GENERALIZED EIKONAL EQUATION IN EXCITABLE MEDIA

Numerical simulations show that, in excitable media, the standard eiko nal equation describing the dependence of a wave front's local velocit y on its curvature fails badly in the presence of significant dispersi on [Pertsov et al. Phys. Rev. Lett. 78, 2656 (1997)]. Here we derive a corrected eikonal equation, valid in an unrestricted frequency range, which includes highly dispersive conditions. The derivation, which us es a finite-renormalization technique, is applied to diffusion-reactio n equations with generic reactivity functions and two diffusivities of arbitrary ratio. In the important case of equal diffusivities alpha, we obtain at low curvature, the following contribution to the speed: [ -1 + (omega/c)(partial derivative c/partial derivative omega)](alpha/r ), where 1/r is the curvature, omega is the frequency, and c = c(omega ) is the speed of a plane wave with that frequency. In the single-diff usivity case there is a further contribution (epsilon/c)(partial deriv ative c/partial derivative epsilon)(alpha/r), where epsilon is the rat io of time scales for diffusing and nondiffusing variables; epsilon is not restricted to a small range. Both cases yield excellent agreement with numerical simulations. Our various formulas are compared with th e classical results of Zykov (Biofizika 25, 888 (1980) [Biophysics 25, 906 (1980)]) and of Keener [SIAM J. Appl. Math. 46, 1039 (1986)].