Symmetry groups of linear partial differential equations and representation theory: The Laplace and axially symmetric wave equation
type: Item
instance of: scholarly article
title: Symmetry groups of linear partial differential equations and representation theory: The Laplace and axially symmetric wave equation
zbMATH Open document ID: 0962.35010
author: Mark Craddock
published in: Journal of Differential Equations
publication date: 2001-05-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 1517168
zbMATH Keywords: Lie point symmetries
zbMATH Keywords: prolongation algorithm
reviewed by: Mark R. Sepanski
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3786
OpenAlex ID: W2001452638
cites work: Symmetries and differential equations
cites work: Symmetry groups of partial differential equations, separation of variables, and direct integral theory
cites work: The symmetry groups of linear partial differential equations and representation theory. I
cites work: Symmetry group methods for heat kernels.
cites work: Q5724171
cites work: Q3741813
cites work: Q4273954
cites work: Lie groups and algebras with applications to physics, geometry, and mechanics
cites work: Uniformly Bounded Representations for the Lorentz Groups
DOI: 10.1006/JDEQ.2000.3786
description: scientific article; zbMATH DE number 1517168