Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys. Display omitted •We develop a mathematical model for the stress tensor of dispersive hyperelastic solids within the theory of simple materials of differential type.•We study the wave propagation in dispersive hyperelastic solids.•In the small but finite regime, we find exact solutions representing solitary and kink waves.