New Symmetry Groups for Generalized Solutions of ODEs

It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions. 0. Idea and Motivation The standard Lie group theory applied to symmetries of solutions of PDEs, Olver [1-3], deals with classical, and specifically, C-smooth solutions of such equations. On the other hand, as is well known, and especially in the case of nonlinear PDEs, there is a major interest in solutions which are no longer classical, thus in particular, fail to be C-smooth, and instead are generalized solutions. In Rosinger [2-13] a characterization and construction was given for the infinitely many differential algebras of generalized functions, each such algebra containing the Schwartz distributions. These algebras prove to be particularly suitable, among others, for finding generalized solutions to large classes of nonlinear PDEs, Rosinger [1,6-11,13,15], Oberguggenberger.