Hirota’s difference equations |
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Authors: | A V Zabrodin |
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Institution: | (1) Joint Institute for Chemical Physics, Moscow, Russia;(2) Institute for Theoretical and Experimental Physics, Moscow, Russia |
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Abstract: | A review of selected topics for Hirota’s bilinear difference equation (HBDE) is given. This famous three-dimensional difference
equation is known to provide a canonical integrable discretization for most of the important types of soliton equations. Similar
to continuous theory, HBDE is a member of an infinite hierarchy. The central point of our paper 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 two-dimensional equations are considered, with discrete counterparts of the
KdV, sine-Gordon, Toda chain, relativistic Toda chain, and other examples.
The article was written by the request of the Editorial Board.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 179–230, November, 1997. |
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