Conservation laws of scaling-invariant field equations

TL;DR

This paper introduces a straightforward formula for deriving conservation laws in scaling-invariant field equations, applicable to various physical systems, by utilizing adjoint-symmetries to generate all local conservation laws with non-zero scaling weight.

Contribution

It presents a new, simple conservation law formula based on adjoint-symmetries for scaling-invariant field equations, unifying the derivation of conservation laws across different physical models.

Findings

Successfully applied to soliton equations, fluid flow, and nonlinear wave equations.

Derived conservation laws for Yang-Mills and Einstein gravitational equations.

Provides a systematic method for generating all local conservation laws with non-zero scaling weight.

Abstract

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravitational field equations are considered.