×

Inverse scattering method for a soliton cellular automaton. (English) Zbl 1160.37411

Summary: A set of action-angle variables for a soliton cellular automaton is obtained. It is identified with the rigged configuration, a well-known object in Bethe ansatz. Regarding it as the set of scattering data an inverse scattering method to solve initial value problems of this automaton is presented. By considering partition functions for this system a new interpretation of a fermionic character formula is obtained.

MSC:

37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37B15 Dynamical aspects of cellular automata
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
68Q80 Cellular automata (computational aspects)
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Takahashi, D., On some soliton systems defined by using boxes and balls, (Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA’93) (1993)), 555-558
[2] Takahashi, D.; Satsuma, J., A soliton cellular automaton, J. Phys. Soc. Jpn., 59, 3514-3519 (1990)
[3] Tokihiro, T.; Nagai, A.; Satsuma, J., Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization, Inverse Problems, 15, 1639-1662 (1999) · Zbl 1138.37341
[4] Tokihiro, T.; Takahashi, D.; Matsukidaira, J.; Satsuma, J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76, 3247-3250 (1996)
[5] Fukuda, K.; Okado, M.; Yamada, Y., Energy functions in box ball systems, Int. J. Mod. Phys. A, 15, 1379-1392 (2000) · Zbl 0980.37029
[6] Torii, M.; Takahashi, D.; Satsuma, J., Combinatorial representation of invariants of a soliton cellular automaton, Physica D, 92, 209-220 (1996) · Zbl 0885.68112
[7] Kuniba, A.; Okado, M.; Takagi, T.; Yamada, Y., Vertex operators and partition functions for the box-ball system, Research Institute for Mathematical Sciences (Kyoto Univ.) Kôkyûroku, 1302, 91-107 (2003), (in Japanese)
[8] Kerov, S. V.; Kirillov, A. N.; Reshetikhin, N. Yu., Combinatorics, Bethe ansatz, and representations of the symmetric group, Zap. Nauchn. Sem. LOMI, 155, 50-64 (1986) · Zbl 0617.20024
[9] Yoshihara, D.; Yura, F.; Tokihiro, T., Fundamental cycle of a periodic box-ball system, J. Phys. A: Math. Gen., 36, 99-121 (2003) · Zbl 1066.82036
[10] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0762.35001
[11] Hikami, K.; Inoue, R.; Komori, Y., Crystallization of the Bogoyavlensky lattice, J. Phys. Soc. Jpn., 68, 2234-2240 (1999) · Zbl 0943.82019
[12] Hatayama, G.; Kuniba, A.; Takagi, T., Soliton cellular automata associated with crystal bases, Nucl. Phys. B, 577, 619-645 (2000) · Zbl 1024.82017
[13] Hatayama, G.; Hikami, K.; Inoue, R.; Kuniba, A.; Takagi, T.; Tokihiro, T., The \(A_M^{(1)}\) automata related to crystals of symmetric tensors, J. Math. Phys., 42, 274-308 (2001) · Zbl 1032.17021
[14] Hatayama, G.; Kuniba, A.; Okado, M.; Takagi, T.; Yamada, Y., Remarks on fermionic formula, Contemp. Math., 248, 243-291 (1999) · Zbl 1032.81015
[15] Hatayama, G.; Kuniba, A.; Okado, M.; Takagi, T.; Tsuboi, Z., Paths, crystals and fermionic formulae, (Kashiwara, M.; Miwa, T., Prog. in Math. Phys. MathPhys Odyssey 2001, Integrable Models and Beyond (2002), Birkhäuser: Birkhäuser Basel), 205-272 · Zbl 1016.17011
[16] Macdonald, I., Symmetric Functions and Hall Polynomials (1995), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0824.05059
[17] Kirillov, A. N.; Reshetikhin, N. Yu., The Bethe ansatz and the combinatorics of Young tableaux, J. Sov. Math., 41, 925-955 (1988) · Zbl 0639.20029
[18] Nakayashiki, A.; Yamada, Y., Kostka polynomials and energy functions in solvable lattice models, Selecta Math., New Ser., 3, 547-599 (1997) · Zbl 0915.17016
[19] Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), Academic Press: Academic Press London · Zbl 0538.60093
[20] Kang, S.-J.; Kashiwara, M.; Misra, K. C.; Miwa, T.; Nakashima, T.; Nakayashiki, A., Affine crystals and vertex models, Int. J. Mod. Phys. A, 7, Suppl. 1A, 449-484 (1992) · Zbl 0925.17005
[21] Andrews, G. E., The Theory of Partitions (1984), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0581.05009
[22] Bethe, H. A., Zur Theorie der Metalle, I: Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys., 71, 205-231 (1931) · Zbl 0002.37205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.