Nonlinear Analysis of Reaction Diffusion Equations for Turing Pattern Selection

[roastedwater]

So if I have a general reaction diffusion equation and perform a liner stability analysis on it I would only get the domains in which Turing Instability occurs but not which pattern I would get (stripes , spots etc.). So I need to use a nonlinear analysis. One technique people use is a multiscale analysis. I understand the method till the point where the Ginzburg-Landau equations come in. Can anyone explain this to me? Where do these equations come from? And what do I do with them? Help would be greatly appreciated. Thanks

 

[gtoguy97]

.The Ginzburg-Landau equation is an important equation in nonlinear analysis, particularly in the study of pattern formation in reaction-diffusion systems. It is derived from a Taylor expansion of a more general reaction-diffusion equation, and is used to describe the behavior of the system in the vicinity of a stationary state. The equation takes the form of a nonlinear diffusion equation, which has a coefficient that depends on the local concentration of the species involved in the reaction. This coefficient is known as the reaction coefficient and is proportional to the rate of reaction. By solving the Ginzburg-Landau equation, one can determine the local concentrations of each of the species and find out the type of patterns (stripes, spots, etc.) that emerge in the system.