Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

Abstract

Convectively unstable, open reactive flows of oscillatory media, whose phase is fixed or periodically modulated at the inflow boundary, are known to result in stationary and traveling waves, respectively. The latter are implicated in biological segmentation. The boundary-controlled pattern selection by this flow-distributed oscillator (FDO) mechanism has been generalized to include differential flow (DIFI) and differential diffusion (Turing) modes. Our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion. To do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing, and show how the solutions are affected by differential transport. We find that the DIFI and FDO modes are closely related and lie in the same frequency range, while the Turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range. Finally, we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes.