Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection-reaction-diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c^* such that for each wave speed c≤c^*, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c^* are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c^*.