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

Stephen Anco

doi:10.3390/sym9030033

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.

Highlights

The most well-known method for finding conservation laws of differential equations (DEs) is Noether’s theorem [1], which is applicable to any system of one or more DEs admitting a variational formulation in terms of a Lagrangian
The conservation law theorem stated by Ibragimov in References [9,12] for “nonlinear self-adjoint”
Is nothing but a re-writing of the condition that a DE system admits an adjoint-symmetry [4,29], and this condition automatically holds for any DE system that admits a local conservation law

Summary

Introduction

The most well-known method for finding conservation laws of differential equations (DEs) is Noether’s theorem [1], which is applicable to any system of one or more DEs admitting a variational formulation in terms of a Lagrangian. Ibragimov’s conservation law formula is a simple re-writing of a special case of the earlier formula using symmetries and adjoint-symmetries; Ibragimov’s “nonlinear self-adjointness” condition in its most general form is equivalent to the existence of an adjoint-symmetry for a general DE system and reduces to the existence of a symmetry in the case of a variational DE system; this formula does not always yield all admitted local conservation laws, and it produces trivial conservation laws whenever the symmetry is a translation and the adjoint-symmetry is translation-invariant; the computation to find adjoint-symmetries (and, to apply the formula) is just as algorithmic as the computation of local symmetries; most importantly, if all adjoint-symmetries are known for a given DE system (whether or not it has a variational formulation), they can be used directly to obtain all local conservation laws, providing a kind of generalization of Noether’s theorem to general DE systems All of these remarks have been pointed out briefly in Reference [23], and Remark (2) has been discussed in References [16,17], but it seems worthwhile to give a comprehensive discussion for all of the remarks (1)–(5), with examples, as the formula continues to be used in recent papers when a complete, general method for finding all local conservation laws is available instead.

Conservation Laws and Symmetries

Ibragimov’s Conservation Law Formula

Adjoint-Symmetries and a Formula for Generating Conservation Laws

Multiplier Determining equations

A Direct Construction Method to Find All Local Conservation Laws

Concluding Remarks