A Hierarchy of Integrable Differential-Difference Equations and Darboux Transformation

By embedding a free function into discrete zero curvature equation, we generalize the original hierarchy of integrable differential-difference equations into a new hierarchy which includes the famous relativistic Toda hierarchy. Infinitely many conservation laws and Darboux transformation for the first nontrivial system in the new hierarchy are constructed with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained Darboux transformation. Finally, the figures of one- and two-soliton solutions with proper parameters are presented to illustrate the structures of soliton solutions.