Dynamics of Slowly Varying Wavefronts

Having thus reformulated Method I into a more systematic form, we now notice that the variety of problems that the theory can treat has become considerably richer. This and the next sections take up two such examples that might possibly have been overlooked without Method II. Both are related to typical phenomena in reaction-diffusion systems but are not necessarily of an oscillatory nature. 

The first class of phenomena we would like to discuss concerns the dynamics of wavefronts of certain types of chemical waves in two dimensional media. (Extension to three-dimensional cases is straightforward, and not discussed here.) Let x and y denote the spatial coordinates. Imagine at first that the composition vector X has no y-dependence, so that we are essentially working with a one-dimensional system  

The system length is supposed to be infinite. Without diffusion, the system is assumed to admit one or more stable equilibrium points but no stable oscillations. Furthermore, these equilibrium states are supposed to remain stable to nonuniform fluctuations. As is well known, it sometimes happens that steadily traveling nonlinear waves arise in such non-oscillatory media. (For a special model, see Sect. 7.3.) Two types of waves are possible, which are schematically 

Method II becomes relevant to the dynamics of such chemical waves when they are extended to form two-dimensional waves. Let X ipc-ct) denote a steadily traveling pulse- or kink solution of (4.3.1), where c is the propagation 

It would be inappropriate if not absolutely impossible, to apply Method I to problem (4.3.2), whereas Method II seems to be much more suited to it. In fact, we shall find a perfect analogy existing between the formulation below and that of Sect. 4.2. 

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Dynamics of Slowly Varying Wavefronts 53 Clearly, the wavefront equation takes the form... 

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... 

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