Darboux Transformation Of The Relativistic Toda Lattices

A Darboux transformation of the discrete spectral problem associated with the relativistic Toda lattices is found with the help of gauge transformation, further, a kind of algebraic algorithm to obtain explicit solutions for the relativistic Toda lattices is given. As an application, explicit solutions for the first two systems of differential-difference equations in the hierarchy of relativistic Toda lattices are given.