Hirota's difference equations
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.
- Publication:
-
Theoretical and Mathematical Physics
- Pub Date:
- November 1997
- DOI:
- 10.1007/BF02634165
- arXiv:
- arXiv:solv-int/9704001
- Bibcode:
- 1997TMP...113.1347Z
- Keywords:
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- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- High Energy Physics - Theory
- E-Print:
- LaTeX, 43 pages, LaTeX figures (with emlines2.sty)