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

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.

Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

“…Although these methods are different ostensibly, the constructions of the conservation laws are related to each other virtually. Recently, Anco [12] has found that Ibragimov's conservation law formula (76) is a simple re-writing of a special situation of the bilinear skew-symmetric identity (18) using symmetries and adjoint symmetries.…”

“…, etc. Condition (12) can be separated in regards to G α [U] and its differential consequences to hold a set of over-determined linear homogeneous PDEs named the determining system (adjoint invariance conditions) for multipliers Λ α [U]. If (1) is self-adjoint system, i.e., PDE System (1) has a Lagrangian function, then its multipliers are generators of its admitted continuous (point, contact, higher order) symmetries in characteristic form subject to additional conditions.…”

“…If (1) is self-adjoint system, i.e., PDE System (1) has a Lagrangian function, then its multipliers are generators of its admitted continuous (point, contact, higher order) symmetries in characteristic form subject to additional conditions. For any multiplier {Λ α [U]} of adjoint invariance conditions (12), one can directly calculate the corresponding fluxes Φ j [u] under an integral formula (see [5,[7][8][9]). Zhang [19] has studied successfully the existence of multipliers for conservation law of PDEs by applying the property of nonlinear self-adjointness with differential substitution.…”

“…From the famous Noether's theorem for variational problems [1][2][3][4][5] introduced in Noether's celebrated 1918 paper, research on orientation has been transferred to the derivation and application of conservation laws for PDEs. Several systematic approaches have been established for the calculation of conservation laws, such as the symmetry/adjoint symmetry pair method (SA method) [5][6][7], the direct construction method (multiplier approach, variational derivatives approach) [3][4][5][7][8][9], symmetry action on a known conservation law method [5,10], Ibragimov's conservation theorem [11] (equivalent to the SA method [12]) and Cheviakov's recursion formul [13]. Wolf, Hereman, Temuerchaolu and Cheviakov [14][15][16][17] have presented some effective computer programmes to calculate conservation laws for PDEs.…”

“…of formal Lagrangian L. Thus, with the help of the multiplier determining Equation (12), one can obtain the solutions of adjoint equation System (78), which are given by:…”