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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References

  1. Bialecki M., Doliwa A., Discrete Kadomtsev-Petviashvili and Korteweg-de Vries equations over finite fields, Theoret. and Math. Phys. 137 (2003), 1412-1418, nlin.SI/0302064.
  2. Bialecki M., Nimmo J.J.C., On pattern structures of the N-soliton solution of the discrete KP equation over a finite field, J. Phys. A: Math. Theor. 40 (2007), 949-959, nlin.SI/0608041.
  3. Date E., Jinbo M., Miwa T., Method for generating discrete soliton equation. II, J. Phys. Soc. Japan 51 (1982), 4125-4131.
  4. Doliwa A., Bialecki M., Klimczewski P., The Hirota equation over finite fields: algebro-geometric approach and multisoliton solutions, J. Phys. A: Math. Gen. 36 (2003), 4827-4839, nlin.SI/0211043.
  5. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  6. Kakei S., Nimmo J.J.C., Willox R., Yang-Baxter maps and the discrete KP hierarchy, Glasg. Math. J. 51 (2009), 107-119.
  7. Nijhoff F.W., Papageorgiou V.G., Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation, Phys. Lett. A 153 (1991), 337-344.
  8. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. (4) 146 (1987), 337-381.
  9. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation PV, Japan. J. Math. (N.S.) 13 (1987), 47-76.
  10. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221-255.
  11. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305-332.
  12. Papageorgiou V.G., Tongas A.G., Veselov A.P., Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys. 47 (2006), 083502, 16 pages, math.QA/0605206.
  13. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  14. Ramani A., Grammaticos B., Satsuma J., Integrability of multidimensional discrete systems, Phys. Lett. A 169 (1992), 323-328.
  15. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  16. Tamizhmani K.M., Private communication, 2012.
  17. Wolfram S., Statistical mechanics of cellular automata, Rev. Modern Phys. 55 (1983), 601-644.


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