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

This article is cited in 56 scientific papers (total in 56 papers)

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.


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

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Received: 20.06.1997

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
\by A.~V.~Zabrodin
\paper A survey of Hirota's difference equations
\jour TMF
\yr 1997
\vol 113
\issue 2
\pages 179--230
\jour Theoret. and Math. Phys.
\yr 1997
\vol 113
\issue 2
\pages 1347--1392

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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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