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 TMF, 1997, Volume 113, Number 2, Pages 179–230 (Mi tmf1074)

A survey of Hirota's difference equations

A. V. Zabrodinab

a N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences
b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov–Shabat equations for $M$-operators realized as difference or pseudo-difference operators. A unified approach to various types of $M$-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.

DOI: https://doi.org/10.4213/tmf1074

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English version:
Theoretical and Mathematical Physics, 1997, 113:2, 1347–1392

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Citation: A. V. Zabrodin, “A survey of Hirota's difference equations”, TMF, 113:2 (1997), 179–230; Theoret. and Math. Phys., 113:2 (1997), 1347–1392

Citation in format AMSBIB
\Bibitem{Zab97} \by A.~V.~Zabrodin \paper A survey of Hirota's difference equations \jour TMF \yr 1997 \vol 113 \issue 2 \pages 179--230 \mathnet{http://mi.mathnet.ru/tmf1074} \crossref{https://doi.org/10.4213/tmf1074} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1608967} \transl \jour Theoret. and Math. Phys. \yr 1997 \vol 113 \issue 2 \pages 1347--1392 \crossref{https://doi.org/10.1007/BF02634165} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000072788800001} 

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