Exact travelling wave solutions of the discrete nonlinear Schrodinger equation and the hybrid lattice equation obtained via the exp-function method

An algorithm is devised for deriving exact travelling wave solutions of differential-difference equations (DDEs) by means of the exp-function method. This method was previously applied to nonlinear partial differential equations (NPDEs) or coupled NPDEs, whereas here it is applied to DDEs. For illustration, we apply this method to solve the discrete nonlinear Schrodinger equation (DNLSE) and the hybrid lattice equation. As some special examples, new exact travelling wave solutions can degenerate into the discrete solitary wave solution reported in the available literature. Moreover, two kinds of new exotic waves are found in the DNLSE and the hybrid lattice equation.