THE INHOMOGENEOUS TODA LATTICE - ITS HIERARCHY AND DARBOUX-BACKLUND TRANSFORMATIONS

In this paper we show how one can construct hierarchies of nonlinear differential difference equations with n-dependent coefficients. Among these equations we present explicitly a set of inhomogeneous Toda lattice equations which are associated with a discrete Schrodinger spectral problem whose potentials diverge asymptotically. Then we derive a new Darboux transformation which allows us to get bounded solutions for the equations presented before and apply it in a specially simple case when the solution turns out to be expressed in terms of Hermite polynomials