Wavefront Instability

We derived in Chap. 4 an evolution equation for slowly varying wavefronts in two-dimensional reaction-diffusion systems. Quite analogously to the dynamics of oscillatory systems with a slowly varying phase pattern, we obtained an asymptotic expansion (4.3.28). If a happened to be small and negative, while the other parameters were of ordinary magnitude and y positive, then the same reasoning advanced in Sect. 7.2 applies, and we get the one-dimensional phase turbulence equation 

One may wonder if there exist any specific reaction-diffusion models giving rise to negative a. Such a model in fact exists, for which one may even prove analytically the possibility of arbitrarily small a. This is a piecewise linear version of the Bonhoeffer - van der Pol model including diffusion, and is given by 

Clearly, a can be positive or negative depending on the parameter values. 

In contrast to the instabihty of uniform oscillations, the physical origin of the wavefront instability seems to have some relation to the conventional diffusion instability. To see this, we first give a brief quaUtative interpretation of the conventional diffusion instabihty. Here the notions activator and inhibitor seem to be helpful, and for simpUcity we imagine a two-component activator-inhibitor system. The instability then turns out to be due to relatively rapid diffusion of the inhibiting substance. Consider the activator-inhibitor kinetics (first, without diffusion)  

The shapes of the nullclines / = 0 and = 0 are supposed to be such as shown in Fig. 7.6a. Let and t] denote the deviations from the intersection point (Xq, To)-Linearization of (7.3.4) about this point leads to 

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Kuramoto, Y. (1980a) Instability and turbulence of wavefronts in reaction-diffusion systems. Prog. Theor. Phys. 63, 1885... 

When conditions are set such that Db Da Dy = Dx, the thermodynamic branch may develop an instability which now drives the system to a new state periodic in both time and space (domain III). As shown in Fig. 3 in the course of one period, the system evolves with the appearance of nonstationary, concentrational wavefronts which propagate in the reaction volume. Temporal, quasi-discontinuous oscil-... 

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