THE EIKONAL EQUATION - STABILITY OF REACTION DIFFUSION WAVES ON A SPHERE

The eikonal approach to excitable reaction-diffusion systems provides an analytical hold on the phenomenon of wave propagation on curved surfaces and the evolution of toroidal scroll structures in the neighbourhood of organising filaments. The extent to which the eikonal equation contains information on the stability of these wave solutions is important from a theoretical and experimental point of view. We initiated this study in 1987 by demonstrating the stability of an expanding spherical wave. Recent experiments on wave propagation on a sphere encouraged us further to investigate stability of waves on surfaces. We analyse in this paper the geometrical stability of symmetric counter-rotating spiral waves propagating on the unit sphere; we demonstrate that these solutions are stable under small perturbations normal to the wave front, lying on the unit sphere.