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


SIGMA 6 (2010), 028, 11 pages      arXiv:0911.2684      http://dx.doi.org/10.3842/SIGMA.2010.028
Contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems”

Yang-Baxter Maps from the Discrete BKP Equation

Saburo Kakei a, Jonathan J.C. Nimmo b and Ralph Willox c
a) Department of Mathematics, College of Science, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
b) Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, UK
c) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Received November 13, 2009, in final form March 19, 2010; Published online March 31, 2010

Abstract
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

Key words: Yang-Baxter map; discrete BKP equation.

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References

  1. Adler V.E., Bobenko A.I., Suris Yu B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  2. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci. 18 (1982), 1077-1110.
  3. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan 51 (1982), 4125-4131.
  4. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. V, J. Phys. Soc. Japan 52 (1983), 766-771.
  5. Etingof P., Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Comm. Algebra 31 (2003), 1961-1973, math.QA/0112278.
  6. Hatayama G., Hikami K., Inoue R., Kuniba A., Takagi T., Tokihiro T., The AM(1) automata related to crystals of symmetric tensors, J. Math. Phys. 42 (2001), 274-308, math.QA/9912209.
  7. Hatayama G., Kuniba A., Takagi T., Soliton cellular automata associated with crystal bases, Nuclear Phys. B 577 (2000), 619-645, solv-int/9907020.
  8. Hirota R., Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation, J. Phys. Soc. Japan 43 (1977), 1424-1433.
  9. Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 3785-3791.
  10. Hirota R., Ultradiscretization of the Sawada-Kotera equation, in Mathematics and Physics in Nonlinear Waves (November 6-8, 2008, Fukuoka, Japan), Reports of RIAM Symposium, Vol. 20ME-S7, Research Institute for Applied Mechanics, Kyushu University, 2009, 76-85 (in Japanese).
  11. Kakei S., Nimmo J.J.C., Willox R., Yang-Baxter maps and the discrete KP hierarchy, Glasg. Math. J. 51 (2009), no. A, 107-119.
  12. Maillet J.M., Nijhoff F.W., Integrability for multidimensional lattice models, Phys. Lett. B 224 (1989), 389-396.
  13. Miwa T., On Hirota's difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9-12.
  14. Nimmo J.J.C., Darboux transformations and the discrete KP equation, J. Phys. A: Math. Gen. 30 (1997), 8693-8704.
  15. Nimmo J.J.C., Darboux transformations for discrete systems, Chaos Solitons Fractals 11 (2000), 115-120.
  16. Papageorgiou V.G., Suris Yu.B., Tongas A.G., Veselov A.P., On quadrirational Yang-Baxter maps, arXiv:0911.2895.
  17. 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.
  18. Suris Yu.B., Veselov A.P., Lax matrices for Yang-Baxter maps, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 223-230, math.QA/0304122.
  19. Takagi T., Soliton cellular automata, in Combinatorial Aspect of Integrable Systems, MSJ Mem., Vol. 17, Math. Soc. Japan, Tokyo, 2007, 105-144.
  20. Takahashi D., Matsukidaira J., Box and ball system with a carrier and ultradiscrete modified KdV equation, J. Phys. A: Math. Gen. 30 (1997), L733-L739.
  21. Takahashi D., Satsuma J., A soliton cellular automaton, J. Phys. Soc. Japan 59 (1990), 3514-3519.
  22. Tokihiro T., Takahashi D., Matsukidaira J., Satsuma J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996), 3247-3250.
  23. Veselov A.P., Yang-Baxter maps and integrable dynamics, Phys. Lett. A 314 (2003), 214-221, math.QA/0205335.
  24. Veselov A.P., Yang-Baxter maps: dynamical point of view, in Combinatorial Aspect of Integrable Systems, MSJ Mem., Vol. 17, Math. Soc. Japan, Tokyo, 2007, 145-167, math.QA/0612814.
  25. Willox R., Tokihiro T., Satsuma J., Darboux and binary Darboux transformations for the nonautonomous discrete KP equation, J. Math. Phys. 38 (1997), 6455-6469.
  26. Willox R., Tokihiro T., Satsuma J., Nonautonomous discrete integrable systems, Chaos Solitons Fractals 11 (2000), 121-135.

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