On the propagation and bifurcation of singular surface shocks under a class of wave equations based on second-sound flux models and logistic growth

Working in the context of hyperbolic reaction–diffusion–acoustic theory, we present a detailed study of singular surface shock phenomena under two hyperbolic versions of the Fisher–KPP equation, both of which are based on flux laws originally used to describe the phenomenon of second-sound. Employing both analytical and numerical methods, we investigate the propagation, evolution, and qualitative behavior of density shocks, i.e., propagating surfaces across which the density exhibits a jump, under the two models considered. In the process, we identify a class of wave equations with the property that members of the class all exhibit the same shock structure, evolution, and velocity. Lastly, possible follow-on investigations are proposed and applications to other areas of continuum physics are noted.