Transformation Theories for Nonlinear Discrete Systems

Transformation theories for nonlinear discrete systems are summarized. As a model system, the special interest is paid on a one-dimensional exponential lattice (Toda lattice) with infinite length. Bäcklund transformation and a canonical transformation for Toda lattice is explicitly presented. It is shown that the Bäcklund transformation is closely related to inverse scattering method and the existence of an infinite number of conservation laws. A remarkable transformation which relates solutions of Toda lattice equation and a nonlinear self-dual network equation is found. The transformation may be thought as a discrete version of the well-known Miura transformation. Furthermore, it is shown that a class of nonlinear differential-difference equations which appear to have N-solution solutions can be derived naturally from the Volterra system. It is suggested that the Volterra system which is first order in time is a fundamental equation in the theory of lattice soliton. In the presentation mathematical similarities between continuous and discrete case are emphasized.