On solutions to evolution equations defined by lattice operators

  • Takatoshi Ikegami
  • Daisuke Takahashi
  • Junta Matsukidaira
Open Access
Original Paper Area 1

Abstract

We discuss a specific form of evolution equations defined by lattice operators. We give exact solutions for a class of those equations and evaluate the complexity of the solutions. Moreover we discuss the relationship between them and binary cellular automata, and analyze their asymptotic behavior utilizing the explicit expression of the solution.

Keywords

Lattice operator Max-plus algebra Ultradiscretization Cellular automaton Integrable system 

Mathematics Subject Classification

39A14 06D99 37B15 68Q80 

References

  1. 1.
    Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices. W. H. Freeman, San Francisco (1971)Google Scholar
  2. 2.
    Goles E., Latapy M., Magnien C., Morvan M., Phan H.D.: Sandpile models and lattices: a comprehensive survey. Theor. Comp. Sci. 322, 383–407 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bellon, M.P., Viallet, C.M. Algebraic entropy. Commun. Math. Phys. 204, 425–437 (1999)Google Scholar
  4. 4.
    Grammaticos B., Ramani A., Viallet C.M.: Solvable chaos. Phys. Lett. A 336, 152–158 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn. CRC Press, Boca Raton (1998)Google Scholar
  6. 6.
    Gaubert S., Plus M.: Methods and applications of (max, +) linear algebra. Lect. Notes Comput. Sci. 1200, 261–282 (1997)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Wolfram, S.: Theory and Applications of Cellular Automata. World Scientific, Singapore (1986)Google Scholar
  8. 8.
    Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)Google Scholar
  9. 9.
    Takahashi D., Satsuma J.: A soliton cellular automaton. J. Phys. Soc. Jpn. 59, 3514–3519 (1990)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Tokihiro T., Takahashi D., Matsukidaira J., Satsuma J.: From soliton equations to integrable cellular automata through a limiting procedure. Phys. Rev. Lett. 76, 3247–3250 (1996)CrossRefGoogle Scholar
  11. 11.
    Nishinari K., Takahashi D.: Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton. J. Phys. A 31, 5439–5450 (1998)CrossRefMATHGoogle Scholar
  12. 12.
    Takahashi, D., Matsukidaira, J., Hara, H., Feng, B.: Max-plus analysis on some binary particle systems. J. Phys. A Math. Theor. 44, 135102 (2011)Google Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Takatoshi Ikegami
    • 1
  • Daisuke Takahashi
    • 1
  • Junta Matsukidaira
    • 2
  1. 1.Department of Applied MathematicsWaseda UniversityTokyoJapan
  2. 2.Department of Applied Mathematics and InformaticsRyukoku UniversityShigaJapan

Personalised recommendations