Reaction-Diffusion Equations

Abstract

Reaction-diffusion (RD) equations arise naturally in systems consisting of many interacting components, (e.g., chemical reactions) and are widely used to describe pattern-formation phenomena in variety of biological, chemical and physical sys-tems. The principal ingredients of all these models are equation of the form ∂ t u = D∇ 2 u + R(u), (8.1) where u = u(r, t) is a vector of concentration variables, R(u) describes a local reac-tion kinetics and the Laplace operator ∇ 2 acts on the vector u componentwise. D de-notes a diagonal diffusion coefficient matrix. Note that we suppose the system (8.1) to be isotropic and uniform, so D is represented by a scalar matrix, independent on coordinates. 8.1 Reaction-diffusion equations in 1D In the following sections we discuss different nontrivial solutions of this sys-tem (8.1) for different number of components, starting with the case of one com-ponent RD system in one spatial dimension, namely u t = D u xx + R(u) , (8.2) where D = const. Suppose, that initial distribution u(x, 0) is given on the whole space interval x ∈ (−∞, +∞). 8.1.1 The FKPP-Equation Investigation in this field starts form the classical papers of Fisher [17] and Kol-mogorov, Petrovsky and Piskunoff [25] motivated by population dynamics issues, 75 where authors arrived at a modified diffusion equation: ∂ t u(x, t) = D ∂ 2 x u(x,t) + R(u) , (8.3) with a nonlinear source term R(u) = u − u 2 . A typical solution of the Eq. (8.3) is a propagating front, separating two non-equilibrium homogeneous states, one of which (u = 1) is stable and another one (u = 0) is unstable [10, 13, 51]. Such fronts behavior is often said to be front propagation into unstable state and fronts as such are referred to as waves (or fronts) of transition from an unstable state. Initially the subject was discussed and investigated mostly in mathematical soci-ety (see, e.g., [16] where nonlinear diffusion equation was discussed in details). The interest in physics in these type of fronts was stimulated in the early 1980s by the work of G. Dee and coworkers on the theory of dendritic solidification [12]. Exam-ples of such fronts can be found in various physical [28, 52], chemical [43, 14] as well as biological [3] systems. Notice that for Eq. (8.3) the propagating front always relaxes to a unique shape and velocity c * = 2 √ D, (8.4) if the initial profile is well-localized [1, 2, 50].