Symmetry groups of linear partial differential equations and representation theory: The Laplace and axially symmetric wave equation
This paper studies the Lie point symmetries of the Laplace and axially symmetric wave equations via Lie's prolongation algorithm. In a previous paper [J. Differ. Equations 116, No. 1, 202-247 (1995; Zbl 0845.35020)] the author examined the Schrödinger, heat, and a particular Fokker-Planck equation. In each case, a global group of symmetries, \(G\), is identified from the group of point symmetries. Moreover, Lie's prolongation algorithm produces a genuine irreducible representation of \(G\) that is identified. Similar analysis is carried out for the Laplace and axially symmetric wave equations. Complete results are given for the axially symmetric wave equation. Only partial results are available for the Laplace equation due to the fact that the group of point symmetries is infinite-dimensional. However, the author conjectures that for general linear PDEs the group of point symmetries produced by Lie's algorithm is equivalent to a representation of the underlying group.
scientific article; zbMATH DE number 1517168