On the Construction and Integration of a Hierarchy for the Periodic Toda Lattice with a Self-Consistent Source

Abstract

In this paper, it is derived a rich hierarchy for the Toda lattice with a selfconsistent 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, Dubrovintype 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.

On the Integration of the Higher Order Toda Lattice with a Self-Consistent Integral Type Source

“…An inverse scattering method was also developed to find solutions for the Toda lattice with self-consistent sources [36][37][38]. Integrability of the periodic Toda lattice and its hierarchy with a source has been shown in previous works [39][40][41][42][43][44].…”