Differential-difference evolution equations. II (Darboux transformation for the Toda lattice)

Toda lattice equation is represented in the form of the condition of compatibility of the system of linear equations corresponding to a non-Hermitian Lax representation. The Darboux invariance of this linear system is defined and proved in the text, and enables us to construct some new formulas for the solutions of the Toda lattice equation. These formulas involving determinants are applicable to an arbitrary initial solution of the Toda equation for example to a solution growing at infinity.Physique Mathématique et Théorique, Equipe de recherche associée au CNRS.This work has been done as part of the program Recherche Cooperative sur Programme No. 264: Etude interdisciplinaire des problèmes inverses.Leningrad State University, U.S.S.R.