Selected Topics in Nonlinear Wave Phenomena: Diffusive Solitons, Singular Surfaces, and Wave Chaos

We explore some recent topics of interest in the field of nonlinear wave phenomena. We do so in the context of problems arising in acoustic/second-sound (e.g., thermal waves) propagation in certain nonlinear media; reaction-diffusion theory, e.g., population and chemical dynamics; and systems described by equations of the nonlinear Klein--Gordon type (e.g., the sine--Gordon equation). We investigate the corresponding governing equations with an emphasis on shock, solitary/traveling, and chaotic wave phenomena. Employing both analytical and numerical techniques, this study is carried out with the purpose of gaining a better understanding of the physical systems represented in the mathematical models. Finally, other applications of this research are noted and discussed, time permitting.