Reaction–diffusion system
The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons.
[7] The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form and therefore describes a permanent decrease of the "free energy"
Note that while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are unstable.
The eigenfunction ψ = ∂x u0(x) should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue λ = 0 cannot be the lowest one, thereby implying instability.
To determine the velocity c of a moving front, one may go to a moving coordinate system and look at stationary solutions: This equation has a nice mechanical analogue as the motion of a mass D with position û in the course of the "time" ξ under the force R with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination of c. When going from one to more space dimensions, a number of statements from one-dimensional systems can still be applied.
An important idea that was first proposed by Alan Turing is that a state that is stable in the local system can become unstable in the presence of diffusion.
These three solution types are also generic features of two- (or more-) component reaction–diffusion equations in which the local dynamics have a stable limit cycle[13] For a variety of systems, reaction–diffusion equations with more than two components have been proposed, e.g. the Belousov–Zhabotinsky reaction,[14] for blood clotting,[15] fission waves[16] or planar gas discharge systems.
[20] The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction–diffusion systems in spite of large discrepancies e.g. in the local reaction terms.
[23][24] Other applications of reaction–diffusion equations include ecological invasions,[25] spread of epidemics,[26] tumour growth,[27][28][29] dynamics of fission waves,[30] wound healing[31] and visual hallucinations.
[11][40] Aside from these generic examples, it has turned out that under appropriate circumstances electric transport systems like plasmas[41] or semiconductors[42] can be described in a reaction–diffusion approach.
[45][46] Reaction-diffusion systems are described to the highest degree of detail with particle based simulation tools like SRSim or ReaDDy[47] which employ among others reversible interacting-particle reaction dynamics.
