Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions

TL;DR

This paper classifies symmetries and conservation laws for multi-dimensional radial wave equations with power nonlinearities, revealing new energy and momentum forms and a novel Morawetz identity beyond Noether's theorem.

Contribution

It provides a comprehensive classification of point symmetries and conservation laws, including new non-variational conserved quantities and a novel Morawetz dilation identity.

Findings

Explicit classification of symmetries and conservation laws.

Identification of generalized energies and radial momenta.

Introduction of a new Morawetz dilation identity.

Abstract

Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two wave equations in a different way than Noether's theorem.