Skip to main content
SpringerLink
Log in

On complete integrability of a hierarchy of finite-dimensional Hamiltonian systems

  • Published:
Acta Mathematicae Applicatae Sinica Aims and scope Submit manuscript

Abstract

A hierarchy of Hamiltonian systems obtained from the Lax pair of KdV hierarchy under the constraint condition on potentialu = 〉q, q〈 is presented. The independent integrals for these Hamiltonian systems are constructed by using recursion operator and shown to be in involution. Thus this hierarchy of Hamiltonian systems is completely integrable in the sense of Liouville, and they commute with each other.

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. J. Moser, Various Aspects of Integrable Hamiltonian Systems, in Progress in Mathematics (Birkhäuser),3 (1980), 233–289.

    Google Scholar 

  2. H.P. Mckean, Springer Lecture Notes in Mathematics,755, 1979, 83–200.

    Google Scholar 

  3. M. Jimbo and T. Miwa, Monodromy Preserving Deformations of Linear Ordinary Differential Equations with Rational Coefficients,Physica,2D (1981), 407–418.

    Google Scholar 

  4. H. Airault, H. P. Mckean, J. Moser, Rational and Elliptic Solutions of the KdV Equation and Related Many-body Problems,Comm. Pure Appl. Math.,30 (1977), 95–198.

    Google Scholar 

  5. H. Flaschka, Relations between Infinte-dimensional and Finite-dimensional Isospectral Equations, in Proceedings of RIMS Symposium on Nonlinear Integrable Systems-Classical Theory and Quantum Theory, Kyoto, 1981, Ed. M. Jimbo and T. Miwa, World Science Publishing Co., Singapore, 1983, 219–240.

    Google Scholar 

  6. Zeng Yunbo and Li Yishen, The Constraints of Potentials and the Finite-dimensional Integrable Systems,J. Math. Phys.,30 (1989), 1679–1689.

    Google Scholar 

  7. P.D. Lax, Periodic Solutions of the KdV Equation,Comm. Pure Appl. Math.,28 (1975), 141–188.

    Google Scholar 

  8. Zeng Yunbo and Li Yishen, Three Kinds of Constraint of Potential for KdV Hierarchy,Acta Math. Sinica,6(3) (1990), 257–272.

    Google Scholar 

  9. V.I. Arnold, Mathematical Methods of Classical Mechanics, MIR (Moscow, 1975), Springer, New York, 1978.

    Google Scholar 

  10. Cao Cewen, A Cubic System which Generates Bargmann Potential andN-gap Potential,Chinese Quarterly J. of Math.,3(1) (1988), 90–96.

    Google Scholar 

  11. S.S. Chern and C.K. Peng, Lie Groups and KdV Equations,Manuscripta, Math.,28 (1979), 207–217.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Science and Technology of China, 230026, Hefei, China

    Zeng Yunbo & Li Yishen

Authors
  1. Zeng Yunbo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Yishen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This project is supported by the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yunbo, Z., Yishen, L. On complete integrability of a hierarchy of finite-dimensional Hamiltonian systems. Acta Mathematicae Applicatae Sinica 8, 193–201 (1992). https://doi.org/10.1007/BF02014576

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02014576

Keywords

Search

Navigation