Hirota's difference equations

Research output: Contribution to journalArticlepeer-review

57 Citations (Scopus)

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 languageEnglish
Pages (from-to)1347-1392
Number of pages46
JournalTheoretical and Mathematical Physics
Volume113
Issue number2
DOIs
Publication statusPublished - Nov 1997
Externally publishedYes

Fingerprint

Dive into the research topics of 'Hirota's difference equations'. Together they form a unique fingerprint.

Cite this