Symmetries of second-order PDEs and conformal Killing vectors

TL;DR

This paper investigates the Lie point symmetries of second-order PDEs, revealing their connection to conformal Killing vectors of associated metrics, with applications to equations like Klein-Gordon and Laplace in Riemannian spaces.

Contribution

It establishes a link between PDE symmetries and conformal algebra of the metric, providing a systematic way to determine symmetries for important equations in geometric contexts.

Findings

Lie symmetries relate to conformal algebra of the metric

Symmetries of Klein-Gordon and Laplace equations are characterized

Hidden symmetries of the wave equation are analyzed

Abstract

We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Important elements in this class are the Klein--Gordon equation and the Laplace equation. We apply the general results and determine the Lie point symmetries of these equations in various general classes of Riemannian spaces. Finally we study the type II\ hidden symmetries of the wave equation in a Riemannian space with a Lorenzian metric.