Two-dimensional Toda lattice equations

We investigate solutions to the two-dimensional Toda lattice equation if asax log(1 + V.)= Vn+l-2 Vn+ Vn-l for n= -oo, .... -1, 0,1, .... oo, and related ones including (i) the cylindrical Toda lattice equation (ii) the two-dimensional Toda lattice equation in the cylindrical coordinates and (iii) the discrete two-dimensional Toda lattice equation where Q.(l, m)=log{l+(<l'c)-'[exp(<l'cVn(/, m)-1]} and the symbols L1+, and L1-m are the forward and backward difference operators, respectively. Exact solutions to all of them are expressed with Casorati determinants.