Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 054, 12 pages      arXiv:1201.5429      https://doi.org/10.3842/SIGMA.2012.054

Discrete Integrable Equations over Finite Fields

Masataka Kanki a, Jun Mada b and Tetsuji Tokihiro a
a) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan
b) College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan

Received May 18, 2012, in final form August 15, 2012; Published online August 18, 2012

Abstract
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.

Key words: integrable system; discrete KdV equation; finite field; cellular automaton.

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