Spiral and concentric waves in active media with soft and hard excitation

Abstract

An approximation method is presented for describing spiral waves with the generalized Ginzburg-Landau equation, which for weak overcriticality is a universal model of active media with soft excitation of self-oscillation. A comparison with numerical results indicates the this method, which is based on direct matching of asymptotic expansions, is valid for large and small values of the radial variable, and yields satisfactory values for the asymptotic radial wave number of a spiral wave for values of the dispersion coefficients that are not too small. Spiral waves are described with a simple model of an active medium with hard excitation. This description is based on the rigorous method of asymptotic expansion matching for small dispersion coefficients and on the direct matching method for the general case. Both methods are used to show that, in contrast to spiral waves, the asymptotic wave number for concentric waves is an arbitrary parameter, and the group velocity is directed from the periphery to the center.