Lie Symmetries of Differential Equations: A Computational Approach to Optimal Systems of Lie Subalgebras

Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of invariant solutions; thus, it is crucial to obtain a classification of subgroups in order to have an \emph{optimal system} of inequivalent solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite--dimensional Lie algebra uses inner automorphisms that are obtained by exponentiating the adjoint groups. In this thesis, after shortly reviewing the basic notions about Lie algebras and Lie groups of transformations of differential equations, we present an effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite--dimensional Lie algebra abstractly assigned by means of its structure constants, or realized in terms of matrices or vector fields, or defined by a basis and the set of non-zero Lie brackets. The algorithm is implemented in the computer algebra system \emph{Wolfram Mathematica}\texttrademark. Various meaningful and non-trivial examples are considered. In particular, we classify the optimal systems of Lie subalgebras of all real Lie algebra of dimension 3, 4 and 5. Also, we analyze the optimal systems of Lie subalgebras of Noether symmetries of some geodesic equations associated to special metrics in a four--dimensional space, as well as the optimal systems of Lie symmetries admitted by some well known PDEs (linear heat equation, Burgers' equation, Korteweg-deVries equation).