Discrete Analogue of a Generalized Toda Equation

Abstract

A discrete analogue of a generalized Toda equation and its Backlund transformations are obtained. The equation is expressed with the bilinear form as follows \begin{aligned} [Z_{1} \exp (D_{1})+Z_{2} \exp (D_{2})+Z_{3} \exp (D_{3})]f \cdot f=0 \end{aligned} 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.