Discrete Analogue of a Generalized Toda Equation

Abstract

A discrete analogue of a generalized Toda equation and its Bäcklund transformations are obtained. The equation is expressed with the bilinear form as follows
[Z₁ exp (D₁)+Z₂ exp (D₂)+Z₃ exp (D₃)]f·f=0
where Z_i and D_i for i=1, 2, 3, are an arbitrary parameter and a linear combination of the binary operators D_t, D_x, D_y, D_n, etc., respectively.
The equation is very generic, namely appropriate combinations of parameters give various types of soliton equations including the Korteweg-de Vries equation, Kadomtsev-Petviashvili equation, modified KdV equation, sine-Gordon equation, nonlinear Klein-Gordon equation, Benjamin-Ono equation and various types of discrete analogues of soliton equations.