## 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.

Original language | English |
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Pages (from-to) | 1347-1392 |

Number of pages | 46 |

Journal | Theoretical and Mathematical Physics |

Volume | 113 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 1997 |

Externally published | Yes |