Skip to main content
SpringerLink
Log in

Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Yu. A. Mitropol’skii, N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, and V. G. Samoilenko, Integrable Dynamical Systems. Spectral and Differential Geometric Aspects [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  2. A. K. Prikarpatskii and I. V. Mikityuk, Algebraic Aspects of Integrability of Nonlinear Dynamical Systems on Manifolds [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  3. A. K. Prikarpatskii, V. G. Samoilenko, R. I. Andrushkiv, Yu. A. Mitropol’skii, and M. M. Prytula, “Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear systems. I,” J. Math. Phys., 35, No. 4, 1763–1777 (1994).

    Article  MathSciNet  Google Scholar 

  4. A. K. Prikarpatskii, V. G. Samoilenko, and R. I. Andrushkiv, “Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear systems. II. The reduction via Dirac and canonical quantization procedure,” J. Math. Phys., 35, No. 8, 4088–4116 (1994).

    Article  MathSciNet  Google Scholar 

  5. L. A. Takhtadzhyan and L. D. Faddeev Hamiltonian Approach in Soliton Theory [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  6. R. Abraham and J. Marsden, Foundations of Mechanics, Addison-Wesley (1978).

  7. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and A. P. Pitaevskii, Soliton Theory. Inverse-Problem Method [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  8. V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).

    MATH  Google Scholar 

  9. F. Magri, “A simple model of the integrable Hamiltonian equations,” J. Math. Phys., 19, No. 3, 1156–1162 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Magri, “On the geometry of soliton equations,” Acta Appl. Math., 41, No. 2, 247–270 (1995).

    MATH  MathSciNet  Google Scholar 

  11. M. Blaszak, “Bi-Hamiltonian formulation for the Korteweg-de Vries hierarchy with sources,” J. Math Phys., 36, No. 9, 4826–4831 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  12. W. Oevel and W. Strampp, “Constrained KP-hierarchy and bi-Hamiltonian structures,” Commun. Math. Phys., 157, No. 1, 51–81 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. K. Prikarpatskii, R. V. Samulyak, D. Blackmore, et al., “Some remarks on Lagrangian and Hamiltonian formalism, related to infinite-dimensional dynamical systems with symmetries,” in: Condensed Matter. Physics. Proc. IFCS., 5, Lviv (1995), pp. 27–40.

Download references

Authors
  1. M. M. Prytula
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 49, No. 11, pp. 1512–1518, November, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prytula, M.M. Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds. Ukr Math J 49, 1697–1704 (1997). https://doi.org/10.1007/BF02487508

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487508

Keywords

Search

Navigation