Conservation laws and variational structure of damped nonlinear wave equations

Author:

Anco Stephen C.1,Márquez Almudena P.2ORCID,Garrido Tamara M.2ORCID,Gandarias Maria L.2ORCID

Affiliation:

1. Department of Mathematics and Statistics Brock University St. Catharines Ontario Canada

2. Department of Mathematics University of Cadiz Cadiz Spain

Abstract

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension.

Publisher

Wiley

Subject

General Engineering,General Mathematics

Reference34 articles.

1. J. D.Kraus ElectromagneticsEdited by(3rd ed.) McGraw‐Hill 1984.

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3. Thermoelasticity with Second Sound: A Review

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