Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-04T17:14:34.063Z Has data issue: false hasContentIssue false
  • English
  • Français

YANG–BAXTER MAPS AND THE DISCRETE KP HIERARCHY

Published online by Cambridge University Press:  01 February 2009

S. KAKEI ,
J. J. C. NIMMO  and
R. WILLOX
S. KAKEI
Affiliation:
Department of Mathematics, Rikkyo University, Nishi-ikebukuro, Toshima-ku, Tokyo 171-8501, Japan e-mail: kakei@rkmath.rikkyo.ac.jp
J. J. C. NIMMO
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G128QQ, UK e-mail: jjcn@maths.gla.ac.uk
R. WILLOX
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan e-mail: willox@ms.u-tokyo.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples.

Keywords

39A1039A12
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue A , February 2009 , pp. 107 - 119
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Berenstein, A. and Kazhdan, D., Geometric and unipotent crystals, Geom. Funct. Anal., Special Volume, ‘Part I’ (2000), 188–236.CrossRefGoogle Scholar
2.Białecki, M., On discrete Sato-like theory with some specializations for finite fields, RIMS kōkyūroku (to appear).Google Scholar
3.Bobenko, A. I. and Suris, Y. B., Integrable systems on quad-graphs, Int. Math. Res. Not. 11 (2002), 573–611.CrossRefGoogle Scholar
4.Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., Transformation groups for soliton equations: Euclidean Lie algebras and reduction of the KP hierarchy, Publ. RIMS, Kyoto Univ. 18 (1982), 1077–1110.CrossRefGoogle Scholar
5.Date, E., Jimbo, M. and Miwa, T., Method for generating discrete soliton equation. II, J. Phys. Soc. Jpn. 51 (1982), 4125–4131.CrossRefGoogle Scholar
6.Drinfeld, V. G., On some unsolved problems in quantum group theory, in Lecture Notes in Mathematics, vol. 1510 (Kulish, P. P., Editor) (Springer, Berlin, 1992), 1–8.Google Scholar
7.Etingof, P., Geometric Crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Comm. Algeb. 31 (2003), 1961–1973.CrossRefGoogle Scholar
8.Hatayama, G., Hikami, K., Inoue, R., Kuniba, A.. Takagi, T. and Tokihiro, T., The A N(1) automata related to crystals of symmetric tensors, J. Math. Phys. 42 (2001), 274–308.CrossRefGoogle Scholar
9.Hirota, R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50 (1981), 3785–3791.CrossRefGoogle Scholar
10.Kuniba, A., Okado, M., Sakamoto, R., Takagi, T. and Yamada, Y., Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection, Nuclear Phys. B 740 (2006), 299–327.CrossRefGoogle Scholar
11.Kuniba, A., Okado, M., Takagi, T. and Yamada, Y., Geometric crystal and tropical R for D (1)n, Int. Math. Res. Not. 48 (2003), 2565–2620.CrossRefGoogle Scholar
12.Miwa, T., On Hirota's difference equations, Proc. Jpn. Acad. Ser. A Math. Sci. 58 (1982), 9–12.CrossRefGoogle Scholar
13.Nijhoff, F. W., The direct linearizing transform for the τ function in three-dimensional lattice equations, Phys. Lett. A 110 (1985), 10–14.CrossRefGoogle Scholar
14.Nijhoff, F. W., Theory of integrable three-dimensional lattice equations, Lett. Math. Phys. 9 (1985), 235–241.CrossRefGoogle Scholar
15.Nijhoff, F. W., Quispel, G. R. W. and Capel, H. W., Direct linearization of nonlinear difference–difference equations Phys. Lett. A 97 (1983), 125–128.CrossRefGoogle Scholar
16.Nimmo, J. J. C., Darboux transformations and the discrete KP equation, J. Phys. A 30 (1997), 8693–8704.CrossRefGoogle Scholar
17.Ohta, Y., Hirota, R., Tsujimoto, S. and Imai, T., Casorati and Disrete gram type determinant representations of solutions to the discrete KP hierarchy, J. Phys. Soc. Jpn. 62 (1993), 1872–1886.CrossRefGoogle Scholar
18.Papageorgiou, V. G., Tongas, A. G. and Veselov, A. P., Yang–Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys. 47 (2006), 083502.CrossRefGoogle Scholar
19.Post, G., Note on ‘N-pseudoreductions’ of the KP hierarchy, J. Phys. A 20 (1987), 6591–6592.CrossRefGoogle Scholar
20.Takahashi, D. and Matsukidaira, J., Box and ball system with a carrier and ultradiscrete modified KdV equation, J. Phys. A 30 (1997), L733–L739.CrossRefGoogle Scholar
21.Takahashi, D. and Satsuma, J., A soliton cellular automaton, J. Phys. Soc. Jpn. 59 (1990), 3514–3519.CrossRefGoogle Scholar
22.Tokihiro, T., Takahashi, D., Matsukidaira, J. and Satsuma, J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996), 3247–3250.CrossRefGoogle ScholarPubMed
23.Tsujimoto, S., On a discrete analogue of the two-dimensional Toda lattice hierarchy, Publ. Res. Inst. Math. Sci. 38 (2002), 113–133.CrossRefGoogle Scholar
24.Tsujimoto, S. and Hirota, R., Ultradiscrete KdV equation, J. Phys. Soc. Jpn. 67 (1998), 1809–1810.Google Scholar
25.Ueno, K. and Takasaki, K., Toda lattice hierarchy, in Advanced Studies in Pure Mathematics, vol. 4 (Okamoto, K., Editor) (North-Holland, Amsterdam, 1984), 1–94.Google Scholar
26.Veselov, A. P., Yang–Baxter maps and integrable dynamics, Phys. Lett. A 314 (2003), 214–221.CrossRefGoogle Scholar
27.Veselov, A. P., Yang–Baxter maps: dynamical point of view. Combinatorial Aspect of Integrable Systems, in MSJ Mem, vol. 17 (Mathematical Society of Japan, Tokyo, 2007), 145–167.Google Scholar
28.Willox, R., A direct linearization of the KP hierarchy and an initial value problem for tau functions, RIMS kōkyūroku 1170 (2000), 111–118.Google Scholar