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 SIGMA, 2012, Volume 8, 054, 12 pages (Mi sigma731)

Discrete integrable equations over finite fields

Masataka Kankia, Jun Madab, Tetsuji Tokihiroa

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

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.

Keywords: integrable system, discrete KdV equation, finite field, cellular automaton.

DOI: https://doi.org/10.3842/SIGMA.2012.054

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ArXiv: 1201.5429
MSC: 35Q53; 37K40; 37P25
Received: May 18, 2012; in final form August 15, 2012; Published online August 18, 2012
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Citation: Masataka Kanki, Jun Mada, Tetsuji Tokihiro, “Discrete integrable equations over finite fields”, SIGMA, 8 (2012), 054, 12 pp.

Citation in format AMSBIB
\Bibitem{KanMadTok12} \by Masataka Kanki, Jun Mada, Tetsuji Tokihiro \paper Discrete integrable equations over finite fields \jour SIGMA \yr 2012 \vol 8 \papernumber 054 \totalpages 12 \mathnet{http://mi.mathnet.ru/sigma731} \crossref{https://doi.org/10.3842/SIGMA.2012.054} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2970774} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000307816600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84865812791} 

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This publication is cited in the following articles:
1. Jimenez A., “Cellular Automata to Describe Seismicity: a Review”, Acta Geophys., 61:6 (2013), 1325–1350
2. Kanki M., Mada J., Tokihiro T., “The Space of Initial Conditions and the Property of an Almost Good Reduction in Discrete Painlevé II Equations Over Finite Fields”, J. Nonlinear Math. Phys., 20:1, SI (2013), 101–109
3. Yura F., “Solitons with a Nested Structure Over Finite Fields”, J. Phys. A-Math. Theor., 47:32 (2014), 325201
4. Isojima Sh., “on Exact Solutions With Periodic Structure of the Ultradiscrete Toda Equation With Parity Variables”, J. Math. Phys., 55:9 (2014), 093509
5. Bialecki M., Czechowski Z., “Random Domino Automaton: Modeling Macroscopic Properties By Means of Microscopic Rules”, Achievements, History and Challenges in Geophysics, Geoplanet-Earth and Planetary Sciences, eds. Bialik R., Majdanski M., Moskalik M., Springer-Verlag Berlin, 2014, 223–241
6. Roberts J.A.G., Tran D.T., “Signatures Over Finite Fields of Growth Properties For Lattice Equations”, J. Phys. A-Math. Theor., 48:8 (2015), 085201
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