A new version of the modified discrete nonlinear Schrödinger (MDNLS) equation that governs wave propagation in discrete dispersive nonlinear transmission lines is solved by the means of the Jacobi elliptic function method. Contribution of the linear dispersive capacitance is appreciated and it is established that this equation possesses exact solutions whose nature and characteristics depend on the modulus of Jacobian elliptic function. These solutions are either periodic waves, peak or buddle solitons used in telecommunication and medical domain. Propagation and stability of these solutions in the network are tested through direct simulations.