Discrete Toda Field Equations

There are two-dimensional Toda field equations corresponding to each (finite or affine) Lie algebra. The question addressed in this note is whether there exist integrable discrete versions of these. It is shown that for certain algebras (such as $A_n$, $A_n^{(1)}$ and $B_n$) there do, but some of these systems are defined on the half-plane rather than the full two-dimensional lattice.

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