Solitary Wave and Periodic Wave Solutions for the Relativistic Toda Lattices

In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partialdifferential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example, several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approachare listed and a discussion is given in conclusion.