Finite-dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem

Abstract

A method is developed that extends the nonlinearization procedure of eigenvalue problems to the discrete case, from which the finite-dimensional discrete systems and finite-dimensional completely integrable systems associated with the Toda hierarchy are obtained. A new model for solving discrete soliton equations is proposed, which enables one to get the solutions of the Toda lattice equations by solving the compatible systems of discrete equations and nonlinear ordinary differential equations.