Advertisement

Ultradiscrete Systems (Cellular Automata)

  • Tetsuji Tokihiro
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 644)

Abstract

Ultradiscretization is a limiting procedure which allows one to obtain a cellular automaton (CA) from continuous equations. Using this method, we can construct integrable CAs from integrable partial difference equations. In this course, we focus on a typical integrable CA, called a Box and Ball system (BBS), and review its peculiar features. Since a BBS is an ultradiscrete limit of the discrete KP equation and discrete Toda equation, we can obtain explicit solutions and conserved quantities for the BBS. Furthermore the BBS is also regarded as a limit (crystallization) of an integrable lattice model. Recent topics, and a periodic BBS in particular are also reviewed.

Keywords

Cellular Automaton Soliton Solution Young Diagram Fundamental Cycle Boltzmann Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1. S. Wolfram: Cellular Automata and Complexity (Addison-Wesley, Reading, MA 1994)Google Scholar
  2. 2. S. Wolfram: Phys. Scr. T9, 170 (1985)Google Scholar
  3. 3. T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma: Phys. Rev. Lett. 76, 3247 (1996).CrossRefGoogle Scholar
  4. 4. J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii: Phys. Lett. A 255, 287 (1997)CrossRefGoogle Scholar
  5. 5. K. Park, K. Steiglitz, and W. P. Thurston: Physica D 19, 423 (1986) CrossRefGoogle Scholar
  6. 6. A. S. Fokas, E. P. Papadopoulou and Y. G. Saridakis: Physica D 41, 297 (1990) CrossRefGoogle Scholar
  7. 7. A. S. Fokas, E. P. Papadopoulou, Y. G. Saridakis and M. J. Ablowitz: Studies in Applied Mathematics 81, 153 (1989)MathSciNetzbMATHGoogle Scholar
  8. 8. M. J. Ablowitz, J. M. Keiser, L. A. Takhtajan: Quaestiones Math. 15, 325 (1992)MathSciNetzbMATHGoogle Scholar
  9. 9. D. Takahashi: ‘On some soliton systems defined by boxes and balls’. In: Proceedings of the International Symposium on Nonlinear Theory and Its Applications, NOLTA’93, p.555 (1991)Google Scholar
  10. 10. D. Takahashi and J. Satsuma: J. Phys. Soc. Jpn. 59, 3514 (1990)Google Scholar
  11. 11. D.Yoshihara, F.Yura and T.Tokihiro: J. Phys. A.FMath. Gen. 36, 99 (2003)CrossRefGoogle Scholar
  12. 12. A. Nagai, D. Takahashi and T. Tokihiro: Physics Letters A 255, 265 (1999)CrossRefGoogle Scholar
  13. 13. D. Takahashi and J. Matsukidaira: J. Phys. A.FMath. Gen. 30, 733 (1997) Google Scholar
  14. 14. T. Tokihiro, A. Nagai and J. Satsuma: Inverse Probl. 15, 1639 (1999)CrossRefGoogle Scholar
  15. 15. T. Tokihiro, D. Takahashi and J. Matsukidaira: J. Phys. A.FMath. Gen. 33, 607 (2000)CrossRefGoogle Scholar
  16. 16. K. Hikami, R. Inoue, and Y. Komori: J. Phys. Soc. Jpn. 68, 2234 (2000)zbMATHGoogle Scholar
  17. 17. K. Fukuda, M. Okado, and Y. Yamada: Int. J. Mod. Phys. A 15, 1379 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18. G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi, and T. Tokihiro: J. Math. Phys. 42, 274 (2001)CrossRefGoogle Scholar
  19. 19. M. Bruschi, P. M. Santini and O. Ragnisco: Physics Letters A 169 151 (1992)Google Scholar
  20. 20. A. Bobenko, M. Bordemann, C. Gunn, U. Pinkall: Comm. Math. Phys. 158, 127 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21. R. Hirota: J. Phys. Soc. Jpn.50, 3785 (1981)Google Scholar
  22. 22. T. Miwa: Proceedings of the Japan Academy 58 A, 9(1982)Google Scholar
  23. 23. E. Date, M. Jimbo, T. Miwa: J. Phys. Soc. Jpn. 51, 4125 (1982)Google Scholar
  24. 24. R. Willox and J. Satsuma: Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems, Lect. Notes Phys. 644, 17 (2004)Google Scholar
  25. 25. M. Sato: RIMS Kokyuroku 439, 30 (1981).Google Scholar
  26. 26. E. Date, M. Jimbo, M. Kashiwara, T. Miwa: ‘Transformation groups for soliton equations’. In: Proceedings of RIMS symposium on Non-linear Integrable Systems-Classical Theory and Quantum Theory, Kyoto, Japan May 13 – May 16, 1981, ed. by M. Jimbo, T. Miwa (World Scientific Publ. Co., Singapore 1983) pp. 39–119Google Scholar
  27. 27. T. Miwa, M. Jimbo and E. Date: Solitons – Differential equations, symmetries and infinite dimensional algebras (Cambridge University Press, UK 2000)Google Scholar
  28. 28. M. Toda: J. Phys. Soc. Jpn. 22, 431 (1967)Google Scholar
  29. 29. A. Nagai, T. Tokihiro and J. Satsuma: Glasgow Math. J. 43A,91 (2001)CrossRefGoogle Scholar
  30. 30. M. Torii, D. Takahashi and J. Satsuma: Physica D 92, 209 (1996)CrossRefGoogle Scholar
  31. 31. W. Fulton: Young Tableaux (Cambridge University Press, UK, 1997)Google Scholar
  32. 32. C. N. Yang: Physical Review Letters 19, 1312 (1967)CrossRefGoogle Scholar
  33. 33. R. J. Baxter: Annals of Physics 70, 193 (1972)Google Scholar
  34. 34. P. P. Kulish and E. K. Sklyanin: Journal of Soviet Mathematics 19, 1596 (1982).Google Scholar
  35. 35. A. Nakayashiki and Y. Yamada: Selecta Mathematica, New Series 30, 547 (1997)Google Scholar
  36. 36. See for example, M. Jimbo: ‘Topics from representations of U q(g)-an introductory guide to physicists’. In: Nankai Lectures on Mathematical Physics (World Scientific, Singapore, 1992), pp. 1-61.Google Scholar
  37. 37. M. Kashiwara: Communications in Mathematical Physics 133, 249 (1990)Google Scholar
  38. 38. F. Yura and T. Tokihrio: J. Phys. A.FMath. Gen. 35, 3787 (2002)CrossRefGoogle Scholar
  39. 39. T. Kimijima and T. Tokihiro: Inverse Problems 18, 1705 (2002)CrossRefGoogle Scholar
  40. 40. J. Mada and T. Tokihiro: J. Phys. A.FMath. Gen.36, 7251 (2003)Google Scholar
  41. 41. A. Ramani, D. Takahashi, B. Grammaticos and Y. Ohta: Physica D 114 185 (1998)Google Scholar

Authors and Affiliations

  • Tetsuji Tokihiro
    • 1
  1. 1.Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914Japan

Personalised recommendations