Algebraic structures for adjoint-symmetries and symmetries of partial differential equations
aa r X i v : . [ m a t h - ph ] A ug ALGEBRAIC STRUCTURES FOR ADJOINT-SYMMETRIESAND SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS
STEPHEN C. ANCO and BAO WANG Department of Mathematics and StatisticsBrock UniversitySt. Catharines, ON L2S3A1, CanadaAbstract.
Symmetries of a partial differential equation (PDE) can be defined as the solu-tions of the linearization (Frechet derivative) equation holding on the space of solutions tothe PDE, and they are well-known to comprise a linear space having the structure of a Liealgebra. Solutions of the adjoint linearization equation holding on the space of solutions tothe PDE are called adjoint-symmetries. Their algebraic structure for general PDE systemsis studied herein. This is motivated by the correspondence between variational symmetriesand conservation laws arising from Noether’s theorem, which has a well-known generaliza-tion to non-variational PDEs, where symmetries are replaced by adjoint-symmetries, andvariational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certainEuler-Lagrange condition). Several main results are obtained. Symmetries are shown tohave three different linear actions on the linear space of adjoint-symmetries. These linearactions are used to construct bilinear adjoint-symmetry brackets, one of which is like apull-back of the symmetry commutator bracket and has the properties of a Lie bracket. Inthe case of variational PDEs, adjoint-symmetries coincide with symmetries, and the linearactions themselves constitute new bilinear symmetry brackets which differ from the com-mutator bracket when acting on non-variational symmetries. Several examples of nonlinearPDEs are used to illustrate all of the results. Introduction
In the study of partial differential equations (PDEs), symmetries are a fundamental intrin-sic (coordinate-free) structure of a PDE and have numerous important uses [1, 2, 3], such asfinding exact solutions, mapping known solutions into new solutions, detecting integrability,and finding linearizing transformations. In addition, when a PDE has a variational principle,then through Noether’s theorem [2, 3] the infinitesimal symmetries of the PDE under whichthe variational principle is invariant — namely, variational symmetries — yield conservationlaws.Like symmetries, conservation laws [2, 3, 6] are another important intrinsic (coordinate-free) structure of a PDE. They provide conserved quantities and conserved norms, which areused in the analysis of solutions; they detect integrability and can be used to find linearizingtransformations; they also can be used to check the accuracy of numerical solution methodsand give rise to discretizations with good properties.A modern form of the Noether correspondence between variational symmetries and con-servation laws has been developed in the past few decades [4, 2, 5, 6] and generalizedto non-variational PDEs. In this generalization, infinitesimal symmetries are replaced by [email protected], [email protected]. djoint-symmetries, and variational symmetries are replaced by multipliers which are adjoint-symmetries satisfying an Euler-Lagrange condition [7, 8, 6]. From a purely algebraic view-point, symmetries of a PDE are the solutions of the linearization (Frechet derivative) equa-tion. Solutions of the adjoint linearization equation are called adjoint-symmetries. As animportant consequence, the problem of finding the conservation laws for a PDE is reducedto a kind of adjoint of the problem of the symmetries of the PDE. In particular, for anyPDE system, conservation laws can be explicitly derived in a similar algorithmic way to thestandard way that symmetries are derived (see Ref. [6] for a review).These developments motivate studying the basic mathematical properties of adjoint-symmetries and their connections to infinitesimal symmetries. As is well known, the setof infinitesimal symmetries of a PDE has the structure of a Lie algebra, in which the subsetof variational symmetries is a Lie subalgebra, and the set of conservation laws of a PDEis mapped into itself under the symmetries of the PDE. This leads to two interesting basicquestions: • How do symmetries act on adjoint-symmetries and multipliers? • Does the set of adjoint-symmetries have any kind of algebraic structure, with the setof multipliers constituting some sub-structure?In Ref. [9, 10], the explicit action of infinitesimal symmetries on multipliers is derived forgeneral PDE systems and used to study invariance of conservation laws under symmetries.For scalar PDEs, recently in Ref. [11], a linear mapping from symmetries into adjoint-symmetries is constructed in terms of any fixed adjoint-symmetry that is not a multiplier.The present paper is addressed to expanding on this work and will give answers to thebasic questions just posed for general PDE systems.Firstly, it will be shown that there are two different actions of infinitesimal symmetrieson adjoint-symmetries. For adjoint-symmetries that also are multipliers, these two actionscoincide with the known action given in Ref. [9, 10]. Furthermore, one of the actions yieldsmultipliers from adjoint-symmetries, while the difference of the two actions produces a thirdaction that vanishes on multipliers. Each of the three actions can be viewed as a generalizedpre-symplectic operator, in analogy with symplectic operators that map symmetries intoadjoint-symmetries for Hamiltonian systems.Secondly, the three actions of infinitesimal symmetries on adjoint-symmetries will be usedto construct associated bracket structures on the subset of adjoint-symmetries given bythe range of each action. Two different constructions will be given: the first bracket isskew-symmetric and can be viewed as a pull-back of the symmetry commutator bracket toadjoint-symmetries; the second bracket is non-symmetric and does not utilize the commuta-tor structure of symmetries. Most significantly, one of the skew-symmetric brackets will beshown to satisfy the Jacobi identity, and thus it gives a Lie algebra structure to a naturalsubset of adjoint-symmetries. In certain situations, this subset will coincide with the wholeset of adjoint-symmetries.Thirdly, for Euler-Lagrange PDEs, adjoint-symmetries coincide with infinitesimal symme-tries, and thus each of the three symmetry actions themselves represents a bracket on theset of infinitesimal symmetries. One of these brackets reduces to the standard commuta-tor when acting on variational symmetries. The other two brackets are new: one of themyields variational symmetries from symmetries, and the other one vanishes on variationalsymmetries. ll of these main results are new and provide important steps in understanding the basicalgebraic structure of adjoint-symmetries. They will be illustrated for several examples: anonlinear diffusion equation, the Korteweg-de Vries equation, a nonlinear wave system, a2D anisotropic elastodynamics equation, the 2D incompressible fluid flow equation, and theZakharov-Kunetsov equation in 3D.The rest of the paper is organized as follows. Section 2 gives a short review of infinitesi-mal symmetries, adjoint-symmetries, and multipliers, from an algebraic viewpoint. Section 3presents the actions of infinitesimal symmetries on adjoint-symmetries and multipliers. Sec-tion 4 derives the bracket structures for adjoint-symmetries and discusses their properties.Sections 5 and 6 specialize the results to Euler-Lagrange PDEs and evolution PDEs. Sec-tion 7 gives the examples. Finally, section 8 contains some concluding remarks.Throughout, the mathematical setting will be calculus in jet space, which is summarizedin an Appendix. Partial derivatives and total derivatives will be denoted using a standard(multi-) index notation. The Frechet derivative will be denoted by ′ . Adjoints of totalderivatives and linear operators will be denoted by ∗ .2. Symmetries and adjoint-symmetries
We will primarily utilize an algebraic perspective (following Ref. [6]), which allows sym-metries and adjoint-symmetries to be defined and handled in a unified way.Consider a general PDE system of order N consisting of M equations G A ( x, u ( N ) ) = 0 , A = 1 , . . . , M (2.1)where x i , i = 1 , . . . , n , are the independent variables, and u α , α = 1 , . . . , m , are the de-pendent variables. The space of formal solutions u α ( x ) of the PDE system will be denoted E . A symmetry is a set of functions P α ( x, u ( k ) ) that is non-singular on E and satisfies G ′ ( P ) A | E = 0 . (2.2)This is the determining equation for P α , called the characteristic functions of the symmetry.Off of the solution space E , the symmetry determining equation is given by G ′ ( P ) A = R P ( G ) A (2.3)where R P = ( R P ) A IB D I is some linear differential operator whose coefficients ( R P ) A IB arefunctions that are non-singular on E .The determining equation for adjoint-symmetries is the adjoint of symmetry determiningequation (2.2). It is obtained by using the Frechet derivative identity Q A G ′ ( P ) A = P α G ′∗ ( Q ) α + D i Ψ i ( P, Q ) . (2.4)An adjoint-symmetry is a set of functions Q A ( x, u ( k ) ) that is non-singular on E and satisfies G ′∗ ( Q ) α | E = 0 . (2.5)Off of the solution space E , this determining equation is given by G ′∗ ( Q ) α = R Q ( G ) α (2.6)where R Q = ( R Q ) Iα B D I is some linear differential operator whose coefficients ( R Q ) Iα B arefunctions that are non-singular on E . he geometrical meaning of symmetries is well known. From the algebraic viewpoint,it comes from the relation G ′ ( P ) A = (pr P α ∂ u α ) G A whereby the symmetry determiningequation (2.2) can be expressed as ((pr P α ∂ u α ) G A ) | E = 0 . (2.7)This is usually the starting point for defining symmetries, since it indicates that X P = P α ∂ u α is a vector field that is tangent to surfaces G A = 0 (and their “prolongations” D k G A = 0, k = 0 , , , . . . ) in jet space.A simple geometrical meaning of adjoint-symmetries is yet to be developed in general. (SeeRef. [6, 14] for some remarks in the case of evolution PDEs.) We will pursue this generalquestion elsewhere.Recall that a multiplier is a set of functions Λ A ( x, u ( k ) ) that are non-singular on E andsatisfy Λ A G A = D i Ψ i off of E , for some vector function Ψ i in jet space. This total divergencecondition is equivalent to E u α (Λ A G A ) = 0 . (2.8)It can be further reformulated through the product rule of the Euler operator, which yieldsthe equivalent condition Λ ′∗ ( G ) α + G ′∗ (Λ) α = 0. Consequently, on E , G ′∗ (Λ) α | E = 0 (2.9)whereby Λ A is an adjoint-symmetry. Off of E , the adjoint-symmetry determining equation(2.6) yields G ′∗ (Λ) α = R Λ ( G ) α (2.10)where R Λ is a linear differential operator. Hence, we see that Λ ′∗ ( G ) α = − G ′∗ (Λ) α = − R Λ ( G ) α . Now suppose that G A = 0 is a regular PDE system, namely it possesses asolved-form with respect to some leading derivatives, and it does not obey any differentialidentities. (See Ref. [6] for more details about regular PDE systems.) Then we can concludethat Λ ′∗ = − R Λ + S I,J ( D I G ) D J where S I,J = − S J,I holds off of E and S I,J is non-singularon E . Furthermore, suppose that Λ A contains no leading derivatives of G A = 0. Then wecan assume without loss of generality that S I,J = 0. Therefore, in this situation, Λ ′∗ = − R Λ holds identically. The adjoint of this equation yields the relationΛ ′ = − R ∗ Λ . (2.11)3. Action of symmetries on adjoint-symmetries
Symmetries of any given PDE system are well-known to form a Lie algebra under commu-tation. From the algebraic viewpoint, if P α , P α are symmetries, then so is their commutator[ P , P ] α = P ′ ( P ) α − P ′ ( P ) α . (3.1)The geometrical formulation is the same:[ X P , X P ] = X [ P ,P ] . (3.2)Stated precisely, the set of symmetries comprises a linear space on which the commutatordefines a bilinear antisymmetric bracket that obeys the Jacobi identity. Any symmetry hasa natural action on this space via the commutator.Symmetries also have a natural action on the set of adjoint-symmetries, since this set isa linear space that is determined by the given PDE system whose solution set is mapped nto itself by a symmetry. Actually, we will now show that there are two distinct actions ofsymmetries on the linear space of adjoint-symmetries.The first symmetry action arises from the adjoint relation between the determining equa-tions (2.2) and (2.5) for symmetries and adjoint-symmetries.As is well known [12, 13, 7, 14], when P α is a symmetry and Q A is an adjoint-symmetry,the adjoint relation (2.4) yields a conservation law since D i Ψ i ( P, Q ) | E = Q A G ′ ( P ) A | E − P α G ′∗ ( Q ) α | E = 0 (3.3)from the determining equations (2.2) and (2.5). Off of E , this formula is given by D i Ψ i ( P, Q ) = Q A R P ( G ) A − P α R Q ( G ) α where R P and R Q are the linear differential op-erators determined by equations (2.3) and (2.6). We now integrate by parts to obtain D i Ψ i ( P, Q ) = ( R ∗ P ( Q ) A − R ∗ Q ( P ) A ) G A + D i F i ( P, Q ; G ) (3.4)and hence ( R ∗ P ( Q ) A − R ∗ Q ( P ) A ) G A is a total divergence in jet space. This implies that thefunctions R ∗ P ( Q ) A − R ∗ Q ( P ) A constitute a conservation law multiplier. Since every multiplieris an adjoint-symmetry, we have a linear mapping Q A X P −→ R ∗ P ( Q ) A − R ∗ Q ( P ) A (3.5)which acts on the linear space of adjoint-symmetries.The second symmetry action is new and arises directly from the action of a symmetry P α applied to the adjoint-symmetry determining equation (2.6). To begin, from the lefthandside of this equation, we get X P ( G ′∗ ( Q ) α ) = G ′∗ ( X P ( Q )) α + X P ( G ′∗ )( Q ) α . (3.6)The last term can be simplified on E by the following steps. First, we have X P ( G ′∗ ) | E = ( X P ( G ′ )) ∗ | E = ( X P ( G )) ′∗ | E since X P commutes with total derivatives. Second,( X P ( G )) ′∗ | E = ( R P ( G )) ′∗ | E through use of the symmetry determining equation (2.3). Thenwe expand the Frechet derivative, ( R P ( G )) ′∗ ( Q ) α | E = ( R P G ′ ) ∗ ( Q ) α | E = G ′∗ ( R ∗ P ( Q )) α | E .Thus, expression (3.6) on E becomes X P ( G ′∗ ( Q ) α ) | E = G ′∗ ( Q ′ ( P ) + R ∗ P ( Q )) α | E . (3.7)Next, from the righthand side of equation (2.6), we have X P ( R Q ( G ) α ) = ( X P R Q )( G ) α + R Q ( X P ( G )) α . (3.8)On E , this yields X P ( R Q ( G ) α ) | E = 0 . (3.9)Finally, from equating expressions (3.9) and (3.7), we get G ′∗ ( Q ′ ( P ) + R ∗ P ( Q )) α | E = 0 (3.10)which shows that Q ′ ( P ) A + R ∗ P ( Q ) A is an adjoint-symmetry. Therefore, we have a linearmapping Q A X P −→ Q ′ ( P ) A + R ∗ P ( Q ) A (3.11)acting on the linear space of adjoint-symmetries.This action is a generalization of a better known action of symmetries on conservation lawmultipliers, which is found in Ref. [9, 10]. Further discussion is given in section 3.1. e now summarize the preceding results and then we will develop some of their conse-quences. Theorem 3.1.
There are two actions (3.5) and (3.11) of symmetries on the linear space ofadjoint-symmetries. The first symmetry action (3.5) maps adjoint-symmetries into conser-vation multipliers. The difference of the first and second actions yields the linear mapping Q A X P −→ Q ′ ( P ) A + R ∗ Q ( P ) A . (3.12)The action (3.12) will be trivial when the adjoint-symmetry is a conservation law multi-plier, as follows from the relation (2.11) which holds under certain mild conditions on theform of the PDE system G A = 0 and the functions Q A . Proposition 3.2.
For a regular PDE system G A = 0 , the symmetry action (3.12) on adjoint-symmetries Q A that contain no leading derivatives (and their differential consequences) inthe PDE system is trivial iff Q A is a conservation law multiplier. The conditions in Proposition 3.2 are satisfied by evolution PDEs, as shown in section 6.3.1.
Symmetry action on multipliers.
The action of a symmetry vector field X P = P α ∂ u α on the multiplier equation Λ A G A = D i Ψ i yields, for the righthand side,pr X P D i Ψ i = D i (pr X P Ψ i ) , (3.13)while for the lefthand side, pr X P (Λ A G A ) = Λ ′ ( P ) A G A + Λ A G ′ ( P ) A . We can simplify thelast term by using the form of the symmetry determining equation (2.3) off of E :Λ A G ′ ( P ) A = Λ A R P ( G ) A = R ∗ P (Λ) A G A + D i F i (3.14)where R ∗ P is the adjoint of the linear differential operator R P . Thus, we havepr X P (Λ A G A ) = (Λ ′ ( P ) A + R ∗ P (Λ) A ) G A modulo total derivatives . (3.15)Now, from equating expressions (3.15) and (3.13), we conclude that (Λ ′ ( P ) A + R ∗ P (Λ) A ) G A is a total derivative. Therefore, Λ ′ ( P ) A + R ∗ P (Λ) A is a multiplier.This yields the following well-known action [9, 10]:Λ A X P −→ Λ ′ ( P ) A + R ∗ P (Λ) A . (3.16)Theorem 3.1 shows that this action extends from conservation law multipliers to adjoint-symmetries through the symmetry action (3.11) on adjoint-symmetries.4. Bracket structures for adjoint-symmetries
The commutator (3.1) of symmetries, and the three actions (3.5), (3.11), (3.12) of sym-metries on adjoint-symmetries, constitute bilinear bracket structures on the linear spaces ofsymmetries and adjoint-symmetriesSymm G := { P α ( x, u ( k ) ) , k ≥ , s.t. G ′ ( P ) A | E = 0 } (4.1)AdjSymm G := { Q A ( x, u ( k ) ) , k ≥ , s.t. G ′∗ ( Q ) α | E = 0 } (4.2)for a given PDE system G A ( x, u ( N ) ) = 0. The linear space of multipliersMultr G := { Λ A ( x, u ( k ) ) , k ≥ , s.t. G ′∗ (Λ) α + Λ ′∗ ( G ) α = 0 } (4.3)is a subspace of the linear space of adjoint-symmetries. n interesting question is whether there exists any bilinear bracket structure on linearspace of adjoint-symmetries. Such a structure would allow the possibility for a pair of knownadjoint-symmetries to generate a new adjoint-symmetry, just as a pair of known symmetriescan generate a new symmetry.It will be useful to begin with a general discussion.Suppose that a PDE system G A ( x, u ( N ) ) = 0 possesses the extra structure G ′ H = J G ′∗ (4.4)where H and J are linear differential operators whose coefficients are non-singular on E .Then, for any adjoint-symmetry Q A , G ′ ( H ( Q )) A | E = J ( G ′∗ ( Q )) A | E = 0 (4.5)whereby H ( Q ) α is a symmetry. Moreover, the adjoint of the structure (4.4) G ′ J ∗ = H ∗ G ′∗ (4.6)shows that J ∗ ( Q ) α is a symmetry. Note that, off of E , R H ( Q ) = J ( R Q ) , R J ∗ ( Q ) = H ∗ ( R Q ) . (4.7)Hence, a structure (4.4) gives rise to two mappings from AdjSymm G into Symm G , whichwe will call generalized pre-Hamiltonian operators in analogy with Hamiltonian operatorsthat map adjoint-symmetries into symmetries for Hamiltonian systems. These mappings canbe combined with the symmetry actions on adjoint-symmetries from Theorem 3.1, yieldingthree brackets on AdjSymm G .Specifically, consider the operator H . The structure (4.4) shows that R ∗ P ( Q ) A − R ∗ Q ( P ) A = R ∗ Q ( J ∗ ( Q )) A − R ∗ Q ( H ( Q )) A , (4.8) Q ′ ( P ) A + R ∗ P ( Q ) A = Q ′ ( H ( Q )) A + R ∗ Q ( J ∗ ( Q )) A , (4.9) Q ′ ( P ) A + R ∗ Q ( P ) A = Q ′ ( H ( Q )) A + R ∗ Q ( H ( Q )) A , (4.10)where each of these expressions defines a bilinear bracket ( Q , Q ) H on AdjSymm G . Similarbrackets ( Q , Q ) J ∗ are obtained by switching H ↔ J ∗ ( J ↔ H ∗ ).For any specific PDE system G A ( x, u ( N ) ) = 0, a generalized pre-Hamiltonian structure(4.4) can be sought by using the same methods for seeking recursion operators. This aspectwill be pursued elsewhere.We will now show how an action of symmetries on adjoint-symmetries can be used itselfto construct a generalized pre-Hamiltonian structure that yields a corresponding bilinearbracket structure on the linear space adjoint-symmetries.Consider, in general, any symmetry action Q A X P −→ S P ( Q ) A on AdjSymm G , where S P is alinear operator which is also linear in P α . Note that S P may be constructed from both totalderivatives D I and partial derivatives ∂ u αI .The action S P ( Q ) A alternatively defines a dual linear operator S Q ( P ) A := S P ( Q ) A (4.11)from Symm G into AdjSymm G , which constitutes a generalized pre-symplectic operator [11] inanalogy with symplectic operators that map symmetries into adjoint-symmetries for Hamil-tonian systems. For a fixed adjoint-symmetry Q A , S Q will have an inverse S − Q which is efined modulo its kernel, ker( S Q ) ⊂ Symm G , and which acts on the linear subspace givenby its range, S Q (Symm G ) ⊆ AdjSymm G .The linear operator S − Q can be used in two different ways to construct a bilinear bracket:the first bracket will be skew-symmetric and is like a pull-back of the symmetry commu-tator (3.1); the second bracket will be non-symmetric and does not involve the symmetrycommutator (3.1).4.1. Adjoint-symmetry commutator brackets from symmetry actions.
The con-struction of the first bracket goes as follows.
Proposition 4.1.
Fix an adjoint-symmetry Q A in AdjSymm G , and let S Q be the linearoperator (4.11) associated to a symmetry action S P on AdjSymm G . If the kernel of S Q isan ideal in Symm G , then Q [ Q , Q ] A := S Q ([ S − Q Q , S − Q Q ]) A (4.12) defines a bilinear bracket on S Q (Symm G ) ⊆ AdjSymm G . This bracket can be expressed as Q [ Q , Q ] A = Q ′ ( S − Q Q ) − Q ′ ( S − Q Q ) − S ′ Q ( S − Q Q )( S − Q Q ) + S ′ Q ( S − Q Q )( S − Q Q ) (4.13) where S ′ Q denotes the Frechet derivative of S Q . Any one of the symmetry actions (3.5), (3.11), (3.12) can be used to write down formallya corresponding bracket (4.12). However, S − Q is well-defined only modulo ker( S Q ), and so inthe absence of any extra structure to fix this arbitrariness, the condition that ker( S Q ) is anideal is necessary and sufficient for the bracket to be well defined (namely, invariant under S − Q → S − Q + ker( S Q )).For ker( S Q ) to be an ideal, it is required to be a subalgebra that is preserved by Symm G .The subalgebra condition [ker( S Q ) , ker( S Q )] ⊆ ker( S Q ) (4.14)states that S Q ([ P , P ]) = 0 is required to hold for all pairs of symmetries X P = P α ∂ u α and X P = P α ∂ u α such that S Q ( P ) A = S P ( Q ) A = 0 and S Q ( P ) A = S P ( Q ) A = 0. We will nowdetermine whether this condition (4.14) holds for each of the three symmetry actions.It will be simplest to begin with the second symmetry action (3.11). Consider0 = S Q ( P ) A = Q ′ ( P ) A + R ∗ P ( Q ) A , S Q ( P ) A = Q ′ ( P ) A + R ∗ P ( Q ) A . (4.15)We apply the symmetries X P and X P respectively to these two equations and subtractthem. This yields0 = X P ( Q ′ ( P ) A + R ∗ P ( Q ) A ) − X P ( Q ′ ( P ) A + R ∗ P ( Q ) A )= Q ′ ( X P ( P ) − X P ( P )) A + X P ( R ∗ P )( Q ) A − X P ( R ∗ P )( Q ) A + R ∗ P ( X P ( Q )) A − R ∗ P ( X P ( Q )) A (4.16)using X P Q ′ ( P ) − X P Q ′ ( P ) = Q ′′ ( P , P ) − Q ′′ ( P , P ) = 0. The first term in equation(4.16) reduces to a commutator expression Q ′ ( X P ( P ) − X P ( P )) A = Q ′ ([ P , P ]) A . (4.17)The middle two terms can be expressed as X P ( R ∗ P )( Q ) A − X P ( R ∗ P )( Q ) A = R ∗ [ P ,P ] ( Q ) A − [ R ∗ P , R ∗ P ]( Q ) A (4.18) y use of the identity R ∗ [ P ,P ] = [ R ∗ P , R ∗ P ] + X P ( R ∗ P ) − X P ( R ∗ P ) (4.19)which can derived straightforwardly from the symmetry determining equation (2.3) off of E .Next, the last term in equation (4.18) can be combined with the last two terms in equation(4.16), yielding R ∗ P ( Q ′ ( P ) + R ∗ P ( Q )) A − R ∗ P ( Q ′ ( P ) + R ∗ P ( Q )) A = 0 (4.20)due to equations (4.15). Hence, after these simplifications, equation (4.16) becomes 0 = Q ′ ([ P , P ]) A + R ∗ [ P ,P ] ( Q ) A = S Q ([ P , P ]) A . This establishes the following result. Lemma 4.2.
For the second symmetry action (3.11) , ker( S Q ) is a subalgebra in Symm G . To continue, we carry out similar steps for the third symmetry action (3.12), starting from0 = S Q ( P ) A = Q ′ ( P ) A + R ∗ Q ( P ) A , S Q ( P ) A = Q ′ ( P ) A + R ∗ Q ( P ) A . (4.21)Respectively applying the symmetries X P and X P to these two equations and subtracting,we obtain 0 = Q ′ ([ P , P ]) A + R ∗ Q ([ P , P ]) A + X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A . (4.22)Hence, we see that S Q ([ P , P ]) = Q ′ ([ P , P ]) A + R ∗ Q ([ P , P ]) A = X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A does not vanish in general. This represents an obstruction to the bracketbeing well-defined. A useful remark is that if Q = Λ is a conservation law multiplier for aregular PDE system, then the relation (2.11) shows that X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A = X P ( Q ′ )( P ) A − X P ( Q ′ )( P ) A = Q ′′ ( P , P ) − Q ′′ ( P , P ) = 0 (4.23)whereby the obstruction vanishes.A similar obstruction arises for the bracket given by the first symmetry action (3.5).Specifically, by the same steps used for the second and third symmetry actions, we ob-tain S Q ([ P , P ]) = R ∗ [ P ,P ] ( Q ) A − R ∗ Q ([ P , P ]) A = X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A + R ∗ P ( S Q ( P )) A − R ∗ P ( S Q ( P )) A . This expression contains the same obstruction terms asfor the third symmetry action, as well as terms that involve the third symmetry action itself.If Q = Λ is a conservation law multiplier for a regular PDE system, then this obstructionvanishes.Consequently, we have the following two results. Lemma 4.3.
For the third symmetry action (3.12) , ker( S Q ) is a subalgebra in Symm G iffthe condition X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A = 0 (4.24) holds for all symmetries X P = P α ∂ u α and X P = P α ∂ u α in ker( S Q ) . Lemma 4.4.
For the first symmetry action (3.5) , ker( S Q ) is a subalgebra in Symm G iff thecondition X P ( R ∗ Q )( P ) A − X P ( R ∗ Q )( P ) A + R ∗ P ( S Q ( P )) A − R ∗ P ( S Q ( P )) A = 0 (4.25) holds for all symmetries X P = P α ∂ u α and X P = P α ∂ u α in ker( S Q ) . The preceding developments can be summarized as follows. heorem 4.5. The adjoint-symmetry commutator bracket (4.12) associated to each of thesymmetry actions (3.5) , (3.11) , (3.12) is well-defined on S Q (Symm G ) ⊆ AdjSymm G if ad(Symm G ) ker( S Q ) ⊆ ker( S Q ) and, for the actions (3.5) and (3.12) , if the respective condi-tions (4.25) and (4.24) hold when dim ker( S Q ) > . These latter conditions are identicallysatisfied when Q is a conservation law multiplier for a regular PDE system. An alternative way to have the bracket be well defined is if the quotient Symm G / ker( S Q )can be naturally identified with a subspace in Symm G . This is equivalent to requiring thatthe symmetry Lie algebra admits an extra structure of a direct-sum decomposition as a linearspace Symm G = ker( S Q ) ⊕ coker( S Q ) (4.26)such that the decomposition is independent of a choice of basis. Then S − Q can be defined asbelonging to the subspace coker( S Q ), and hence the bracket will be well defined.We will now show that the extra structure (4.26) will typically exist for a symmetry Liealgebra that contains a scaling symmetry: X P scal . = P α scal . ∂ u α , P α scal . = q ( α ) u α − r ( i ) x i u αi (4.27)where the constants q ( α ) and r ( i ) are the scaling weights of the variables u α and x i .Every symmetry in Symm G can be decomposed into a sum of symmetries that are scalinghomogeneous. Consequently, there will exist a basis for Symm G consisting of P scal . and { P k } k =1 ,..., dim(Symm G ) − , such that [ P scal . , P k ] = w ( k ) P k where the constant w ( k ) is the scalingweight of the symmetry P k . Here there exists a direct-sum decompositionSymm G = span( P scal . ) ⊕ X k ⊕ span( P k ) . (4.28)which is basis independent. This will provide the extra structure (4.26) if the subspacesker( S Q ) and coker( S Q ) can be uniquely characterized in terms of their scaling weights. Theorem 4.6.
Suppose
Symm G contains a scaling symmetry (4.27) . For each of the sym-metry actions (3.5) , (3.11) , (3.12) , if ker( S Q ) is a scaling-homogeneous subspace in Symm G ,then the adjoint-symmetry commutator bracket (4.12) is well-defined on S Q (Symm G ) ⊆ AdjSymm G by taking S − Q to belong to a sum of scaling-homogeneous subspaces with dif-ferent scaling weights than that of ker( S Q ) . This result can be generalized if ker( S Q ) is a direct sum of scaling homogeneous subspacesthat have no scaling weights in common with any scaling homogeneous subspace in coker( S Q ).Now we will look at the basic properties of the general adjoint-symmetry commutatorbracket (4.12). Recall that the underlying symmetry commutator bracket is antisymmetricand obeys the Jacobi identity. This implies that the same properties are inherited by thebracket (4.12). Proposition 4.7.
The adjoint-symmetry commutator bracket (4.12) is antisymmetric Q [ Q , Q ] A + Q [ Q , Q ] A = 0 (4.29) and obeys the Jacobi identity Q [ Q , Q [ Q , Q ]] A + Q [ Q , Q [ Q , Q ]] A + Q [ Q , Q [ Q , Q ]] A = 0 . (4.30) he linear space of adjoint-symmetries will be a Lie algebra when there exists an adjoint-symmetry Q A such that S Q (Symm G ) = AdjSymm G where ker( S Q ) satisfies the conditions ineither of Theorems 4.5 and 4.6. Since S Q is a linear mapping, the condition S Q (Symm G ) = AdjSymm G can be expressedequivalently as dim AdjSymm G + dim ker( S Q ) = dim Symm G . (4.31)Hence, dim Symm G ≥ dim AdjSymm G is a necessary condition. This version is most usefulwhen the dimensions are finite.4.2. Adjoint-symmetry non-symmetric brackets from symmetry actions.
The con-struction of the second bracket avoids the symmetry commutator but lacks the attendantproperties.
Proposition 4.8.
Fix an adjoint-symmetry Q A in AdjSymm G , and let S Q be the dual linearoperator (4.11) associated to a symmetry action S P on AdjSymm G . If the kernel of S Q satisfies S P = 0 for all P ∈ ker( S Q ) , (4.32) then a bilinear bracket from AdjSymm G × S Q (Symm G ) into AdjSymm G is defined by Q ( Q , Q ) A := ( S S − Q Q Q ) A . (4.33)Any one of the symmetry actions (3.5), (3.11), (3.12) can be used to write down formallya corresponding bracket (4.33). Note that, unlike the situation for the commutator bracket(4.12), the condition (4.32) only involves the properties of the symmetry action S Q and doesnot depend on the Lie algebra structure of Symm G . This condition can be eliminated whena scaling symmetry (4.27) is contained in the symmetry Lie algebra. Proposition 4.9.
Suppose
Symm G contains a scaling symmetry (4.27) . For any symmetryaction, if ker( S Q ) is a scaling-homogeneous subspace in Symm G , then the adjoint-symmetrybracket (4.33) is well-defined on S Q (Symm G ) ⊆ AdjSymm G by taking S − Q to belong to a sumof scaling-homogeneous subspaces In contrast to the commutator bracket (4.12), the bracket (4.33) is non-symmetric. Itsonly general property is that Q ( Q, Q ) = Q (4.34)for all Q in S Q (Symm G ).5. Results for Euler-Lagrange PDEs
We will next specialize the preceding general results to Euler-Lagrange PDEs.Consider a general system of Euler-Lagrange PDEs for u α ( x ), with a Lagrangian L ( x, u ( N/ ): E u α ( L ) = G α ( x, u ( N ) ) = 0 . (5.1)Here the number of PDEs and the number of dependent variables in the system are equal, M = m , which allows the corresponding indices to be identified, A = α , with a transpose intheir up/down position. s is well known [2, 6], a PDE system has an Euler-Lagrange form iff the Frechet derivativeof the system is self-adjoint, G ′ = G ′∗ . (5.2)Consequently, adjoint-symmetries coincide with symmetries, Q α = P α , (5.3)and thereby AdjSymm G = Symm G .The necessary and sufficient condition for the components of a symmetry P α to be aconservation law multiplier is that they satisfy E u α ( P β G β ) = 0 (5.4)off of E . This condition is the same as X P = P α ∂ u α being a variational symmetry,namely pr X P ( L ) = L ′ ( P ) = D i Φ i ( x, u ( k )) for some vector function Φ i ( x, u ( k )), since E u α (pr X P ( L )) = 0 where pr X P ( L ) = L ′ ( P ) = P α E u α ( L ) + D i Ψ i ( x, u ( k ) ).Through the product rule for the Euler operator combined with the self-adjoint property(5.2), we see that the variational symmetry condition (5.4) can be expressed as G ′ ( P ) = − P ′∗ ( G ) = R P ( G ) . (5.5)5.1. Symmetry brackets.
From the preceding preliminaries, the symmetry actions in The-orem 3.1 and Proposition 3.2 can be formulated as follows.
Theorem 5.1.
The symmetry actions (3.5) , (3.11) , (3.12) yield corresponding bilinear brack-ets on Symm G , which are respectively given by ( P α , P α ) −→ R ∗ P ( P ) α − R ∗ P ( P ) α , (5.6)( P α , P α ) −→ P ′ ( P ) α + R ∗ P ( P ) α , (5.7)( P α , P α ) −→ P ′ ( P ) α + R ∗ P ( P ) α . (5.8) The first bracket (5.6) maps symmetries into variational symmetries. If a PDE system (5.1) is regular, then for variational symmetries that contain no leading derivatives (and theirdifferential consequences) in the PDE system, the third bracket (5.8) is trivial while thebrackets (5.6) and (5.7) reduce to the commutator ( P α , P α ) −→ [ P , P ] α . (5.9)Since the bracket (5.8) is equal to the difference of the two brackets (5.6) and (5.7), we seethat there are three natural bilinear brackets on Symm G : (5.9), which is antisymmetric andobeys the Jacobi identity; (5.6), which is antisymmetric; and (5.7), which has no inherentalgebraic properties except that its antisymmetrization is equal to the difference of theprevious two brackets. In general, the latter two brackets differ from the commutator bracket(5.9) only when acting on non-variational symmetries.In addition to these natural symmetry brackets, more brackets on symmetries are providedby the results in Proposition 4.1 along with Theorems 4.5 and 4.6. Specifically, we have thefollowing counterpart of those results. Theorem 5.2.
Fix a symmetry P α and consider the linear operator P α −→ S P ( P ) α = ( P α , P α ) (5.10) efined by any one of the symmetry actions (5.6) – (5.9) . If the kernel of S P either is anideal in Symm G , or is uniquely characterized by its homogeneity with respect to a scalingsymmetry in Symm G , then P [ P , P ] α := S P ([ S − P P , S − P P ]) α (5.11) defines a commutator bracket on S P (Symm G ) ⊆ Symm G . We will explore this bracket (5.11) for the case when S P = ad( P ) is given by the commu-tator (5.9), with ad denoting the adjoint action on the Lie algebra Symm G . In this case, wehave P [ P , P ] α = ad( P )([ad( P ) − P , ad( P ) − P ]) α = [ P , ad( P ) − P ] α + [ad( P ) − P , P ] α (5.12)where P α , P α belong to the range of ad( P ) in Symm G . The bracket (5.12) will, in general,differ from the usual commutator (5.9). They will be closely related if X P is scaling symmetry(4.27), since then both [ P , ad( P ) − P ] α and [ad( P ) − P , P ] α will be constant multiples of[ P , P ] α . An example will be given in section 7.Additional symmetry brackets arise from Propositions 4.8 and 4.9. Specifically, for a fixedsymmetry P α , if the kernel of S P either satisfies S ker( S P ) = 0 or is uniquely characterized byits homogeneity with respect to a scaling symmetry in Symm G , then P ( P , P ) α := ( S S − P P P ) α (5.13)defines a bilinear bracket from Symm G × S P (Symm G ) into Symm G . This bracket (5.13) isnon-symmetric and obeys P ( P, P ) = P for all symmetries P in S P (Symm G ).6. Results for evolution PDEs
We will now concentrate on developing the general results in Theorems 3.1 and 4.5 morespecifically for evolution PDEs.Consider a general system of evolution PDEs for u α ( t, x ), u αt = g α ( x, u, ∂ x u, . . . , ∂ Nx u ) (6.1)where x now denotes the spatial independent variables x i , i = 1 , . . . , n , while t is the timevariable. In this setting, the number of PDEs and the number of dependent variables in thesystem are equal, M = m . Hence, we can identify the corresponding indices A = α . Thus,we now have G α ( t, x, u ( N ) ) = u αt − g α ( x, u, ∂ x u, . . . , ∂ Nx u ) . (6.2)It will be useful to note that all t -derivatives of u α can be eliminated in any expressionthrough substituting the system (6.1) and its spatial derivatives. This demonstrates, inparticular, that any evolution system is regular [6].Symmetries will thereby be functions P α ( t, x, u, ∂ x u, . . . , ∂ kx u ) that satisfy the determiningequation (2.2), which is given by ( D t P α − g ′ ( P ) α ) | E = 0. The first term can be expressedas D t P α = ∂ t P α + P ′ ( u t ) α = ∂ t P α + P ′ ( g ) α + P ′ ( G ) α . Hence, the symmetry determiningequation takes the explicit form ∂ t P α + P ′ ( g ) α − g ′ ( P ) α = ∂ t P α + [ g, P ] α = 0 (6.3)which implies that G ′ ( P ) α = P ′ ( G ) α off of E . Consequently, we have R P = P ′ . (6.4) n components, ( R P ) αβ = ( P α ) u βI D I .Likewise, adjoint-symmetries will be functions Q α ( t, x, u, ∂ x u, . . . , ∂ kx u ) that satisfy thedetermining equation (2.5), which is given by ( − D t Q α − g ′∗ ( Q ) α ) | E = 0. This equation hasthe explicit form − ( ∂ t Q α + Q ′ ( g ) α + g ′∗ ( Q ) α ) = 0 . (6.5)Hence, off of E , we have G ′∗ ( Q ) α = − Q ′ ( G ) α , which yields R Q = − Q ′ . (6.6)In components, ( R Q ) αβ = − ( Q α ) u βI D I .The necessary and sufficient condition for an adjoint-symmetry to be a conservation lawmultiplier is that its Frechet derivative is self-adjoint Q ′ = Q ′∗ . (6.7)In components, this condition can be expressed as the system of equations Q u αI = ( − | I | E Iu α ( Q ) , | I | = 0 , , . . . (6.8)in terms of the higher Euler operators (A.7).Thus, the determining system for multipliers consists of equations (6.8) and (6.5). Weremark that the latter equation (6.5) can be replaced by ∂ t Q α + { Q, g } ∗ α = 0 in terms of theanti-commutator { A, B } = A ′ ( B ) + B ′ ( A ).6.1. Symmetry actions on adjoint-symmetries.
The symmetry actions in Theorem 3.1can be simplified by use of the relations (6.4) and (6.6). Combined with Proposition 3.2,this yields the following result.
Theorem 6.1.
The actions (3.5) and (3.11) of symmetries on the linear space of adjoint-symmetries are respectively given by Q α X P −→ Q ′∗ ( P ) α + P ′∗ ( Q ) α = E u α ( P β Q β ) , (6.9) Q α X P −→ Q ′ ( P ) α + P ′∗ ( Q ) α . (6.10) The action (3.12) given by the difference of these two actions consists of Q α X P −→ Q ′ ( P ) α − Q ′∗ ( P ) α (6.11) which is non-trivial iff Q α is not a conservation law multiplier. In component form, the actions (6.9)–(6.11) expressed as linear operators on AdjSymm G are given by S P ( Q ) α = D ∗ I (( P β Q β ) u αI ) := S Q ( P ) α , (6.12) S P ( Q ) α = ( Q α ) u βI D I P β + D ∗ I ( Q β ( P β ) u αI ) := S Q ( P ) α , (6.13) S P ( Q ) α = ( Q α ) u βI D I P β − D ∗ I ( P β ( Q β ) u αI ) := S Q ( P ) α , (6.14) here S Q denotes the corresponding dual linear operator (4.11) from Symm G intoAdjSymm G . The components of these dual operators are explicitly given by( S Q ) αβ = D ∗ I ∂ u αI Q β , (6.15)( S Q ) αβ = ( Q α ) u βI D I + D ∗ I Q β ∂ u αI , (6.16)( S Q ) αβ = ( Q α ) u βI D I − D ∗ I ( Q β ) u αI , (6.17)acting on functions P β , where the righthand sides are viewed as operator compositions.For each of these operators, let ( S − Q ) γα denote the components of the formal inverseoperator S − Q . This operator gives a linear mapping from S Q (Symm G ) ⊆ AdjSymm G intoSymm G modulo ker( S Q ). Note that, despite the apparent complexity of S − Q in general,in computations the structure constants of the map S Q can be used directly to find thepre-image of any given adjoint symmetry.6.2. Adjoint-symmetry commutator bracket.
For any symmetry action expressed asan operator S Q from Symm G to AdjSymm G , the resulting adjoint-symmetry commutatorbracket (4.12) in Proposition 4.1 is given by the components Q [ Q , Q ] α = ( Q ′ ) αβ ( S − Q ) βγ Q γ − ( Q ′ ) αβ ( S − Q ) βγ Q γ + ( S ′ Q ) αβ ( S − Q Q )( S − Q ) βγ Q γ − ( S ′ Q ) αβ ( S − Q Q )( S − Q ) βγ Q γ (6.18)where ( S ′ Q ) αβ is the Frechet derivative of ( S Q ) αβ . In particular,( S ′ Q ) αβ ( f ) = D ∗ I ∂ u αI Q ′ ( f ) β , (6.19)( S ′ Q ) αβ ( f ) = Q ′ u βI ( f ) α D I + D ∗ I Q ′ ( f ) β ∂ u αI , (6.20)( S ′ Q ) αβ ( f ) = Q ′ u βI ( f ) α D I − D ∗ I Q ′ u αI ( f ) β , (6.21)acting on functions f α , where the righthand sides are viewed as operator compositions.The bracket (6.18) is well-defined when ker( S Q ) satisfies the conditions in either of Theo-rems 4.5 and 4.6.In the case of Theorem 4.5, these conditions can be expressed entirely in terms of Q α and a pair of symmetries P α , P α , by means of the relations (6.6) and (6.4). In particular,condition (4.25) takes the form X P ( Q ′∗ )( P ) α − X P ( Q ′∗ )( P ) α + P ′ ∗ ( Q ′ ( P ) − Q ′∗ ( P )) α − P ′ ∗ ( Q ′ ( P ) − Q ′∗ ( P )) α = 0 (6.22)and condition (4.24) takes the form X P ( Q ′∗ )( P ) α − X P ( Q ′∗ )( P ) α = 0 (6.23)for all symmetries X P = P α ∂ u α and X P = P α ∂ u α in ker( S Q ) when dim ker( S Q ) >
1. When Q α is a conservation law multiplier, each condition is identically satisfied, which can be seenfrom the properties (6.7) and Q ′′ ( P , P ) α = Q ′′ ( P , P ) α .6.3. Adjoint-symmetry non-symmetric bracket.
The non-symmetric bracket (4.33) inProposition 4.8 is given by the components Q ( Q , Q ) α = ( S S − Q Q ) αβ Q β (6.24)where ( S − Q Q ) α = ( S − Q ) αβ Q β , in terms of any symmetry action S Q . This bracket is well-defined when ker( S Q ) satisfies the conditions in either of Propositions 4.8 and 4.9. . Examples
We will apply the main results on symmetry actions and adjoint-symmetry brackets toseveral examples of nonlinear PDEs: nonlinear diffusion equation, Korteweg-de Vries equa-tion, nonlinear wave system, 2D anisotropic elastodynamics equation, 2D incompressiblefluid flow. In each example, we list the point symmetries (in evolutionary form) and theiralgebra, the low-order adjoint-symmetries, the three actions of the point symmetries on theseadjoint-symmetries, and the corresponding adjoint-symmetry commutator brackets. All ofthe symmetries and adjoint-symmetries are obtained by solving the determining equations(2.2) and (2.5) through a standard method (see Ref. [2, 6]).We will also illustrate the novel symmetry brackets (5.6), (5.7), and (5.12) by consider-ing the Zakharov-Kunetsov equation as an example of an Euler-Lagrange equation. Thesebrackets will be compared to the commutator bracket (5.9).7.1.
Nonlinear diffusion equation. u t = ( k ( u ) u x ) x . (7.1)This is a scalar evolution equation, where k ( u ) is a diffusivity function. Its point sym-metries and adjoint-symmetries are well known [3]. Here we will take k ( u ) to be a generalfunction (and so we will not consider special cases for which additional symmetries or adjoint-symmetries could exist).A basis for the linear space of point symmetries consists of P = u t , P = u x , P = 2 tu t + xu x , (7.2)which represent generators for a time-translation, a space-translation, and a spatial scaling.Their algebra is given by the non-zero commutators[ P , P ] = − P , [ P , P ] = − P . (7.3)A basis of the linear space of adjoint-symmetries is given by Q = 1 , Q = x (7.4)which are also multipliers for conservation laws of mass and center of mass. Consequently,from Theorem 6.1, the third symmetry action (6.11) is trivial, while the other two symmetryactions (6.9)–(6.10) are given by the linear operator S P ( Q ) = E u ( P Q ) . (7.5)Its action is summarized in Table 1. Note that, for evaluating the symmetry actions, all t -derivatives of u are replaced through equation (7.1). Table 1.
Nonlinear diffusion equation: symmetry action (7.5) on adjoint-symmetries P P P Q − Q Q − Q − Q The dual operator S Q ( P ) = E u ( P Q ) will have a maximal range if we choose Q = Q = x .Then we have ker( S Q ) = span( P ), which is an ideal, and ran( S − Q ) = span( P , P ) modulo er( S Q ). From Table 1, we obtain S − Q ( Q ) = − P , S − Q ( Q ) = − P . Hence, the adjoint-symmetry commutator bracket (6.18) can be directly computed by Q [ Q , Q ] = S Q ([ − P , − P ]) = − S Q ( P ) = Q (7.6)through the symmetry commutator (7.3).This bracket (7.6) is a non-trivial Lie bracket for all diffusivities k ( u ). Table 2.
Nonlinear diffusion equation: adjoint-symmetry commutator bracket Q Q Q Q Q Q = Q = x also to construct a non-symmetric symmetry bracket (6.24), since S ker( S Q ) = S P = 0. By direct computation, we obtain Q ( Q , Q ) = − S P ( Q ) = 0 , Q ( Q , Q ) = − S P ( Q ) = Q , (7.7a) Q ( Q , Q ) = − S P ( Q ) = Q , Q ( Q , Q ) = − S P ( Q ) = Q . (7.7b)7.2. Korteweg-de Vries equation. u t + uu x + u xxx = 0 . (7.8)This is an integrable scalar evolution equation, which arises in shallow water wave theory(see, e.g. Ref. [15]).We will first consider its point symmetries and counterpart adjoint-symmetries, and after-ward we will consider its higher-order symmetries and higher-order adjoint-symmetries.A basis of the linear space of point symmetries is well-known to consist of P = u t , P = u x , P = tu x − , P = 3 tu t + xu x + 2 u, (7.9)which are generators representing a time-translation, a space-translation, a Galilean boost,and a scaling. Their algebra is given by the non-zero commutators[ P , P ] = − P , [ P , P ] = − P , [ P , P ] = − P , [ P , P ] = 2 P . (7.10)Counterparts of the point symmetries are the first-order adjoint-symmetries Q ( t, x, u, u t , u x ). A basis consists of Q = 1 , Q = u, Q = tu − x. (7.11)These adjoint-symmetries are also multipliers for conservation laws representing mass, mo-mentum, and Galilean momentum. (In the integrable systems literature, multipliers are oftencalled cosymmetries in the case when an integrable evolution equation has a bi-Hamiltonianformulation.)From Theorem 6.1, the third symmetry action (6.11) is trivial, and the other two symmetryactions (6.9)–(6.10) are given by the linear operator (7.5). Its action is summarized inTable 3. In evaluating the symmetry actions, note that all t -derivatives of u are replacedthrough equation (7.8).To obtain a maximal range for the dual operator S Q ( P ) = E u ( P Q ), we can choose either Q = Q = u or Q = Q = tu − x . able 3. KdV equation: symmetry action (7.5) on adjoint-symmetries P P P P Q Q Q − Q Q Q − Q Q S Q ) = span( P , P ), which is an ideal. Then, fromTable 3, we find ran( S Q ) = span( Q , Q ), where ran( S − Q ) = span( P , P ) modulo ker( S Q ),with S − Q ( Q ) = − P and S − Q ( Q ) = P . Hence, for Q = Q = u , the adjoint-symmetrycommutator bracket (6.18) is well-defined on span( Q , Q ). It can be directly computed by Q [ Q , Q ] = S Q ([ − P , P ]) = − S Q (2 P ) = Q . (7.12) Table 4.
KdV equation: adjoint-symmetry commutator bracket Q Q Q Q Q S Q ) = span( P , P ), which is a subalgebra but is not an ideal,as shown by the symmetry structure (7.10). Nevertheless, since P is a scaling symme-try, ker( S Q ) is uniquely characterized by the pair of scaling weights (2 , S − Q ( Q ) = P and S − Q ( Q ) = − P . The resulting adjoint-symmetry commutator bracket (6.18) is well-definedon span( Q , Q ). It is given by Q [ Q , Q ] = S Q ([ P , − P ]) = 0 (7.13)since P and P commute.It is not possible to construct a non-symmetric symmetry bracket (6.24) because, as seenfrom Table 3, there is no adjoint-symmetry for which condition (4.32) holds. In particular, S ker( S Q ) = 0 for all adjoint-symmetries (7.11).Next, we look at higher symmetries and higher adjoint-symmetries. The KdV equationpossesses the recursion operator R = D x + u + D x uD − x (7.14)which generates a hierarchy of symmetries given byˆ P k = R k ( u x ) , k = 0 , , , . . . (7.15)starting from a translation symmetry. In explicit form, ˆ P = u x , ˆ P = uu x + u xxx = − u t ,ˆ P = u u x + u x u xx + uu xxx + u xxxxx , etc. The linear space of these symmetries is anabelian algebra under the commutator (5.9).The adjoint of this recursion operator generates a corresponding hierarchy of adjoint-symmetries ˆ Q k = R ∗ k ( u ) , k = 0 , , , . . . . (7.16) heir explicit form is given by ˆ Q = u , ˆ Q = u + u xx , ˆ Q = u + uu xx + u x + u xxxx , etc.Each of these adjoint-symmetries is a conservation law multiplier. Moreover, every productˆ Q k ˆ P l , k, l = 0 , , , . . . , is a total divergence.Consequently, each of the three symmetry actions (6.9)–(6.11) is trivial, and hence thereis no Q for which ran( S − Q ) = span( ˆ P , ˆ P , . . . ).However, we can consider the scaling adjoint-symmetry Q = v + xu − t ( u xx + u ) (7.17)where v is potential defined by u = v x . This adjoint-symmetry comes from the scalingsymmetry of the KdV equation in potential form, as shown in Ref. [11]. In particular, eachsymmetry of that equation is an adjoint-symmetry of the KdV equation. Unlike the localadjoint-symmetries, the scaling adjoint-symmetry (7.17) is not a conservation law multiplier,and thus the third symmetry action (6.11) which is given by the linear operator S P ( Q ) = Q ′ ( P ) − Q ′∗ ( P ) (7.18)will be non-trivial.We obtain S ˆ P k ( Q ) = − (3 + 2 k ) ˆ Q k = S Q ( ˆ P k ) (7.19)and hence, S − Q ( ˆ Q k ) = − k ˆ P k . The resulting adjoint-symmetry bracket (6.18) is computedby Q [ ˆ Q k , ˆ Q l ] = k )(3+2 l ) S Q ([ ˆ P k , ˆ P l ]) = 0 , k, l = 0 , , , . . . (7.20)which vanishes.Therefore, the algebra of the adjoint-symmetries (7.16) is abelian, similarly to the algebraof the symmetries (7.15). In particular, these algebras are isomorphic through the linearmapping (7.19).The same result is obtained if we consider the non-symmetric symmetry bracket (6.24): Q ( ˆ Q k , ˆ Q l ) = − l S ˆ P l ( ˆ Q k ) = 0 , k, l = 0 , , , . . . (7.21)since, for any symmetry P , S P ( ˆ Q k ) = 0 due to ˆ Q k being a multiplier.Finally, we will apply the third symmetry action (7.18) to the low-order adjoint-symmetries, which comprise the first-order adjoint-symmetries Q , Q , Q , the second-orderadjoint symmetry ˆ Q , and the scaling adjoint-symmetry (7.17). This action is non-trivialonly on the scaling adjoint-symmetry: S P ( Q ) = − Q , S P ( Q ) = 2 Q , S P ( Q ) = 2 Q , S P ( Q ) = 2 Q. (7.22)Since ker( S Q ) is empty, the adjoint-symmetry commutator bracket (6.18) is automaticallywell defined on ran( S Q ) = span( Q , Q , ˆ Q , Q ). A direct computation yields the Lie bracketshown in Table 5.7.3. Nonlinear wave system. u t = v x , v t = au p u x + bu (7.23)This is a system of coupled evolution equations, where p = 0 is an arbitrary nonlinearitypower, and where a, b are non-zero constant coefficients. able 5. KdV equation: third adjoint-symmetry commutator bracket Q Q ˆ Q QQ − Q Q − Q Q ˆ Q − ˆ Q Q p to be a general power (and so we will not consider specialvalues of p for which the system could admit extra symmetries or adjoint-symmetries).A basis of the linear space of point-symmetries is given by P = (0 , , P = ( u t , v t ) , P = ( u x , v x ) ,P = ( ptu t + 2 pxu x − u, ptv t + 2 pxv x − ( p + 2) v ) (7.24)which represent generators for a shift, a time-translation, a space-translation, and a scaling.The non-zero commutators in the symmetry algebra consist of[ P , P ] = − ( p + 2) P , [ P , P ] = − pP , [ P , P ] = − pP . (7.25)A straightforward computation yields a basis of the linear space of adjoint-symmetries Q = (1 , , Q = ( bt, , Q = ( x + bt , t ) . (7.26)For evaluating the symmetry actions, all t derivatives of ( u, v ) are replaced through thesystem (7.23).The first symmetry action (6.9) is given by the linear operator S P ( Q ) = ( E u ( P t Q ) , E v ( P t Q )) . (7.27)(Here t denotes the transpose.) This action is summarized in Table 6. Table 6.
Nonlinear wave system: symmetry action (6.9) on adjoint-symmetries P P P P Q − (2 p + 2) Q Q − bQ − (3 p + 2) Q Q − Q − Q − (4 p + 2) Q From Theorem 3.1, we know that these adjoint-symmetries are also conservation law mul-tipliers. It is straightforward to see that the resulting conservation laws describe conservationof mass, mass-flux, and center of mass, respectively, for Q , Q , Q .As a consequence, we see by Theorem 6.1 that the third symmetry action (6.11) is trivial,while the second symmetry action (6.10) coincides with the symmetry action (7.27).A maximal range for the dual of the linear operator (7.27) is obtained if we choose Q = Q = ( x + bt , t ). Then, from Table 6, we have ker( S Q ) = span( P ), which is an ideal, andran( S Q ) = span( Q , Q , Q ), where S − Q ( Q ) = − P , S − Q ( Q ) = − P , S − Q ( Q ) = − p +2 P . ence, the adjoint-symmetry commutator bracket (6.18) can be directly computed by Q [ Q , Q ] = S Q ([ P , P ]) = 0 , (7.28a) Q [ Q , Q ] = p +2 S Q ([ P , P ]) = − p p +2 S Q ( P ) = p p +1 Q , (7.28b) Q [ Q , Q ] = p +2 S Q ([ P , P ]) = − p p +2 S Q ( P ) = p p +1) Q . (7.28c)This bracket is non-trivial for all p = 0 , − . It is a Lie bracket on the whole linear space ofadjoint-symmetries (7.26). Table 7.
Nonlinear wave system: adjoint-symmetry commutator bracket Q Q Q Q p p +1 Q Q p p +1) Q Q
0A non-symmetric symmetry bracket (6.24) also can be obtained by use of Q = Q =( x + bt , t ), since S ker( S Q ) = S P = 0. By direct computation, we obtain Q ( Q , Q ) = − S P ( Q ) = 0 , Q ( Q , Q ) = − S P ( Q ) = 0 , Q ( Q , Q ) = − p +2 S P ( Q ) = p +12 p +1 Q , (7.29a) Q ( Q , Q ) = − S P ( Q ) = 0 , Q ( Q , Q ) = − S P ( Q ) = bQ , Q ( Q , Q ) = − p +2 S P ( Q ) = p +24 p +2 Q , (7.29b) Q ( Q , Q ) = − S P ( Q ) = Q , Q ( Q , Q ) = − S P ( Q ) = Q , Q ( Q , Q ) = − p +2 S P ( Q ) = Q . (7.29c) Table 8.
Nonlinear wave system: adjoint-symmetry non-symmetric bracket Q Q Q Q p +12 p +1 Q Q bQ p +24 p +2 Q Q Q Q Q
2D elastodynamics wave equation. u tt + ku t = a ( u xp ) x + b ( u yq ) y (7.30)This is a quasilinear damped wave equation, with wave speed constants a, b = 0; nonlinearitypowers p, q = 0 , −
1; and damping constant k >
0. It arises in the theory of anisotropicelastodynamics. While it is not an Euler-Lagrange equation as written, it does come from aLagrangian via a variational integrating factor: e kt ( u tt + ku t − a ( u xp ) x − b ( u yq ) y ) = E u ( L ) (7.31)where L = e kt ( − u t + a p +1 u xp +1 + b q +1 u yq +1 ). y a straightforward computation, we obtain for equation (7.30) a basis of the linear spaceof point symmetries P = 1 , P = e − kt , P = u t , P = u x , P = u y , P = u − p − p +1 xu x − q − q +1 yu y . (7.32)These symmetries represent generators for a shift, a damped shift, a time-translation, space-translations, and a scaling. Their non-zero commutators are given by[ P , P ] = P , [ P , P ] = − kP , [ P , P ] = P , [ P , P ] = p − p +1 P , [ P , P ] = q − q +1 P . (7.33)The variational integrating factor (7.31) provides a one-to-one correspondence betweenadjoint-symmetries and symmetries: e kt P = Q. (7.34)Hence, a basis of the corresponding linear space of adjoint-symmetries consists of Q = 1 , Q = e kt , Q = e kt u t , Q = e kt u x , Q = e kt u y ,Q = e kt ( u − p − p +1 xu x − q − q +1 yu y ) . (7.35)The first symmetry action (3.5) is given by the linear operator S P ( Q ) = R ∗ P ( Q ) − R ∗ Q ( P ) . (7.36)A summary of its action is shown in Table 9. Table 9.
Elastodynamics wave equation: first symmetry action on adjoint-symmetries P P P P P P Q pq + p + q − p +1)( q +1) Q Q − kQ pq + p + q − p +1)( q +1) Q Q kQ kQ kQ pq + p + q )( p +1)( q +1) Q + kQ Q − kQ q +1 q +1 Q Q − kQ p +1 p +1 Q Q − pq + p + q − p +1)( q +1) Q − pq + p + q − p +1)( q +1) Q − pq + p + q )( p +1)( q +1) Q − kQ − q +1 q +1 Q − p +1 p +1 Q Q and Q , which yield elastic momentum and weighted elasticmomentum; Q and Q , which yield spatial momenta; Q + pq + p + q ) k ( p +1)( q +1) Q , which yields energy.These multipliers can also be obtained through solving the determining condition (2.8).Hence, the multiplier subspace is 5 dimensional. From Proposition 3.2, we see that thethird symmetry action (3.12) is trivial on this subspace, and moreover, the second symmetryaction (3.11) will coincide with the first symmetry action (7.36) when they are restricted tothis subspace.For the symmetry action (7.36), the maximal range is obtained for Q = Q , but then wehave ker( S Q ) = span( P ) which is not an ideal, since P is a scaling symmetry. The nextlargest range is obtained for Q = Q , which has ran( S Q ) = span( Q , Q , Q , ˜ Q ) where˜ Q = Q + k ( p +1)( q +1)2(2 pq + p + q ) Q . (7.37) ith the choice Q = Q , we see ker( S Q ) = span( P , P ), which is an ideal, as shown bythe symmetry structure (7.33). As a consequence, from Table 9 we get S − Q ( Q ) = k P , S − Q ( Q ) = k P , S − Q ( Q ) = k P , S − Q ( ˜ Q ) = ( p +1)( q +1)2(2 pq + p + q ) P , (7.38)whereby ran( S − Q ) = span( P , P , P , P ) modulo ker( S Q ). The resulting adjoint-symmetrycommutator bracket (4.12) is well-defined and can be computed straightforwardly: its non-zero components consist of Q [ Q , ˜ Q ] = ( p +1)( q +1)2(2 pq + p + q ) k S Q ([ P , P ]) = ( p +1)( q +1)2(2 pq + p + q ) k S Q ( p − p +1 P ) = ( p − q +1)2(2 pq + p + q ) Q , (7.39a) Q [ Q , ˜ Q ] = ( p +1)( q +1)2(2 pq + p + q ) k S Q ([ P , P ]) = ( p +1)( q +1)2(2 pq + p + q ) k S Q ( q − q +1 P ) = ( q − p +1)2(2 pq + p + q ) Q . (7.39b)This defines a Lie bracket on the linear subspace of adjoint-symmetries span( Q , Q , Q , ˜ Q ). Table 10.
Elastodynamics wave equation: first and second adjoint-symmetrycommutator brackets Q Q Q ˜ Q Q Q ( p − q +1)2(2 pq + p + q ) Q Q ( q − p +1)2(2 pq + p + q ) Q ˜ Q Q , Q , Q , Q , Q , Q ) because, as seen from Table 9, there is no adjoint-symmetryfor which condition (4.32) holds. However, if we consider the previous subspace of adjoint-symmetries span( Q , Q , Q , ˜ Q ), then the adjoint-symmetry Q = Q satisfies S ker( S Q ) = 0and consequently the corresponding non-symmetric symmetry bracket (6.24) is well-definedon this subspace. Its non-zero components are given by Q ( Q , ˜ Q ) = ( p +1)( q +1)2(2 pq + p + q ) S P ( Q ) = pq + p + q − pq + p + q ) Q , Q ( ˜ Q , Q ) = k S P ( ˜ Q ) = pq + p + q − pq + p + q ) Q , (7.40a) Q ( Q , ˜ Q ) = ( p +1)( q +1)2(2 pq + p + q ) S P ( Q ) = ( p +1)(3 q +1)2(2 pq + p + q ) Q , Q ( ˜ Q , Q ) = k S P ( ˜ Q ) = ( p − q +1)2(2 pq + p + q ) Q , (7.40b) Q ( Q , ˜ Q ) = ( p +1)( q +1)2(2 pq + p + q ) S P ( Q ) = ( q +1)(3 p +1)2(2 pq + p + q ) Q , Q ( ˜ Q , Q ) = k S P ( ˜ Q ) = ( p +1)( q − pq + p + q ) Q , (7.40c) Q ( ˜ Q , ˜ Q ) = ( p +1)( q +1)2(2 pq + p + q ) S P ( ˜ Q ) = ˜ Q . (7.40d)Next we consider the third symmetry action (3.12). Since the multiplier subspace hascodimension 1 in the space of adjoint-symmetries, this action will be non-trivial on the1 dimensional quotient subspace span( Q , Q , Q , Q , Q , Q ) / span( Q , Q , Q , Q , ˜ Q ). Wewill choose Q = Q for the basis element. The resulting symmetry action is given by thelinear operator S P ( Q ) = Q ′ ( P ) + R ∗ Q ( P ) (7.41)which is shown in Table 11. able 11. Elastodynamics wave equation: third symmetry action on adjoint-symmetries P P P P P P Q − kQ − kQ − kQ − kQ − kQ − kQ For this action (7.41), we have ker( S Q ) = ∅ and ran( S Q ) = span( Q , Q , Q , Q , Q , Q ),and thus ran( S − Q ) = span( P , P , P , P , P , P ). The resulting adjoint-symmetry commuta-tor bracket (6.18) is shown in Table 12 which is readily computed from S − Q ( Q ) = − k P , S − Q ( Q ) = − k P , S − Q ( Q ) = − k P ,S − Q ( Q ) = − k P , S − Q ( Q ) = − k P , S − Q ( Q ) = − k P . (7.42)This defines a Lie bracket on the whole linear space of adjoint-symmetries. Table 12.
Elastodynamics wave equation: third adjoint-symmetry commu-tator bracket Q Q Q Q Q Q Q Q − k Q Q − k Q Q Q − p − k ( p +1) Q Q − q − k ( q +1) Q Q Q = Q , the non-symmetric symmetry bracket (6.24) is well-defined andhas non-zero components Q ( Q , Q i ) = Q i , i = 1 , . . . , Q ∈ span( Q , Q , Q , Q , ˜ Q ) cor-respond to variational symmetries in characteristic form P ∈ span( P , P , P , P , ˜ P ), with˜ P = P + k ( p +1)( q +1)2(2 pq + p + q ) P . Variational symmetries comprise a subalgebra in the symmetryalgebra.7.5.
2D incompressible inviscid fluid equations. ∆ u t + u x ∆ u y − u y ∆ u x = 0 (7.43)This is a scalar equation which governs the vorticity ω = ∆ u for incompressible inviscid fluidflow in two spatial dimensions [17].By a straightforward computation, we obtain a basis of the linear space of point symmetries P = u t , P = tu t + u, P = yu x − xu y , (7.44) P = 2 tu t + xu x + yu y , P = 2 t ( yu x − xu y ) + x + y , (7.45) P ( f ) = f ( t ) , P ( f ) = f ( t ) u x + f ′ ( t ) y, P ( f ) = f ( t ) u y − f ′ ( t ) x, (7.46)where each f ( t ) is an arbitrary function. These symmetries are generators for a time-translation, a time dilation, a rotation, a scaling, a Galilean rotational boost, a time-dependent shift, and generalized spatial translations. The non-zero commutators in their lgebra are given by[ P , P ] = − P , [ P , P ] = − P , [ P , P ] = − P , [ P , P ] = − P , [ P , P ] = − P , [ P , P ( f )] = − P ( f ′ ) , [ P , P ( f )] = − P ( f ′ ) , [ P , P ( f )] = − P ( f ′ ) , [ P , P ( f )] = − P ( f + tf ′ ) , [ P , P ( f )] = − P ( tf ′ ) , [ P , P ( f )] = − P ( tf ′ ) , [ P , P ( f )] = − P ( f ) , [ P , P ( f )] = P ( f ) , [ P , P ( f )] = − P ( tf ′ ) , [ P , P ( f )] = P ( f − tf ′ ) , [ P , P ( f )] = P ( f − tf ′ ) , [ P , P ( f )] = − P ( tf ) , [ P , P ( f )] = 2 P ( tf ) , [ P ( f ) , P ( f )] = P (( f f ) ′ ) . (7.47)Counterparts of the point symmetries for equation (7.43) are first-order adjoint-symmetries Q ( t, x, y, u, u t , u x , u y ). A straightforward computation yields a basis of the associated linearspace, which consists of the adjoint-symmetries Q = x + y , Q = u, Q ( f ) = f ( t ) , Q ( f ) = xf ( t ) , Q ( f ) = yf ( t ) (7.48)where each f ( t ) is an arbitrary function. These adjoint-symmetries can be checked to beconservation law multipliers through the determining condition (2.8).Consequently, from Theorem 3.1, we see that the third symmetry action (3.12) is trivial,and that the first and second symmetry actions reduce to the linear operator (7.36). Theaction of this operator is given in Table 13. Table 13.
2D incompressible fluid equation: first symmetry action onadjoint-symmetries P P P P P P ( f ) P ( f ) P ( f ) Q Q − Q − Q ( f ) − Q ( f ) Q Q Q Q ( f ) Q ( f ′ ) − Q ( f ′ ) Q ( F ) − Q ( F ′ ) Q ( F − tF ′ ) 0 − Q ( tF ′ ) 0 0 0 0 Q ( F ) − Q ( F ′ ) Q ( F − tF ′ ) − Q ( F ) − Q ( F + 2 tF ′ ) − Q ( tF ) 0 − Q ( f F ) 0 Q ( F ) − Q ( F ′ ) Q ( F − tF ′ ) Q ( F ) − Q ( F + 2 tF ′ ) 2 Q ( tF ) 0 0 − Q ( f F )A maximal range for the dual of the linear operator (7.36) is obtained by the choice Q = Q = u . From Table 13, we then have ran( S Q ) = span( Q , Q , Q ( f ) , Q ( f ) , Q ( f )),and ker( S Q ) = span( P , P , P ) which is a subalgebra but is not an ideal, as shown by thesymmetry structure (7.47). The inverse action is given by S − Q ( Q ) = P , S − Q ( Q ) = P ,S − Q ( Q ( f )) = P ( f ) , S − Q ( Q ( f )) = − P ( ∫ f dt ) , S − Q ( Q ( f )) = P ( ∫ f dt ) , (7.49)modulo span( P , P , P ), from which the adjoint-symmetry commutator bracket (6.18) canbe directly computed. Because span( P , P , P ) is not an ideal, some components of thebracket are not well defined. For instance: Q [ Q , Q ] = S Q ([ P + span( P , P , P ) , P + span( P , P , P )])= S Q ( P ) + S Q ([ P , span( P , P , P )]) + S Q ([span( P , P , P ) , P ])= S Q ( P ) + S Q (span(2 P , P , − P ))= S Q ( P ) + span( S Q ( P ) , S Q ( P ) , S Q ( P )) = cQ (7.50) here c is an arbitrary constant.Nevertheless, this arbitrariness can be fixed by taking advantage of the scaling symmetry P , as shown by Theorem 4.6. The subspaces span( P ) and span( P , P ) are characterizeduniquely by the respective scaling weights 2 and 0, as seen from the commutator structure(7.47). Hence, by defining S − Q to belong to a sum of scaling-homogeneous subspaces thatdo have not those scaling weights, we can use the relations (7.49) without the addition ofker( Q ) = span( P , P , P ). Then we obtain the adjoint-symmetry commutator bracket shownin Table 14. Table 14.
2D incompressible fluid equation: first adjoint-symmetry (scaling)commutator bracket Q Q Q ( f ) Q ( f ) Q ( f ) Q Q Q ( tf + ∫ f dt ) − Q ( tf + ∫ f dt ) Q − Q ( f + tf ′ )) − Q ( f + tf ′ ) − Q ( f + tf ′ ) Q ( f ) 0 0 0 Q ( f ) 0 Q (( ∫ f dt ∫ f dt ) ′ ) Q ( f ) 0Likewise, the non-symmetric symmetry bracket (6.24) for the choice Q = Q = u is welldefined on span( Q , Q , Q ( f ) , Q ( f ) , Q ( f )) only if we define S − Q by means of scalinghomogeneity.7.6. Zakharov-Kunetsov equation. u tx + u x u xx + u xxxx + u xxyy + u xxzz = 0 (7.51)This is the potential form of the Zakharov-Kunetsov (ZK) equation [18] v t + vv x + v xxx + v xyy + v xzz = 0, with v = u x .The ZK equation is a generalization of the KdV equation to three spatial dimensions. Inpotential form, it is the Euler-Lagrange equation of the Lagrangian L = − u t u x − u x + ( u xx + u xy + u xz ).By a straightforward computation, we obtain for equation (7.51) a basis of the linear spaceof point symmetries P = u t , P = u x , P = u y , P = u z , P = x − tu x ,P = zu y − yu z , P = u + 3 tu t + xu x + yu y + zu z , P ( f ) = f ( t, y, z ) (7.52)where f is an arbitrary function of its arguments. These symmetries represent generators fora time-translation, space-translations, a Galilean boost, a rotation, a scaling, and a gaugetransformation, all of which can be verified to be variational symmetries. Their algebra issummarized in Table 15.We first consider the novel symmetry bracket (5.12) which is given in terms of the linearoperator S P = ad( P ) . (7.53)The range for this operator is maximal for the choice P = P , which is the scal-ing symmetry. Then we have ran( S P ) = span( P , P , P , P , P , P ( f )) and ker( S P ) =span( P , P , t − / P ( f ker ( t/y , t/z )), where f satisfies f + 3 tf t + yf y + zf z = 0, and where f ker is an arbitrary function of its two arguments. It is straightforward to check that ker( S P ) able 15. ZK equation: point-symmetry algebra P P P P P P P P ( f ) P P − P − P ( f t ) P − P (1) 0 − P P P − P − P ( f y ) P − P − P − P ( f z ) P P P − P ( zf y − yf z ) P − P ( f + 3 tf t + yf y + zf z ) P ( f ) 0is a subalgebra, but it is not an ideal. However, the condition that ker( S P ) be an ideal canbe by-passed by utilizing the scaling symmetry, as shown by Theorem 5.2.From Table 15, we see that ker( S P ) has scaling weight 0, whereas the scaling weightsof span( P ), span( P , P , P ), span( P ) are non-zero, and span( P ( f )) is not scaling homo-geneous for a general function f ( t, y, z ). Hence, we can define S − Q to belong to a sum ofscaling-homogeneous subspaces whose scaling weights are non-zero. The resulting symmetrybracket (5.12) on span( P , P , P , P , P ) is readily computed fromad( P ) − P = P ad( P ) − P = P , ad( P ) − P = − P , ad( P ) − P = P , ad( P ) − P = P , (7.54)through Table 15. This yields P [ P , P ] = [ P , − P ] = − P , (7.55a) P [ P , P ] = [ P , − P ] = P (1) (7.55b)for the non-zero components. Table 16.
ZK equation: symmetry bracket (5.12) P P P P P P − P P P (1) P P P u tx + u x u xx + u xxxx + u xxyy = 0 . (7.56)This equation retains the translation symmetries P , P , P , and the Galilean boost P ,while the gauge transformation symmetry and the scaling symmetry have the form˜ P = u + 3 ptu t + xu x + yu y , ˜ P ( f ) = f ( t, y ) . (7.57) nly the scaling symmetry is non-variational. The algebra of these point sym-metries span( P , P , P , P , ˜ P , ˜ P ( f )) is the same as the corresponding subalgebra ofspan( P , P , P , P , P , P ( f )) in Table 15.To compute the new brackets (5.6) and (5.7), we first determine R P for each of the pointsymmetries: R P = D t , R P = D x , R P = D y , R P = − tD x ,R ˜ P = 3 tD t + xD x + yD y + 5 , R ˜ P = 0 . (7.58)Then we compute the expressions (5.6) and (5.7) for each pair of point symmetries. Theresulting brackets are shown in Tables 17 and 18. For comparison, the commutator bracketis summarized in Table 19. Table 17.
2D ZK equation: symmetry bracket (5.6) P P P P ˜ P ˜ P ( f ) P P − P − ˜ P ( f t ) P − ˜ P (1) − P P − P − ˜ P ( f y ) P − P ˜ P (1) 0 0 P P P P P − P − ˜ P (3 tf t + yf y )˜ P ( f ) ˜ P ( f t ) 0 ˜ P ( f y ) 0 ˜ P (3 tf t + yf y ) 0 Table 18.
2D ZK equation: symmetry bracket (5.7) P P P P ˜ P ˜ P ( f ) P P − P − ˜ P ( f t ) P − ˜ P (1) − P P − P − ˜ P ( f y ) P − P ˜ P (1) 0 0 2 P P P P P − P P − ˜ P (3 tf t + yf y )˜ P ( f ) ˜ P ( f t ) 0 ˜ P ( f y ) 0 ˜ P ( f + 3 tf t + yf y ) 0 Table 19.
2D ZK equation: point-symmetry algebra P P P P ˜ P ˜ P ( f ) P P − P − ˜ P ( f t ) P − ˜ P (1) − P P − P − ˜ P ( f y ) P − P ˜ P (1) 0 0 2 P P P P P − P − ˜ P ( f + 3 tf t + yf y )˜ P ( f ) ˜ P ( f t ) 0 ˜ P ( f y ) 0 ˜ P ( f + 3 tf t + yf y ) 0Notice that, in accordance with Theorem 6.1, the bracket (5.6) is antisymmetric, thebracket (5.7) is non-symmetric, and these two brackets differ from the commutator bracketonly for scaling symmetry ˜ P which is non-variational. . Concluding remarks
We have initiated a mathematical study of the algebraic structure of adjoint-symmetriesfor general PDE systems, G A ( x, u ( N ) ) = 0. Several main results have been obtained.We have derived three linear actions of symmetries on adjoint-symmetries. The firstaction arises from a well-known formula that yields a conservation law multiplier, Λ A ∈ Multr G , from a pair consisting of a symmetry, P α ∈ Symm G , and an adjoint-symmetry, Q A ∈ AdjSymm G . Since multipliers are adjoint-symmetries that satisfy certain extra Helmholtz-type conditions, the formula gives an action S P : AdjSymm G P −→ Multr G ⊆ AdjSymm G .The second action S P : AdjSymm G P −→ AdjSymm G comes from applying a symmetry tothe determining equation for adjoint-symmetries. It yields a generalization of a better knownaction of symmetries on conservation law multipliers, Multr G P −→ Multr G . A third action S P := S P − S P has the feature that it is non-trivial only on adjoint-symmetries that arenot multipliers.For each of these linear actions, we have constructed two different bilinear brackets onadjoint-symmetries. The construction uses the dual linear action S Q ( P ) := S P ( Q ) for afixed adjoint-symmetry. The first bracket is like a pull-back of the symmetry commutatorbracket and has the properties of a Lie bracket, whereas the second bracket does involve thecommutator structure of symmetries and is non-symmetric. Certain algebraic conditions areneeded on S Q so that the brackets are well-defined.For the first bracket, in the case of the action S Q , the needed condition is simply thatker( S Q ) ⊂ Symm G is invariant under the commutator action on Symm G . This conditioncan be eliminated when Symm G contains a scaling symmetry such that ker( S Q ) is the uniquesubspace having a fixed scaling weight.Moreover, in the case of Euler-Lagrange PDEs, the three symmetry actions themselvesdefine bilinear brackets on symmetries. On variational symmetries, two of these bracketsreduce to the commutator bracket, while the third bracket vanishes. Otherwise, the threebrackets are different than the commutator bracket.As shown by the examples we have considered, all of these brackets typically have anon-trivial structure, which indicates a very rich interplay among conservation laws andsymmetries beyond the connection provided by Noether’s theorem. Exploring this interplaymore deeply will be an interesting problem for future work.Another open problem will be to seek a geometrical meaning for adjoint-symmetries aswell as the new bracket structures. Appendix A. Calculus in jet space
General references are provided by Ref. [2, 6].The following notation is used: x i , i = 1 , . . . , n , are independent variables; u α , α = 1 , . . . , m , are dependent variables; u αi = ∂u α ∂x i are partial derivatives; ∂ k u is the set of all partial derivatives of u of order k ≥ u ( k ) is set of all partial derivatives of u with all orders up to k ≥ ulti-indices ( I = ∅ , u αI = u α , | I | = 0 I = { i , . . . , i N } , u αI = u αi ...i N , | I | = N ≥ x i , u α , u αi , . . . ), and J ( k ) = ( x, u ( k ) ) is the finitesubspace of order k ≥ D i = ∂ x i + u αi ∂ u α + · · · , i = 1 , . . . , n (A.1)The Frechet derivative of a function f on jet space is defined by( f ′ ) α = f u αI D I (A.2)which acts on functions F α . The Frechet second-derivative is given by the expression f ′′ ( F , F ) = f u αI u βJ ( D I F α )( D J F β ) (A.3)which is symmetric in the pair of functions ( F α , F α ). The adjoint of the Frechet derivativeof f is defined by ( f ′∗ ) α = D ∗ I f u αI = ( − | I | D I f u αI (A.4)which acts on functions F , where the righthand side is a composition of operators.The Euler operator (variational derivative) is defined by E u α = ( − | I | D I ∂ u αI (A.5)It has the property that E u α ( f ) = 0 holds identically iff f = D i F i for some vector function F i ( x, u ( k ) ). The product rule for the Euler operator is given by E u α ( f f ) = f ′ ∗ ( f ) α + f ′ ∗ ( f ) α (A.6)The higher Euler operators are defined similarly E Iu α = (cid:0) IJ (cid:1) ( − | J | D J ∂ u αIJ (A.7)See Ref. [2, 6] for their properties.Some useful relations: f ′ ( F ) = F α E u α ( f ) + D i Γ i ( F ; f ) , Γ i ( F ; f ) = ( D I F α ) E u αiI ( f ) (A.8) f ′ ( F ) = pr X F ( f ) , X F = F α ∂ u α , pr X F = ( D I F α ) ∂ u αI (A.9) References [1] L.V. Ovsiannikov,
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