Conservation Laws and Symmetries of Semilinear Radial Wave Equations

TL;DR

This paper classifies symmetries and conservation laws of various radial wave equations with power nonlinearities across different dimensions, identifying special cases with unique symmetries and conserved quantities.

Contribution

It provides a comprehensive classification of local point symmetries and conserved densities for several radial wave equations, including special cases with dilational and conformal energies.

Findings

Classification of all local point symmetries.

Identification of conserved densities depending on derivatives.

Determination of special cases with unique symmetries and energies.

Abstract

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrodinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg--de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved H^s norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.