After reviewing recent key results for a specific class of linear dispersive peridynamics models, I will introduce and analyze the semilinear generalization of the dispersive relation in the presence of a quartic term in the energy space. The anticipated energy transition across scales will be examined through Fourier analysis of numerical solutions and quantitatively assessed. Finally, I will discuss the energy cascade toward smaller scales, focusing on its general characteristics and potential occurrence in other formulations of the semilinear counterpart.