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 TMF, 2003, Volume 136, Number 1, Pages 30–51 (Mi tmf214)

Spectral Curves and Parameterization of a Discrete Integrable Three-Dimensional Model

S. Z. Pakulyaka, S. M. Sergeevb

a Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics
b Australian National University

Abstract: We consider a discrete classical integrable model on a three-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of various three-dimensional spin models. We find the general solution of this model constructed in terms of the theta functions defined on an arbitrary compact algebraic curve. Imposing periodic boundary conditions fixes the algebraic curve. We show that the curve then coincides with the spectral curve of the auxiliary linear problem. For a rational curve, we construct the soliton solution of the model.

Keywords: three-dimensional integrable systems, Bäcklund transformations, spectral curves

DOI: https://doi.org/10.4213/tmf214

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English version:
Theoretical and Mathematical Physics, 2003, 136:1, 917–935

Bibliographic databases:

Citation: S. Z. Pakulyak, S. M. Sergeev, “Spectral Curves and Parameterization of a Discrete Integrable Three-Dimensional Model”, TMF, 136:1 (2003), 30–51; Theoret. and Math. Phys., 136:1 (2003), 917–935

Citation in format AMSBIB
\Bibitem{PakSer03} \by S.~Z.~Pakulyak, S.~M.~Sergeev \paper Spectral Curves and Parameterization of a Discrete Integrable Three-Dimensional Model \jour TMF \yr 2003 \vol 136 \issue 1 \pages 30--51 \mathnet{http://mi.mathnet.ru/tmf214} \crossref{https://doi.org/10.4213/tmf214} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2025782} \elib{https://elibrary.ru/item.asp?id=13436753} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 136 \issue 1 \pages 917--935 \crossref{https://doi.org/10.1023/A:1024541320960} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000184767700003} 

• http://mi.mathnet.ru/eng/tmf214
• https://doi.org/10.4213/tmf214
• http://mi.mathnet.ru/eng/tmf/v136/i1/p30

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. M. Sergeev, “Evidence for a Phase Transition in Three-Dimensional Lattice Models”, Theoret. and Math. Phys., 138:3 (2004), 310–321
2. von Gehlen G, Pakuliak S, Sergeev S, “The Bazhanov-Stroganov model from 3D approach”, Journal of Physics A-Mathematical and General, 38:33 (2005), 7269–7298
3. Von Gehlen G., Pakuliak S., Sergeev S., “3-dimensional integrable lattice models and the Bazhanov-Stroganov model”, Differential Geometry and Physics, Nankai Tracts in Mathematics, 10, 2006, 210–220
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