Static and swinging chemical waves in a two-interface dynamics on a ring

An existing scaling model of a reaction–diffusion system is extended to a circular trajectory. The equations describe the evolution of a slow (X) and a fast (Y) concentration variable. The fast variable jumps between extreme values across a reaction interface as the rate parameter becomes very large. The model is reduced to one equation for the dynamics of the smooth (slow) variable while the Y-jumps occur at two interfaces spatially located on a ring. The equation is solved subject to 2π-periodicity conditions on the ring and continuity conditions at the interfaces. Both static (with zero velocity) and moving (with velocity v) wave solutions are found. An analogy is then drawn between our reaction–diffusion system and oscillating chemical reactions such as the Belousov–Zhabotinskii (BZ) reagent, confined to a torus-shaped container. A toroidal thin tube with a very small diameter could simulate the ring geometry. The conjectured waves capture the oscillations of the catalyst (ferroin), with the maxima and minima corresponding to the ferroin and ferriin, spatial domains in the doughnut, respectively. The non-stationary wave solutions predict a migration of those domains yielding swinging (back and forth) patterns along the ring. The azimuthal position of the interfaces exhibits temporal oscillations. Thus these simulations suggest interesting experiments on spatio–temporal patterns in excitable chemical media in annular reactors.