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On two integrable cellular automata

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Abstract

We describe two simple cellular automata (CA) models which exhibit the essential attributes of soliton systems. The first one is an invertible, 2-state, 1-dimensional CA or, in other words, a nonlinearZ 2-valued dynamical system with discrete space and time. Against a vacuum state of 0, the system exhibits light cone particles in both spatial directions, which interact in a soliton-like fashion. A complete solution of this system is obtained. We also consider another CA, which is described by the Hirota equation over a finite field, and present a Lax representation for it.

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Author notes

  1. Alexander Bobenko

    Present address: St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, Russia

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623, Berlin, Germany

    Alexander Bobenko, Charlie Gunn & Ulrich Pinkall

  2. Fachbereich Physik der Universität Freiburg, Hermann-Herder-Str. 3, D-79104, Freiburg, Germany

    Martin Bordemann

Authors
  1. Alexander Bobenko
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  2. Martin Bordemann
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  3. Charlie Gunn
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  4. Ulrich Pinkall
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Additional information

Communicated by N. Yu. Reshetikhin

Supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288

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Bobenko, A., Bordemann, M., Gunn, C. et al. On two integrable cellular automata. Commun.Math. Phys. 158, 127–134 (1993). https://doi.org/10.1007/BF02097234

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  • DOI: https://doi.org/10.1007/BF02097234

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