Variational symmetries and conservation laws of the wave equation in one space dimension

TL;DR

This paper uses the direct method and Noether's theorem to analyze conservation laws and variational symmetries of the (1+1)-dimensional wave equation in both light-cone and standard coordinates.

Contribution

It provides a comprehensive computation of conservation laws and variational symmetries for the wave equation using direct methods and Noether's theorem.

Findings

Conservation laws are explicitly characterized in light-cone coordinates.

Variational symmetries are identified for the wave equation.

Results are also extended to the standard space-time form.

Abstract

The direct method based on the definition of conserved currents of a system of differential equations is applied to compute the space of conservation laws of the (1+1)-dimensional wave equation in the light-cone coordinates. Then Noether's theorem yields the space of variational symmetries of the corresponding functional. The results are also presented for the standard space-time form of the wave equation.