An integral invariant for scroll rings in a reaction-diffusion system (Q1119197)

The evolution of scroll rings with filaments of arbitrary shape has been studied analytically for reaction-diffusion systems with all the state variables having the same diffusion coefficients. To this class of systems belong, in particular, models of the Belousov-Zhabotinskii (BZ) chemical reaction. It has been found that the speed of decrease of an area bounded by the planar scroll filament is given by \(\dot S=-2\pi D\) (D is the diffusion coefficient) and is independent of the kind of model used to describe an active medium, time and filament shape; i.e., the integral \(\oint v ds\equiv -2\pi D\), where ds is the differential of the filament length and v is the drift velocity, always remains valid. Using this integral invariant a number of problems on the drift of closed and unclosed three- dimensional vortices have been solved. The theoretical predictions have been verified in experiments with the BZ reaction.