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 c ircular 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 2 pi-periodicity conditions on the ring and continuity conditions at the interfaces. Both static (with zero v elocity) and moving (with velocity v) wave solutions are found. An analogy is then drawn between our reaction-diffusion system and oscillating chemica l reactions such as the Belousov-Zhabotinskii (BZ) reagent, confined to a t orus-shaped container. A toroidal thin tube with a very small diameter coul d simulate the ring geometry. The conjectured waves capture the oscillation s of the catalyst (ferroin), with the maxima and minima corresponding to th e ferroin and ferriin, spatial domains in the doughnut, respectively. The n on-stationary wave solutions predict a migration of those domains yielding swinging (back and forth) patterns along the ring. The azimuthal position o f the interfaces exhibits temporal oscillations. Thus these simulations sug gest interesting experiments on spatio-temporal patterns in excitable chemi cal media in annular reactors.