Symmetry operators and generation of symmetry transformations of partial differential equations

Abstract

The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations, they present certain conceptual difficulties in the case of matrix-valued PDEs; for example, the usual differential-operator representation of the symmetry-generating vector fields is not possible in this case. In this article an algebraic approach to the symmetry problem of PDEs - both scalar and matrix-valued - is described, based on abstract operators (characteristic derivatives) that admit a standard differential-operator representation in the case of scalar-valued PDEs. A number of examples are given.