Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

Abstract

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.

On the paper “Symmetry analysis of wave equation on sphere” by H. Azad and M.T. Mustafa

“…We observe that Lemma 4 is a particular case of the main result obtained in [6]. In this work the authors carried out the group classification of Eq.…”

“…In [4,3,5,10] the Lie point symmetries, the Noether symmetries and the conservation laws of the Kohn-Laplace equations were studied. In [6] the symmetry analysis of Eq. (2) was carried out on an arbitrary (pseudo) Riemannian manifold.…”

Group invariant transformations for the Klein–Gordon equation in three dimensional flat spaces

“…Whilst recently the connection between collineations and symmetries was established for a system of quasilinear PDEs [23]. A similar result has been proved for the Poisson equation [24].…”

Lie Symmetries of Klein-Gordon and Schrödinger Equations

“…As we know, Klein Gordon equation is an appropriate case of Poisson equation and Schrödinger equation is an important form of the heat equation. Since Lie point symmetries of Poisson and heat equations within Riemannian space have been studied in [4] [5], however in this paper we utilize these consequences to conclude the Lie point symmetries of Klein-Gordon equation and Schrödin-ger equation within universal Riemannian space.…”