Group classification via mapping between classes: an example of semilinear reaction–diffusion equations with exponential nonlinearity

There exist relatively few equations describing natural phenomena among a great number of partial differential equations (PDEs). This begs the question what mathematical properties differ equations describing physical processes from other possible ones? It appears that large majority of equations of mathematical physics has nontrivial symmetry properties (see a number of examples e.g. in [1]). It means that manifolds of their solutions are invariant with respect to multi-parameter groups of continuous transformations (Lie groups of transformations) with a number of parameters. Therefore, the presence of nontrivial symmetry properties is one of such distinctive features (and very important one)! In some cases the requirement of invariance of equations under a group enables us to select these equations from a wide set of other admissible ones. For example, there is the only one system of Poincaré-invariant partial differential equations of first order for two real vectors E(x0,x) and B(x0,x), and this is the system of Maxwell equations [1]. The problem arises to single out equations having high symmetry properties from a given class of PDEs. A solution of so-called group classification problem gives an exhaustive solution of this problem. There exist two main approaches of solving group classification problems. The first one is more algebraic and based on subgroup analysis of the equivalence group of a class of differential equations under consideration (see [2, 3, 4, 5] for details). The second approach involves the investigation of compatibility and the direct integration of determining equations implied by the infinitesimal invariance criterion [6]. Unfortunately it is efficient only for classes of a simple structure, e.g., which have a few arbitrary elements of one or two same arguments. A number of results on group classification problems investigated within the framework of this approach are collected in [7] and other books on the subject. To solve more group classification problems different tools have been recently developed. One of them is to carry out group classification using appropriate mapping of a given class to a one having a simpler structure. See the theoretical background of this approach and the first example of its implementation in [8]. In this paper we perform the group classification of the class of semilinear reaction–diffusion equations with exponential nonlinearity