Case of the Ginzburg-Landau Equation

It would be instructive here to illustrate the theory presented above with a simple reaction-diffusion model. A suitable model would be the Ginzburg-Landau equation in the form of (2.4.13). As noted in Sect. 2.4, it is expressed as a two-component reaction-diffusion system, although the diffusion matrix D then involves an antisymmetric part  

The nonlinear phase diffusion equation (3.3.5) now takes the explicit form dy/ 

For the unperturbed orbit, dy//dt = 0 and d0/dt = ojo, which implies the relation 0= o)o t+ y/) for a weak perturbation, where 0 is the phase of W. It follows, therefore, that whenever the orbital deformation due to the diffusion coupling is negligible, the Ginzburg-Landau equation is contracted to 

This reduced form breaks down if 1 + CjC2 0, which actually occurs, at least for a certain hypothetical chemical reaction model as shown in Appendix B. A negative phase diffusion coefficient implies turbulence, and this problem will be revisited in Chap. 7. 

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