Institute for Mathematical Physics Group Analysis of Diierential Equations and Generalized Functions Group Analysis of Diierential Equations and Generalized Functions

We present an extension of the methods of classical Lie group analysis of diierential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the diierential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the innnitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear diierential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.