Integrable three-dimensional lattices

System of equations for stimulated combination scattering and the related double periodic A n (1) Toda chains

“…We thus see that as a result of the replacements in (4), (8), (10), and (11), system of equations (3) is transformed into system of equations (13), which is a Bäcklund transformation for the sine-Gordon equation. Therefore, solutions of system (3) are expressed in terms of two solutions of the sine-Gordon equation that are related by (13).…”

C-Series Discrete Chains

“…In the limit as h → 0, this condition (up to inessential point transformations) passes into the truncation condition f k0−1 = −f k0 for chain (2). Lax pairs for finite chains are written in Sec.…”

“…The degenerate and periodic reductions noted above correspond to the A n andà n algebras. The problem of finding discrete analogues of finite-field Toda chains (2) corresponding to other Lie algebras of finite growth remains open and attracts some interest (see, e.g., [5] and [6]). Examples of truncations related to special solutions of chain (1) were discussed in [5].…”

The Discrete Korteweg—de Vries Equation

“…a system in which both the spatial-as well as the time-variable is discrete. Such systems were systematically studied in a number of papers, e.g., in [10][11][12][13][14][15][16][17][18] (cf., also [19][20][21][22][23]). The lattice model is, in a sense, more fundamental than the original continuous equation, as one can always retrieve the latter by applying an appropriate continuum limit on the lattice system.…”