Lie Symmetries of Differential Equations in a Generalized Functions Setting

Abstract

Group analysis of differential equations provides a vast variety of methods for analyzing linear and nonlinear PDEs, with applications ranging from the construction of explicit or invariant solutions to particular equations, Lie Bäcklund symmetries, variational symmetries and conservation laws to the study of geometric properties of PDEs, to name only a few. Starting from the 1950s research has also been directed towards utilizing symmetry methods for the study of differential equations within the framework of generalized functions. The first systematic approach towards a generalization of Lie group analysis of linear differential equations into a distributional framework is due to Berest and Ibragimov. A direct translation of results from classical group analysis to the distributional setting is again impossible, basically due to the fact that distributions don’t possess graphs and can therefore not be treated ’geometrically’.