dorsal/arxiv
View SchemaA survey of Hirota's difference equations
| Authors | A. Zabrodin |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9704001 |
| URL | https://arxiv.org/abs/solv-int/9704001 |
| DOI | 10.1007/BF02634165 |
Abstract
A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition 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 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.
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"abstract": "A review of selected topics in Hirota\u0027s bilinear difference equation (HBDE)\nis given. This famous 3-dimensional difference equation is known to provide a\ncanonical integrable discretization for most important types of soliton\nequations. Similarly to the continuous theory, HBDE is a member of an infinite\nhierarchy. The central point of our exposition is a discrete version of the\nzero curvature condition explicitly written in the form of discrete\nZakharov-Shabat equations for M-operators realized as difference or\npseudo-difference operators. A unified approach to various types of M-operators\nand zero curvature representations is suggested. Different reductions of HBDE\nto 2-dimensional equations are considered. Among them discrete counterparts of\nthe KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical\nexamples are discussed in detail.",
"arxiv_id": "solv-int/9704001",
"authors": [
"A. Zabrodin"
],
"categories": [
"solv-int",
"hep-th",
"nlin.SI"
],
"doi": "10.1007/BF02634165",
"title": "A survey of Hirota\u0027s difference equations",
"url": "https://arxiv.org/abs/solv-int/9704001"
},
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