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Letters in Mathematical Physics

, Volume 60, Issue 3, pp 211–219 | Cite as

Discrete Dynamical Systems with W(A(1)m−1 × A(1)n−1) Symmetry

  • Kenji Kajiwara
  • Masatoshi Noumi
  • Yasuhiko Yamada
Article

Abstract

We give a birational realization of affine Weyl group of type A(1)m−1 × A(1)n−1. We apply this representation to construct some discrete integrable systems and discrete Painlevé equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.

discrete Painlevé equation Bäcklund transformation affine Weyl group discrete Toda equation 

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References

  1. 1.
    Kajiwara, K., Noumi, M. and Yamada, Y.: A study on the fourth q-Painlevé equation, J. Phys. A: Math. Gen. 34 (2001), 8563-8581.Google Scholar
  2. 2.
    Kirillov, A. N.: Introduction to tropical combinatorics, In: A. N. Kirillov and N. Liskova (eds), Physics and Combinatorics 2000, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 82-150.Google Scholar
  3. 3.
    Noumi, M. and Yamada, Y.: Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281-295.Google Scholar
  4. 4.
    Noumi, M. and Yamada, Y.: Birational Weyl group action arising from a nilpotent Poisson algebra, In: A. N. Kirillov, A. Tsuchiya, H. Umemura (eds), Physics and Combinatorics 1999, Proc. Nagoya 1999 Internat. Workshop, World Scientific, Singapore, 2001, pp. 287-319.Google Scholar
  5. 5.
    Tokihiro, T., Takahashi, D., Matsukidaira, J. and Satsuma, J.: From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996), 3247-3250.Google Scholar
  6. 6.
    Takahashi, D. and Satsuma, J.: A soliton cellular automaton, J. Phys. Soc. Japan 59 (1990), 3541-3519.Google Scholar
  7. 7.
    Hatayama, G., Kuniba, A. and Takagi, T.: Soliton cellular automata associated with finite crystals, Nuclear Phys. B 577 (2000), 619-645.Google Scholar
  8. 8.
    Hirota, R.: Nonlinear partial difference equations. II. Discrete-time Toda equation, J. Phys. Soc. Japan 43 (1977), 2074-2078.Google Scholar
  9. 9.
    Hirota, R. and Tsujimoto, S.: Conserved quantities of a class of nonlinear difference-difference equations, J. Phys. Soc. Japan 64 (1995), 3125-3127.Google Scholar
  10. 10.
    Saito, S., Saitoh, N., Yamamoto, J. and Yoshida, K.: A characterization of discrete time soliton equations, Preprint, arXiv:nlin.SI/0108019.Google Scholar
  11. 11.
    Yamada, Y.: A birational representation of Weyl group, combinatorial R-matrix andd iscrete Toda equation, In: A. N. Kirillov and N. Liskova (eds), Physics and Combinatorics 2000, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 305-319.Google Scholar
  12. 12.
    Masuda, T.: On the rational solutions of q-Painlevé V equation, Preprint, arXiv:nlin.SI/ 0107050, to appear in Nagoya Math. J. (2002).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Kenji Kajiwara
    • 1
  • Masatoshi Noumi
    • 2
  • Yasuhiko Yamada
    • 2
  1. 1.Graduate School of MathematicsKyushu UniversityHigashi-ku, FukuokaJapan
  2. 2.Department of MathematicsKobe UniversityKobeJapan

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