Discrete Two-Dimensional Toda Molecule Equation

A discrete analogue of the two-dimensional Toda molecule equation is obtained, which is expressed as follows {aligned} ₓyQₙ(x,y){=}Vₙ₊₁(x,y)-Vₙ(x+δ,y)-Vₙ(x,y+ε)+Vₙ₋₁(x+δ,y+ε), {aligned} where {aligned} Vₙ(x,y){=}(δε)⁻¹log[1+δεexp[Qₙ(x,y)]], {aligned} with the boundary conditions V 0 ( x , y )= V n +1 ( x , y )=0. The symbol Δ z denotes the forward difference operator with respect to z , and δ and ε are parameters relating to infervals. Exact solutions to it are expressed with Casorati determinants.