Reaction–diffusion systems are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form
∂ t q = D _ _ ∇ 2 q + R ( q ) , {\displaystyle \partial _{t}\mathbf {q} ={\underline {\underline {\mathbf {D} }}}\,\nabla ^{2}\mathbf {q} +\mathbf {R} (\mathbf {q} ),}
where q(x, t) represents the unknown vector function, D is a diagonal matrix of diffusion coefficients, and R accounts for all local reactions. 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. Such patterns have been dubbed "Turing patterns". Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable.
One-component reaction–diffusion equations The simplest reaction–diffusion equation is in one spatial dimension in plane geometry,
∂ t u = D ∂ x 2 u + R ( u ) , {\displaystyle \partial _{t}u=D\partial _{x}^{2}u+R(u),}
is also referred to as the Kolmogorov–Petrovsky–Piskunov equation. If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is Fick's second law. The choice R(u) = u(1 − u) yields Fisher's equation that was originally used to describe the spreading of biological populations, the Newell–Whitehead-Segel equation with R(u) = u(1 − u2) to describe Rayleigh–Bénard convection, the more general Zeldovich–Frank-Kamenetskii equation with R(u) = u(1 − u)e-β(1-u) and 0 < β < ∞ (Zeldovich number) that arises in combustion theory, and its particular degenerate case with R(u) = u2 − u3 that is sometimes referred to as the Zeldovich equation as well. The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form
∂ t u = − δ L δ u {\displaystyle \partial _{t}u=-{\frac {\delta {\mathfrak {L}}}{\delta u}}}
and therefore describes a permanent decrease of the "free energy" L {\displaystyle {\mathfrak {L}}} given by the functional
L = ∫ − ∞ ∞ [ D 2 ( ∂ x u ) 2 − V ( u ) ] d x {\displaystyle {\mathfrak {L}}=\int _{-\infty }^{\infty }\left[{\tfrac {D}{2}}\left(\partial _{x}u\right)^{2}-V(u)\right]\,{\text{d}}x}
with a potential V(u) such that R(u) = dV(u)/du.
In systems with more than one stationary homogeneous solution, a typical solution is given by travelling fronts connecting the homogeneous states. These solutions move with constant speed without changing their shape and are of the form u(x, t) = û(ξ) with ξ = x − ct, where c is the speed of the travelling wave. 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. For c = 0, there is a simple proof for this statement: if u0(x) is a stationary solution and u = u0(x) + ũ(x, t) is an infinitesimally perturbed solution, linear stability analysis yields the equation
… excerpt ends here. Continue reading the full article.
