Group analysis of differential equations and generalized functions

We present an extension of the methods of classical Lie group analysis of d ifferential equations to equations involving generalized functions ( in par ticular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We sho w that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreov er, we introduce a generalization of the infinitesimal methods of group ana lysis that allows us to compute symmetries of linear and nonlinear differen tial equations containing generalized function terms. Thereby, the group ge nerators and group actions may be given by generalized functions themselves .