A one-dimensional reaction-diffusion dynamical system is considered in a bounded domain. Specifically, we examine the effects produced by different sources of symmetry-breaking in the system. Such a source might be a through-flow, responsible for breaking the system’s reflection symmetry.
In addition, we consider the behavior of our system, initially homogeneous, under two sets of boundary conditions: first, Robin, and then periodic. These boundaries are characterized by the absence, in the Robin case, and then the presence, in the periodic case, of translational symmetry.
Reaction-diffusion systems present a broad range of applications, particularly when they are driven outside of equilibrium. Examples of such behaviors are provided by a variety of chemical systems, and particularly autocatalytic reactions, along with cardiac and neuronal activities.
With Robin conditions we observe travelling waves for a transient and then only steady-state solutions. In the periodic domain, in contrast, we find travelling waves to be unstable at first, and then to gain stability. Real-world systems present inhomogeneities, and so we introduce into our system an inhomogeneity characterized by spatially varying forcing. This results in a break in translation symmetry. As the amplitude of the inhomogeneity increases sufficiently, it gives rise to a pinning of the travelling waves.
We look at the PDE as well as the ODE resulting from the PDE’s first nontrivial truncation. In both cases, pinning of traveling waves occurs via different mechanisms, which will be discussed.