Front dynamics in heterogeneous diffusive media

We herein consider two-component reaction-diffusion systems with a specific bistable and odd symmetric nonlinearity, which have the bifurcation structure of pitchfork type traveling front solutions with opposite velocities. We introduce a spatial heterogeneity, for example, a Heaviside-like abrupt change at the origin in the space, into diffusion coefficients. Numerically, the responses of traveling fronts via the heterogeneity can be classified into four types of behavior depending on the strength of the heterogeneity, which, in the present paper, is represented by the height of the jump: passage, stoppage, and two types of reflection. The goal of the present paper is to reduce the PDE dynamics to finite-dimensional ODE systems on a center manifold and show the mathematical mechanism for producing the four types of response in the PDE systems using finite-dimensional ODE systems. The reduced ODE systems include the terms (referred to as heterogeneous perturbations) originating from the interaction between traveling front solutions and the heterogeneity, which is very important for determining the dynamics of the ODE systems. In the present paper, we succeed in calculating these heterogeneous perturbations exactly and explicitly.