Mathematical properties of models of the reaction–diffusion type

Nonlinear systems of the reaction–diffusion (RD) type, including Gierer–Meinhardt models of autocatalysis, are studied using Lie algebras coming from their prolongation structure. Depending on the form of the functions of the fields characterizing the reactions among them, we consider both quadratic and cubic RD equations. On the basis of the prolongation algebra associated with a given RD model, we distinguish the model as a completely linearizable or a partially linearizable system. In this classification a crucial role is played by the relative sign of the diffusion coefficients, which strongly influence the properties of the system. In correspondence to the above situations, different algebraic characterizations, together with exact and approximate solutions, are found. Interesting examples are the quadratic RD model, which admits an exact solution in terms of the elliptic Weierstrass function, and the cubic Gierer–Meinhardt model, whose prolongation algebra leads to the similitude group in the plane.