Abstract: Firstly, the simple model which describes the one-dimensional longitudinal wave propagation in dissipation-dispersion nonlinear solid is derived based on the Mindlin theory and the Murnaghan model. Secondly, the microstructural effect on bell and kink solitary wave evolution is simulated, by using the finite difference method. The results show that with the weakening of microstructural effect, the amplitude attenuation and asymmetric characteristic of bell solitary wave become more and more obvious; with the enhancement of microstructural effect, the “hat” shape change emerging at the top of kink solitary wave and the corresponding asymmetric characteristic become more and more obvious.