Seminarium Gamma

We formalize the notion of class of systems of differential equations within the framework of group analysis of differential equations. Point (resp. contact) transformations in certain underlying spaces, which are associated with a class of differential equations, form various algebraic objects. The basic object is the equivalence groupoid of the class, which consists of all admissible point transformations within the class. We discuss classical notions of group analysis, like the point symmetry group of a system of differential equations or the equivalence group of a class of such systems, in the context of groupoid theory. As an example, we essentially generalize some Lie’s results by proving that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein–Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.