos, eds.), Springer, Heidelberg (1997), p. 256–296.CrossRefGoogle Scholar
  4. 4.
    I. M. Gelfand and I. Zakharevich, Selecta Math., n.s., 6, 131–183 (2000).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. V. Mikhailov and V. V. Sokolov, Comm. Math. Phys., 211, 231–251 (2000).MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    A. G. Reyman and M. A. Semenov-Tian-Shansky, Phys. Lett. A, 130, 456–460 (1988).MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    A. V. Odesskii and V. V. Sokolov, J. Phys. A, 39, 12447–12456 (2006).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Pichereau and G. Van de Weyer, J. Algebra, 319, 2166–2208 (2008).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    M. Aguiar, J. Algebra, 244, 492–532 (2001).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    T. Schedler, Math. Res. Lett., 10, 301–321 (2003).MathSciNetMATHGoogle Scholar
  11. 11.
    A. G. Élashvili, Funct. Anal. Appl., 16, No. 4, 326–328 (1982).CrossRefGoogle Scholar
  12. 12.
    M. Kontsevich, “Formal (non)commutative symplectic geometry,” in: The Gelfand Mathematical Seminars, 1990–1992 (L. Corwin, I. Gelfand, and J. Lepowsky, eds.), Birkhäuser, Boston, Mass. (1993), pp. 173–187.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • A. V. Odesskii
    • 1
  • V. N. Rubtsov
    • 2
    • 3
  • V. V. Sokolov
    • 4
  1. 1.Brock UniversitySt. CatharinesCanada
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.LAREMA, CNRSUniversité d’AngersAngersFrance
  4. 4.Landau Institute for Theoretical PhysicsRASMoscowRussia

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