Complete Integrability of General Nonlinear Differential-Difference Equations Solvable by the Inverse Method. I

TLDR: In this article, the Toda lattice is shown to be a completely integrable Hamiltonian system, and the generalized Volterra system is also a complete Hamiltonian.

Abstract: In recent years a large number of nonlinear differential-difference equations which describe discrete systems such as lattices, ladder networks and competition processes have been solved by the method of inverse scattering. The variety of those equations now becomes comparable with that of nonlinear evolution equations which describe continuous systems and are also solvable by the inverse method. For the continuous cases it has been well established that the system governed by an evolution equation to which the inverse method can be applied is nothing but a completely integrable Hamiltonian system.1l~3l The discrete counterpart of this statement is believed to be correct, but so far it has been proved only for two relevant systems; Flaschka and McLaughlin4l .5l have shown that the Toda lattice is a completely integrable Hamiltonian system and the present authors have proved in a previous paper6l that the generalized Volterra system is also completely integrable. In the present and a subsequent papers we want to put into this category two classes of general nonlinear differential-difference equations solvable by the method of inverse scattering. The first class which we here deal with consists of equations generated by a linear eigenvalue equation