On the incompleteness of Ibragimov's conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries

TL;DR

The paper demonstrates that Ibragimov's conservation law theorem is a special case of a standard symmetry-adjoint symmetry formula, clarifies its limitations, and advocates a more general, algorithmic method using adjoint-symmetries to find all local conservation laws.

Contribution

It shows Ibragimov's theorem is a specific instance of a standard formula and introduces a comprehensive, algorithmic approach using adjoint-symmetries to identify all local conservation laws.

Findings

Ibragimov's theorem is a special case of a standard symmetry-adjoint symmetry formula.

The standard formula can generate trivial conservation laws and may miss some non-trivial ones.

A general, algorithmic method using adjoint-symmetries can find all local conservation laws.

Abstract

A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known Fr\'echet derivative identity. Also, the connection of this formula (and of Ibragimov's theorem) to the standard action of symmetries on conservation laws is explained, which accounts for a number of major drawbacks that have appeared in recent work using the formula to generate conservation laws. In particular, the formula can generate trivial conservation laws and does not always yield all non-trivial conservation laws unless the symmetry action on the set of these conservation laws is transitive. It is emphasized that all local conservation laws for any given system of differential equations can be found instead by a general method using adjoint-symmetries. This general method is a kind of adjoint version of the standard Lie method to find all local symmetries and is completely algorithmic. The relationship between this method, Noether's theorem, and the symmetry/adjoint-symmetry formula is discussed.