Hirota's difference equations

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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.

Original languageEnglish
Pages (from-to)1347-1392
Number of pages46
JournalTheoretical and Mathematical Physics
Issue number2
Publication statusPublished - Nov 1997
Externally publishedYes


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