Discrete Soliton Equation Hierarchy

It is known that the continuous soliton equations such as the KdV equation have hierarchy structures. A higher order KdV equation constituting the hierarchy includes higher order derivative terms in the equation. The Toda equation is characterized by discreteness in the space dimension. In the hierarchy for the discrete soliton equation such as the Toda equation, neither spatial derivatives nor higher order spatial derivatives appear. Despite of this difference, we show both continuous and discrete soliton equation hierarchies starting with the noncommutative zero curvature equation. Especially, we show the explicit form of the higher order Toda equation.