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Symmetry 2017, 9(3), 33; doi:10.3390/sym9030033

On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries

Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada
Academic Editor: Roman M. Cherniha
Received: 1 November 2016 / Revised: 23 January 2017 / Accepted: 17 February 2017 / Published: 27 February 2017
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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 Fréchet derivative identity. Furthermore, 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. View Full-Text
Keywords: conservation law; symmetry; adjoint-symmetry; Fréchet derivative identity; Ibragimov’s theorem conservation law; symmetry; adjoint-symmetry; Fréchet derivative identity; Ibragimov’s theorem
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Anco, S.C. On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries. Symmetry 2017, 9, 33.

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