Traveling Autosolitons and Autowaves (KΩ and Ω-Systems)

Abstract

One-dimensional traveling autosoliton (pulse) is a simplest autowave that can be excited in KΩ and Ω-systems — that is, in systems described by equations of type (9.1), (9.2) with α = τ _η /τ _θ ≪ 1. In the limit ε = l/L → ∞ (more precisely, at L → 0), Ω-systems go over into Fitz-Hugh — Nagumo (FHN) models, which are described by (9.1), (9.2) with L = 0:

$${\tau _\theta }\frac{{\partial \theta }}{{\partial t}} = {l^2}\Delta \theta - q\left( {\theta ,\eta ,A} \right),{\tau _\eta }\frac{{\partial \eta }}{{\partial t}} = - Q\left( {\theta ,\eta ,A} \right).$$

(17.1)

In other words, FHN models represent active systems in which diffusion of inhibitor is negligibly small (Ch. 3). From preceding chapters we know that it is not possible to excite static or pulsating autosolitons in systems of this kind. It is possible, however, to excite traveling autosolitons or complicated autowaves: spiral (reverberating) waves, vortices, vortex rings, scrolls, etc. (see, for example, Winfree 1980, 1987; Mikhailov 1990).