On the construction and integration of a hierarchy for the periodic Toda lattice with a self-consistent source

In this paper, it is derived a rich hierarchy for the Toda lattice with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.