Abstract
The \(U_q(\widehat{sl}_2)\) vertex model at q = 0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann theta function with rational characteristics.
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References
Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM Stud. Appl. Math. 4 (1981)
Baxter R.J. (1982). Exactly solved models in statistical mechanics. Academic, London
Bethe H.A. (1931). Zur Theorie der Metalle, I., Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Physik 71: 205–231
Date E. and Tanaka S. (1976). Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice. Prog. Theor. Phys. Suppl. 59: 107–125
Dubrovin B.A., Matveev V.B. and Novikov S.P. (1976). Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties. Russ. Math. Surv. 31: 59–146
Fukuda K., Okado M. and Yamada Y. (2000). Energy functions in box ball systems. Int. J. Mod. Phys. A 15: 1379–1392
Gardner C.S., Greene J.M., Kruskal M.D. and Miura R.M. (1967). Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19: 1095–1097
Hatayama G., Hikami K., Inoue R., Kuniba A., Takagi T., Tokihiro T., (2001). The \(A^{(1)}_M\) automata related to crystals of symmetric tensors. J. Math Phys. 42, 274–308
Hatayama G., Kuniba A. and Takagi T. (2000). Soliton cellular automata associated with crystal bases. Nucl. Phys. B 577 [PM]: 619–645
Jimbo M. and Miwa T. (1983). Solitons and infinite dimensional Lie algebras. Publ. RIMS. Kyoto Univ. 19: 943–1001
Kac M. and Moerbeke P. (1975). A complete solution of the periodic Toda problem. Proc. Nat. Acad. Sci. USA. 72: 2879–2880
Kashiwara M. (1991). On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63: 465–516
Kirillov A.N., Reshetikhin N. and Yu. (1988). The Bethe ansatz and the combinatorics of Young tableaux. J. Sov. Math. 41: 925–955
Kuniba A. and Nakanishi T. (2000). The Bethe equation at q = 0, the Möbius inversion formula and weight multiplicities: I The sl(2) case. Prog. Math. 191: 185–216
Kuniba, A., Sakamoto, R.: The Bethe ansatz in a periodic box–ball system and the ultradiscrete Riemann theta function. J. Stat. Mech. P09005 (2006)
Kuniba, A., Sakamoto, R., Yamada, Y.: Tau functions in combinatorial Bethe ansatz. preprint math.QA/0610505
Kuniba A., Takagi T. and Takenouchi A. (2006). Bethe ansatz and inverse scattering transform in a periodic box–ball system. Nucl. Phys. B [PM]: 354–397
Kuniba A. and Takenouchi A. (2006). Bethe ansatz at q = 0 and periodic box–ball systems. J. Phys. A Math. Gen. 39: 2551–2562
Mumford D. (1984). Tata Lectures on Theta II. Birkhäuser, Boston
Takahashi D. and Satsuma J. (1990). A soliton cellular automaton. J. Phys. Soc. Jpn. 59: 3514–3519
Tokihiro T., Takahashi D., Matsukidaira J. and Satsuma J. (1996). From soliton equations to integrable cellular automata through a limiting procedure. Phys. Rev. Lett. 76: 3247–3250
Yoshihara D., Yura F. and Tokihiro T. (2003). Fundamental cycle of a periodic box–ball system. J. Phys. A Math. Gen. 36: 99–121
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Kuniba, A., Sakamoto, R. Combinatorial Bethe Ansatz and Ultradiscrete Riemann Theta Function with Rational Characteristics. Lett Math Phys 80, 199–209 (2007). https://doi.org/10.1007/s11005-007-0162-2
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DOI: https://doi.org/10.1007/s11005-007-0162-2