Sophus Lie introduced his theory of continuous groups in order to study
symmetry properties of differential equations. His approach allowed a
unification of existing methods for solving ordinary differential equations
as well as classifications of symmetry groups of partial differential
equations. Here a symmetry group of a differential equation is a Liegroup
action on some space of independent and dependent variables transforming
solutions of a given PDE into other solutions.
Lie's methods have been developed into powerful tools for examining
partial differential equations through group analysis. In many cases,
symmetry groups are the only known means for finding concrete solutions to
complicated equations.
Moreover, group analysis of differential equations has revealed far
reaching connections between symmetry groups of PDEs and conservation
laws of mathematical physics (Nöthertheorems).
Recently, the immense amount of computations needed for determining
symmetry groups of concrete systems has been greatly reduced by the
implementation of computer algebra packages for symmetry analysis of PDEs.
Our interest in group analysis of differential equations lies in a synthesis
of symmetry methods with results from the theory of algebras of generalized
functions. A description of our research in this direction can be found
at M. Kunzinger's
research page.
