Thanks to visit codestin.com
Credit goes to link.springer.com

Skip to main content
Springer Nature Link
Log in

On integrability of Hirota-Kimura type discretizations

  • Special Issue: Algebraic Integrability
  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

We give an overview of the integrability of the Hirota-Kimura discretizationmethod applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota-Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.

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

Access this article

Log in via an institution

Subscribe and save

Springer+
from £29.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

References

  1. Kahan, W., Unconventional Numerical Methods for Trajectory Calculations, Lecture notes, University of California, Berkeley, CA, October 1993.

    Google Scholar 

  2. Petrera, M., Pfadler, A., and Suris, Yu. B., On Integrability of Hirota-Kimura-Type Discretizations: Experimental Study of the Discrete Clebsch System, Experiment. Math., 2009, vol. 18, no. 2, pp. 223–247.

    Article  MathSciNet  MATH  Google Scholar 

  3. Kahan, W. and Li, R.-C., Unconventional Schemes for a Class of Ordinary Differential Equations (with Applications to the Korteweg-de Vries Equation), J. Comput. Phys., 1997, vol. 134, pp. 316–331.

    Article  MathSciNet  MATH  Google Scholar 

  4. Hirota, R. and Kimura, K., Discretization of the Euler Top, J. Phys. Soc. Japan, 2000, vol. 69, no. 3, pp. 627–630.

    Article  MathSciNet  MATH  Google Scholar 

  5. Kimura, K. and Hirota, R., Discretization of the Lagrange Top, J. Phys. Soc. Japan, 2000, vol. 69, no. 10, pp. 3193–3199.

    Article  MathSciNet  MATH  Google Scholar 

  6. Suris, Yu. B., The Problem of Integrable Discretization: Hamiltonian Approach, Progr. Math., vol. 219, Basel: Birkhäuser, 2003.

    MATH  Google Scholar 

  7. Sanz-Serna, J.M., An Unconventional Symplectic Integrator of W. Kahan, Appl. Numer. Math., 1994, vol. 16, pp. 245–250.

    Article  MathSciNet  MATH  Google Scholar 

  8. Adler, M., van Moerbeke, P., and Vanhaecke, P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergeb. Math. Grenzgeb. (3), vol. 47, Berlin-Heidelberg: Springer, 2004.

    MATH  Google Scholar 

  9. Quispel, G. R. W., Roberts, J.A.G., and Thompson, C. J., Integrable Mappings and Soliton Equations, Phys. D, 1989, vol. 34, pp. 183–192.

    Article  MathSciNet  MATH  Google Scholar 

  10. Suris, Yu. B., Integrable Mappings of the Standard Type, Funktsional. Anal. i Prilozhen., 1989, vol. 23, no. 1, pp. 84–85 [Funct. Anal. Appl., 1989, vol. 23, no. 1, pp. 74–76].

    Article  MathSciNet  Google Scholar 

  11. Suslov, G., Theoretical Mechanics, Moscow-Leningrad: Gostekhizdat, 1946.

    Google Scholar 

  12. Dragović, V. and Gajić, B., Hirota-Kimura Type Discretization of the Classical Nonholonomic Suslov Problem, Reg. Chaot. Dyn., 2008, vol. 13, no. 4, pp. 250–256.

    Article  Google Scholar 

  13. Hitchin, N. J., Manton, N. S., and Murray, M.K., Symmetric Monopoles, Nonlinearity, 1995, vol. 8, no. 5, pp. 661–692.

    Article  MathSciNet  Google Scholar 

  14. Petrera, M. and Suris, Yu. B., On the Hamiltonian Structure of Hirota-Kimura Discretization of the Euler Top, Math. Nachr., 2010, no. 11, pp. 1654–1663.

  15. Volterra, V., Sur la théorie des variations des latitudes, Acta Math., 1899, vol. 22, pp. 201–357.

    Article  MathSciNet  Google Scholar 

  16. Basak, I., Explicit Solution of the Zhukovski-Volterra Gyrostat, Reg. Chaot. Dyn., 2009, vol. 14, no. 2, pp. 223–236.

    Article  MathSciNet  Google Scholar 

  17. Veselov, A.P. and Shabat, A. B., Dressing Chains and Spectral Theory of the Schrödinger Operator, Funktsional. Anal. i Prilozhen., 1993, vol. 27, no. 2, pp. 1–21 [Funct. Anal. Appl., 1993, vol. 27, no. 2, pp. 81–96].

    Article  MathSciNet  Google Scholar 

  18. Golse, F., Mahalov, A., and Nicolaenko, B., Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains, in Instability in Models Connected with Fluid Flows: 1, Int. Math. Ser. (N. Y.), vol.6, New York: Springer, 2008, pp. 300–338.

    Google Scholar 

  19. Alber, M., Luther, G.G., Marsden, J.E., and Robbins, J., Geometric Phases, Reduction and Lie-Poisson Structure for the Resonant Three-Wave Interaction, Phys. D, 1998, nos. 1–4, vol. 123, pp. 271–290.

    Article  MathSciNet  MATH  Google Scholar 

  20. Marsden, J. and Ratiu, T., Introduction to Mechanics and Symmetry, Texts Appl. Math., vol. 17, 2nd ed., New York: Springer, 1999.

    MATH  Google Scholar 

  21. Kirchhoff, G., Über die Bewegung eines Rotationskörpers in einer Flüssigkeit, J. Reine Angew. Math., 1870, vol. 71, pp. 237–262.

    Article  Google Scholar 

  22. Clebsch, A., Über die Bewegung eines Körpers in einer Flüssigkeit, Math. Ann., 1870, vol. 3, pp. 238–262.

    Article  MathSciNet  Google Scholar 

  23. Perelomov, A.M., Integrable Systems of Classical Mechanics and Lie Algebras: Vol. 1, Basel: Birkhäuser, 1990.

    Google Scholar 

  24. Reyman, A.G. and Semenov-Tian-Shansky, M.A., Group Theoretical Methods in the Theory of Finite Dimensional Integrable Systems, in Dynamical Systems VII: Integrable Systems. Nonholonomic Dynamical Systems, V. I. Arnol’d, S. P. Novikov (Eds.), Encyclopaedia Math. Sci., vol. 16, Berlin: Springer, 1994, pp. 116–225.

    Google Scholar 

  25. Ratiu, T., Nonabelian Semidirect Product Orbits and Their Relation to Integrable Systems, Oberwolfach Rep., 2006, vol. 3, no. 1.

  26. Gaudin, M., Diagonalisation d’une classe d’Hamiltoniens de spin, J. Physique, 1976, vol. 37, no. 10, pp. 1089–1098.

    MathSciNet  Google Scholar 

  27. Petrera, M. and Suris, Yu. B., An Integrable Discretization of the Rational su(2) Gaudin Model and Related Systems, Comm. Math. Phys., 2008, vol. 283, pp. 227–253.

    Article  MathSciNet  MATH  Google Scholar 

  28. Musso, F., Petrera, M., Ragnisco, O., and Satta, G., A Rigid Body Dynamics Derived from a Class of Extended Gaudin Models: An Integrable Discretization, Regul. Chaot. Dyn., 2005, vol. 10, no. 4, pp. 363–380.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 7-2, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Matteo Petrera, Andreas Pfadler & Yuri B. Suris

Authors
  1. Matteo Petrera
    View author publications

    Search author on:PubMed Google Scholar

  2. Andreas Pfadler
    View author publications

    Search author on:PubMed Google Scholar

  3. Yuri B. Suris
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Matteo Petrera.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petrera, M., Pfadler, A. & Suris, Y.B. On integrability of Hirota-Kimura type discretizations. Regul. Chaot. Dyn. 16, 245–289 (2011). https://doi.org/10.1134/S1560354711030051

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue date:

  • DOI: https://doi.org/10.1134/S1560354711030051

MSC2010 numbers

Keywords