Geodesic
A survey of Hirota's difference equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 2, pp. 179-230

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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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     author = {A. V. Zabrodin},
     title = {A survey of {Hirota's} difference equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--230},
     publisher = {mathdoc},
     volume = {113},
     number = {2},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_113_2_a0/}
}
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A. V. Zabrodin. A survey of Hirota's difference equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 113 (1997) no. 2, pp. 179-230. http://geodesic.mathdoc.fr/item/TMF_1997_113_2_a0/