Pulse trains (PTs) generated by sinusoidal phase masks emerged as a popular tool in the field of coherent control and have been applied successfully in many experiments. Although many attempts were made to throw light on the mechanism of the induced processes, it is not yet fully understood. Based on nonperturbative quantum dynamical calculations in the grid representation, we will elucidate the mechanism of PT excitation between anharmonic bound molecular electronic states. We extract general rules for the prediction of induced dynamics and the resulting spectra. The results allow us to outline perspectives for new applications, especially in the field of nonlinear spectroscopy.