aa r X i v : . [ m a t h - ph ] M a y Self-adjointness and conservation laws of differenceequations
Linyu Peng
Abstract
A general theorem on conservation laws for arbitrary difference equations isproved. The theorem is based on an introduction of an adjoint system relatedwith a given difference system, and it does not require the existence of a differenceLagrangian. It is proved that the system, combined by the original system and itsadjoint system, is governed by a variational principle, which inherits all symmetriesof the original system. Noether’s theorem can then be applied. With some specialtechniques, e.g. self-adjointness properties, this allows us to obtain conservation lawsfor difference equations, which are not necessary governed by Lagrangian formalisms.
Symmetries of variational principles are naturally symmetries of the associatedEuler-Lagrange equations. A connection between such type of symmetries andconservation laws for differential equations is estabilished via Noether’s theorem[3, 12, 22, 23]. However, Noether’s theorem has difficulty in applications to ar-bitrary differential equations. This is overcome by Ibragimov [13] by defining anadjoint system for arbitrary differential equations, and constructing a Lagrangianfor a given differential system together with its adjoint system. Symmetries of agiven differential system can then be extended to variational symmetries for theLagrangian. At this stage, Noether’s theorem is hence appliable.Geometric methods, especially symmetry analysis, as are used for investigatingdifferential equations, have been applied to difference equations over the last decades,see [11, 17, 31]. A discrete version of Noether’s theorem exists, see for example[5, 7, 19, 25]. In the present paper, we attempt to generalize Ibragimov’s methodfor constructing conservation laws from differential systems to difference systems.For a system of difference equations, its transformation groups of symmetries canbe extended to groups of variational symmetries for a Lagrangian governing theoriginal system itself together with its adjoint system. Thus, Noether’s theoremcan be used to construct conservation laws for the combined system. For (strict,quasi, weak) self-adjoint systems, it is possible to transfer such conservation laws toconservation laws of the original system. This procedure can also be realised oncespecial solutions of the adjoint system are known. Ibragimov’s conservation laws for differential equations
Let x = ( x , x , . . . , x p ) and u = ( u , u , . . . , u q ) be p independent variables and q dependent variables, respectively. Let J = ( j , j , . . . , j p ), and let u αJ denote | J | th -order partial derivatives of u . Here | J | = j + j · · · + j p and u αJ := ∂ | J | u α ( ∂x ) j ( ∂x ) j · · · ( ∂x p ) j p . (1)Consider a linear differential operator (the Lie-B¨acklund operator), X = ξ i D i + (cid:0) η α − ξ i u αi (cid:1) ∂∂u α + · · · + X α,J D J (cid:0) η α − ξ i u αi (cid:1) ∂∂u αJ + · · · = ξ i ∂∂x i + η α ∂∂u α + · · · . (2)Here ξ i = ξ i ( x, [ u ]) and η α = η α ( x, [ u ]) are smooth functions, where [ u ] denotes u and its derivatives. The operator D i is the total derivative with respect to x i and D J is a composite of total derivatives. The set of all such differential operators is aLie algebra equipped with the usual Lie bracket between two vector fields.For a system of partial differential equations F α ( x, [ u ]) = 0 , α = 1 , , . . . , q, (3)the adjoint system is given by0 = F ∗ α ( x, [ u ] , [ v ]) := E u α (cid:0) v β F β (cid:1) . (4)The Euler operator E u α is defined as E u α := ∂∂u α − D i ∂∂u αi · · · + ( − D ) J ∂∂u αJ + · · · (5)where ( − D ) J = ( − | J | D J . The system of differential equations (3) and (4) corre-sponds to a Lagrangian L ( x, [ u ] , [ v ]) = v α F α ( x, [ u ]), and we have E u α ( L ) = F ∗ α ( x, [ u ] , [ v ]) , E v α ( L ) = F α ( x, [ u ]) . (6)A differential system is said to be self-adjoint if the system F ∗ α ( x, [ u ] , [ u ]) = 0 isidentical to the original one.If a differential operator X = ξ ∂∂x i + η α ∂∂u α is a symmetry generator for the equa-tions (3), there always exists a variational symmetry generator for the Lagrangian L , namely Y = ξ ∂∂x i + η α ∂∂u α + η α ∗ ∂∂v α . The coefficient η α ∗ is determined by the gen-erator X . Hence, by using Noether’s theorem and the generator Y , we can constructconservation laws for the combination of (3) and (4). With knowledge of particularsolutions of v or self-adjointness of the original system, we can get conservation lawsof the original equations (3). For more details, consult [13]. Example 2.1 ([13]) . Consider the Korteweg-de Vries equation u t = u xxx + uu x (7) nd its adjoint equation v t = v xxx + uv x . (8) It is easy to see that the KdV equation is self-adjoint. These two equations aregoverned by the following Lagrangian L = v ( u t − uu x − u xxx ) . (9) Symmetries of a scaling transformation with generator X = − t ∂∂t − x ∂∂x + 2 u ∂∂u (10) will be extended to variational symmetries of the Lagrangian L , that is, Y = − t ∂∂t − x ∂∂x + 2 u ∂∂u − v ∂∂v . (11) Conservation law D t P + D x P obtained through Noether’s theorem is hence givenby P = (3 tu xxx + 3 tuu x + xu x + 2 u ) v (12) and P = − (2 u + xu t + 3 tuu t + 4 u xx + 3 tu txx ) v + (3 u x + 3 tu tx + xu xx ) v x − (2 u + 3 tu t + xu x ) v xx . (13) Setting v = u and transfering the terms of the form D x ( . . . ) from P to P , we get P = u , P = u x − uu xx − u . (14) Let us consider, in general, a given difference system with independent variables n = ( n , n , . . . , n p ) ∈ Z p , and dependent variables u = ( u , u , . . . , u q ) ∈ U ⊂ R q .Its solutions u = f ( n ) can be viewed as lying on sections s ( n ) = ( n, f ( n )) of thetrivial bundle π : Z p × U → Z p with π ( n, u ) = n , implying that Z p is viewed as thebase space. The shift operator (or map) S is defined as S k : n i n i + δ ik , k = 1 , , . . . , p, (15)with δ ik the Kronecker delta. Let 1 k be the p -tuple with only one nonzero entry,which is 1, at the k th place. Then the k th -shift operator and the naturally extendedshift operator to a function f ( n ) are respectively given by S k : n n + 1 k (16)and S k : f ( n ) f ( n + 1 k ) . (17)We sometimes use the notation S k instead of S k , and the composite of shifts usingmulti-index notation is given by S J = S j S j · · · S j p p , where J = ( j , j , . . . , j p ) is a p -tuple. Moreover, we can define the inverse of the shift map S k as S − k : n → n − k . he inverse map S − J of the composite of shifts is similarly defined. We constructthe prolongation bundle, pr ( ∞ ) ( Z p × U ), which has induced local coordinates [24, 25]( n, u, u { } , u { } , . . . , u {− } , u {− } , . . . ) , (18)where u { k } denotes all k th -order shifts of u . For example, we have u { } = { u α i } ,where u α i = S i f α ( n ).Consider a system of difference equations A = { F α ( n, [ u ]) = 0 , α = 1 , , . . . , q } , (19)where F α are analytic functions with respect to [ u ], which denotes u and its shifts.A one-parameter group G of transformations related to this system is a symmetrygroup, if and only if the associated infinitesimal generator X satisfies pr ( ∞ ) X ( F α ) = 0 on solutions of (19) . (20)Often we write an infinitesimal generator as X = Q α ( n, [ u ]) ∂∂u α , (21)where Q α ( n, [ u ]) are called characteristics with respect to a group of symmetries.Its prolongation is pr ( ∞ ) X = X + · · · + X α,J S J Q α ∂∂u αJ + · · · . (22) Remark 3.1.
Since F α are analytic functions, the symmetry criterion (20) can bere-stated as that a vector field X generates a group of symmetries for a differencesystem A if and only if there exist difference operators B αβ = B Jαβ ( x, [ u ]) S J suchthat pr ( ∞ ) X ( F α ) = X β B αβ ( F β ) . (23) Theorem 3.2.
Assume that X = Q α ∂∂u α and X = Q α ∂∂u α are two infinitesimalgenerators of symmetries for (19), then so does [ X , X ] .Proof. First we prove that pr ( ∞ ) [ X , X ] = [ pr ( ∞ ) X , pr ( ∞ ) X ]. Its right-handside is[ pr ( ∞ ) X , pr ( ∞ ) X ] = X J ,J ( S J Q α ) ∂ ( S J Q β ) ∂u αJ − ( S J Q α ) ∂ ( S J Q β ) ∂u αJ ! ∂∂u βJ , while [ X , X ] is given by[ X , X ] = X J ( S J Q α ) ∂Q β ∂u αJ − ( S J Q α ) ∂Q β ∂u αJ ! ∂∂u β . t is not difficult to find out that for any J , the following calculation is valid S J X J ( S J Q α ) ∂Q β ∂u αJ − ( S J Q α ) ∂Q β ∂u αJ ! = X J ( S J + J Q α ) ∂ ( S J Q β ) ∂u αJ + J − ( S J + J Q α ) ∂ ( S J Q β ) ∂u αJ + J ! = X J ( S J Q α ) ∂ ( S J Q β ) ∂u αJ − ( S J Q α ) ∂ ( S J Q β ) ∂u αJ ! , that is, pr ( ∞ ) [ X , X ] = [ pr ( ∞ ) X , pr ( ∞ ) X ]. Therefore, the symmetry criterionimplies that pr ( ∞ ) [ X , X ]( F α ) = [ pr ( ∞ ) X , pr ( ∞ ) X ]( F α ) , whose right-hand side can be written as the form P β,J B Jαβ S J ( F β ). This finishesthe proof.The following equality is used during the proof that, for any J , pr ( ∞ ) X ( S J ( F α )) = X β,I Q βI ∂ ( S J F α ) ∂u βI = X β,I S J Q βI − J ∂F α ∂u βI − J ! = S J (cid:16) pr ( ∞ ) X ( F α ) (cid:17) . (24)A conservation law is defined as the vanishment of a difference divergence ex-pression Div △ P := p X i =1 ( S i − id) P i (25)on solutions of A .For a difference variational problem L [ u ] = X n L ( n, [ u ]) , (26)the invariance criterion reads infinitesimally as pr ( ∞ ) X ( L n ) = Div △ R, (27)for some p -tuple R . Here we write L n = L ( n, [ u ]). The associated difference Euler-Lagrange equations E △ u α ( L n ) = 0 are obatained by using the difference Euler oper-ator E △ u α := X J S − J ∂∂u αJ . (28)The invariance of L [ u ] implies the invariance of the difference Euler-Lagrange equa-tions [25]. A discrete version of Noether’s theorem exists, which establishs a connec-tion between variational symmetries andconservation laws of the difference Euler-Lagrange equations. It has been proved that, for any finite tuple J , two equations △ u α ( L n ) = 0 and E △ u α ( S J L n ) = 0 are equivalent to each other [26]. For Lagrangiansbeing independent from backward shifts, a general form of conservation laws ob-tained from Noether’s theorem is Div △ P = 0 with [25] P i = X α,J ≥ i Q αJ − i S − i (cid:16) E △ u αJ ( L n ) (cid:17) − R i . (29)Here the operator E △ u αJ is given by E △ u αJ := X I ≥ S − I ∂∂u αI + J . (30)Let H be a linear operator, that is, it can be written as a polynomial of shiftoperators whose coefficients are functions on the prolongation bundle. Its adjointoperator H ∗ is defined by the following equality v H [ u ] = u H ∗ [ v ] + Div △ P. (31)The equation H ∗ [ v ] = 0 is called the adjoint equation of H [ u ] = 0. If for any u ( n ), H ∗ [ u ] = H [ u ] holds, the operator H is said to be (strictly) self-adjoint. Nevertheless,for many cases, H ∗ [ u ] = 0 and H [ u ] = 0 are equivalent to rather than equal to eachother. Example 3.3.
The adjoint equation of u n +2 − u n = 0 is v n − − v n = 0 . Thesetwo equations are equivalent to each other. If we rewrite the original equation as u n +1 − u n − = 0 , the adjoint equation is then v n − − v n +1 = 0 . Definition 3.4.
Consider a system of difference equations (19), and assume thefunctions F α ( n, [ u ]) are independent from backward shifts of u . We define the systemof its adjoint equations as F ∗ α ( n, [ u ] , [ v ]) := E △ u α (cid:0) v βn F β (cid:1) . (32)In the case of linear equations, this definition is equivalent to the one in (31).When F α are linear equations, the adjoint equations are linear with respect to [ v ]and independent from [ u ]. Otherwise, the adjoint equations are linear with respectto [ v ], but can be nonlinear in the coupled variables [ u ] and [ v ]. For linear differencesystems, the order of the adjoint system are the same as that of the original system.Nevertheless, in the nonlinear case, the order of the adjoint system is usually higherthan the original system. Definition 3.5.
A system of difference equations (19) is said to be self-adjoint ifits adjoint system by the substitution v = u , F ∗ α ( n, [ u ] , [ u ]) = 0 (33) holds for all solutions of the original system (19). Example 3.6.
In Example 3.3, it is easy to show that the equation u n +2 − u n = 0 is self-adjoint, since by the substitution v = u , the adjoint equation becomes u n − − u n = 0 , which is equivalent to the original one. xample 3.7. In general, let us consider the following second-order linear differenceequation a ( n ) u n +2 + a ( n ) u n +1 + a ( n ) u n = 0 . (34) Here a ( n ) a ( n ) = 0 for all n . Its adjoint equation is a ( n ) v n + a ( n − v n − + a ( n − v n − = 0 . (35) If the following equalities hold a ( n + 2) = a ( n ) , a ( n + 1) = a ( n ) , a ( n ) = a ( n ) , (36) the difference equation (34) is self-adjoint. Namely, a sufficient (but not necessary)condition for the equation (34) to be self-adjoint is that a ( n ) = a ( n ) = C − ( − n C − n , a ( n ) = C , (37) where C i are constants and C C = 0 . In the differential case, the generalizations of self-adjointness to quasi and weakself-adjointnesses are aslo very useful in finding conservation laws [6, 14]. Weakself-adjointness is also named nonlinear adjointness in [15]. We adjust such ideas todifference equations.
Definition 3.8.
A difference system (19) is said to be quasi self-adjoint if by anontrivial substitution v α = f α ([ u ]) , the adjoint system F ∗ α ( n, [ u ] , [ f ([ u ])]) = 0 (38) holds for all solutions u of the original system. The original system is said to be weak self-adjoint if by a nontrivial substitution v α = f α ( n, [ u ]) , the adjoint systemis satisfied for all solutions u of the original system. Here by nontrivial we meanthat not all the functions f α vanish simultaneously. Example 3.9.
Consider the second-order difference equation (34) again. If wechange the condition (37) to a ( n ) = a ( n ) = C − ( − n C − n , a ( n ) = ( − n C , (39) then the difference equation is weak self-adjoint that can be verified by a substitution v n = ( − n u n . Theorem 3.10.
Any difference system (19) together with its adjoint system (32)are governed by a Lagrangian.Proof.
Let us define a Lagrangian L n ( n, [ u ] , [ v ]) = v αn F α ( n, [ u ]) . (40)Direct calculation shows that E △ v α ( L n ) = F α ( n, [ u ]) , E △ u α ( L n ) = F ∗ α ( x, [ u ] , [ v ]) . (41)Thus, the difference Euler-Lagrange equations cover the original system (19) and itsadjoint system (32). he equation u n +2 − u n = 0 and its adjoint equation v n − − v n = 0 are differenceEuler-Lagrange equations with respect to a Lagrangian L n = v n ( u n +2 − u n ) . (42) Theorem 3.11.
Consider a system of difference equations (19) and its adjoint sys-tem (32). If the original system (19) admits a transformation group of symmetrieswith infinitesimal generator X = Q α ( n, [ u ]) ∂∂u α , (43) then the combined system admits a transformation group of symmetries with anextended generator Y = Q α ( n, [ u ]) ∂∂u α + Q α ∗ ( n, [ u ] , [ v ]) ∂∂v α . (44) The functions Q α ∗ ( n, [ u ] , [ v ]) are to be determined (see (49)).Proof. Write the Lagrangian as L n = v β F β ( n, [ u ]) , (45)and consider the variational symmetry criterion, that is pr ( ∞ ) Y ( L n ) = Y ( v β ) F β + v β pr ( ∞ ) X ( F β ) . (46)Recall that pr ( ∞ ) X ( F β ) = B βα ( F α ) . (47)The equation (46) hence turns out to be pr ( ∞ ) Y ( L n ) = Q β ∗ F β + X α v β B βα ( F α )= X α (cid:0) Q α ∗ + B ∗ βα ( v β ) (cid:1) F α + Div △ R. (48)Here B ∗ βα , the adjoint operator of B βα , and the tuple R are obtained through thediscrete version of integration by parts. Therefore, the proof finishes by setting Q α ∗ = − B ∗ βα ( v β ) . (49) Theorem 3.12.
For the difference system (19), each infinitesimal generator ofsymmetries X = Q α ∂∂u α provides a conservation law for a system combined by theoriginal system and its adjoint system.Proof. The proof is immediate. Theorem 3.11 implies that a symmetry generator ofthe original system can be extended to a variational symmetry generator for a La-grangian, which governs the combined system. Therefore, from Noether’s theorem,we can get a conservation law, see (29). xample 3.13. Consider the following ordinary difference equation u n +2 = u n u n +1 u n − u n +1 . (50) It has been found that it admits a three-dimensional group of Lie point symmetries,whose characteristics are [9] Q = u n , Q = nu n , Q = u n . (51) Define a Lagrangian L n = v n (cid:18) u n +2 − u n u n +1 u n − u n +1 (cid:19) . (52) The adjoint equation is given by v n u n +1 (2 u n − u n +1 ) − v n − u n − (2 u n − − u n ) + v n − = 0 . (53) For Q , we have that (cid:18) Q ∂∂u n + ( SQ ) ∂∂u n +1 + ( S Q ) ∂∂u n +2 (cid:19) (cid:18) u n +2 − u n u n +1 u n − u n +1 (cid:19) = u n +2 − u n u n +1 u n − u n +1 . (54) From (49), this implies a group of variational symmetries for the Lagrangian, namely Y = Q ∂∂u n + Q ∗ ∂∂v n with Q ∗ = − v n . (55) This hence provides a conservation law (first integral) for a system combined by theoriginal equation and its adjoint equation (see (29)), namely P = u n +1 v n − + u n v n − − u n u n − (2 u n − − u n ) v n − . (56) Here R = 0 , since pr ( ∞ ) Y ( L n ) = 0 . (57) Similarly, the other two extended characteristics with respect to the new coordinate v n are respectively Q ∗ = − ( n + 2) (cid:18) u n +2 + u n u n +1 u n − u n +1 (cid:19) v n ,Q ∗ = − (cid:18) u n +2 + u n u n +1 u n − u n +1 (cid:19) v n . (58) We have R = R = 0 , and two first integrals P = ( n + 1) u n +1 v n − + nu n v n − − nu n u n − (2 u n − − u n ) v n − ,P = u n +1 v n − + u n v n − − u n u n − (2 u n − − u n ) v n − . (59) xample 3.14 ( The discrete KdV equation).
A lattice version of the potentialKdV equation u t = u xxx + 3 u x (60) is given by a partial difference equation (see for example [18, 21]) ( u , − u , )( u , − u , ) + β − α = 0 , (61) which belongs to the ABS classification (equation H1) [1]. Here u , = u m,n is thevalue of the dependent variable at the point ( m, n ) ∈ Z , and u i,j denotes shifts of u , . The arbitrary functions α and β are assumed to be constants for simplicity. Itadmits a group of Lie point symmetries with the following infinitesimal generators[29] X = ∂∂u , , X = ( − m + n ∂∂u , , X = ( − m + n u , ∂∂u , ,X = u , ∂∂u , + 2 α ∂∂α + 2 β ∂∂β . (62) Let L n = v , [( u , − u , )( u , − u , ) + β − α ] , (63) and we get the adjoint equation ( u , − u , ) v , − ( u , − − u , ) v , − +( u − , − u , ) v − , − ( u , − − u − , ) v − , − = 0 . (64) From those generators, we get that Q ∗ = Q ∗ = Q ∗ = 0 , Q ∗ = − v n . (65) It is not difficult to get that R = R = R = R = 0 and the following conservationlaws are obtained correspondingly ( P = S − ξ + S − ξ ,P = S − ξ + S − ξ , ( P = ( − m + n ( S − ξ − S − ξ ) ,P = ( − m + n ( S − ξ − S − ξ ) , ( P = ( − m + n ( u , S − ξ − u , S − ξ ) ,P = ( − m + n ( u , S − ξ − u , S − ξ ) , ( P = u , S − ξ + u , S − ξ ,P = u , S − ξ + u , S − ξ . (66) Here S − i denotes the first-order backward shift along the i th -direction, and we write ξ = ( u , − u , ) v , − ( u , − − u , ) v , − ,ξ = − ( u , − u , ) v , − ( u , − u − , ) v − , ,ξ = − ( u , − u , ) v , . (67) Five-point symmetries are provided in [29], which will also lead to conservation laws.One may use even higher order symmetries.
In this section, our method is applied to several ordinary and partial differenceequations. By using known symmtries, we can get symmetries and conservation laws or the system combined by a difference system and its adjoint system. Conservationlaws of the original system can be obtained if it satisfies a certain self-adjointnessproperty. As well it is possible to construct conservation laws of the original systemif special solutions of the adjoint system can be obtained. Example 4.1.
Consider a nonlinear difference equation u n +2 u n − u n +1 = 0 , (68) which admits a group of symmetries with characteristics [11] (see also [25]) Q = u n , Q = nu n . (69) A Lagrangian can be defined as L n = v n ( u n +2 u n − u n +1 ) (70) and hence we get its adjoint equation v n u n +2 − v n − u n + v n − u n − = 0 . (71) The original difference equation is quasi self-adjoint, that can be verified by substi-tuting v n = 1 /u n in its adjoint equation. For Q , we have that (cid:18) Q ∂∂u n + ( SQ ) ∂∂u n +1 + ( S Q ) ∂∂u n +2 (cid:19) (cid:0) u n +2 u n − u n +1 (cid:1) = 2 (cid:0) u n +2 u n − u n +1 (cid:1) . (72) From (49), this implies an infinitesimal generator for the Lagrangian, namely Y = Q ∂∂u n + Q ∗ ∂∂v n with Q ∗ = − v n (73) and hence a first integral is obtained via (29) ( R = 0 here), P = − u n v n − + u n +1 u n − v n − + u n u n − v n − (74) such that SP = P on solutions of the Euler-Lagrange equations with respect to theLagrangian (70). Letting v n = 1 /u n , the first integral becomes P = − u n u n − + u n u n − + u n +1 u n − , (75) which is unfortunately trivial, that is, P = 0 on solutions of the original equation.Similarly, from Q , we get that Q ∗ = − n + 1) v n (76) and P = nu n ( − u n v n − + u n − v n − ) + ( n + 1) u n +1 u n − v n − . (77) By setting v n = 1 /u n , the first integral becomes P = − n u n u n − + n u n u n − + ( n + 1) u n +1 u n − , (78) which can be simplified into a nontrivial first integral P = u n u n − . (79) xample 4.2. Consider a second-order linear ordinary difference equation (cid:18) − n + 12 (cid:19) u n +2 + (cid:18) n − (cid:19) u n +1 − nu n = 0 . (80) A simple infinitesimal generator can be found that X = u n ∂∂u n . Introduce anothervariable v n , and define a Lagrangian L n = v n (cid:20)(cid:18) − n + 12 (cid:19) u n +2 + (cid:18) n − (cid:19) u n +1 − nu n (cid:21) . (81) Its adjoint equation is hence − nv n + (cid:18) n − (cid:19) v n − + (cid:18) − n + 52 (cid:19) v n − = 0 . (82) The generator X is extended to a variational symmetry generator for the Lagrangian,namely Y = X + Q ∗ ∂∂v n with Q ∗ = − v n . Therefore, a first integral via Noether’stheorem is constructed (here R = 0 ) P = (cid:20)(cid:18) n − (cid:19) v n − + (cid:18) − n + 52 (cid:19) v n − (cid:21) u n + (cid:18) − n + 32 (cid:19) v n − u n +1 . (83) It is not difficult to see that the adjoint equation has a constant solution, that is v n = C . Substituting this into the first integral P , we get a first integral of theoriginal equation, that is, nu n + (cid:18) − n + 32 (cid:19) u n +1 . (84)The procedure in Example 4.2 can be applied to linear partial difference equationsas well, since again their adjoint equations are independent from [ u ]. Example 4.3.
Consider the following linear partial difference equation α ( u , + u − , ) − β ( u , + u , − ) = 0 , (85) where α and β are arbitrary positive constants. It is a multisymplectic scheme forthe nonlinear wave equation u tt − u xx = 0 [4, 20, 27]. Provided that the constants α and β are fixed, it admits a two-dimensional group of Lie point symmetries, whoseinfinitesimal generators are X = u , ∂∂u , , X = ( − m + n u , ∂∂u , . (86) Its adjoint equation is α ( v − , + v , ) − β ( v , − + v , ) = 0 , (87) and the governing Lagrangian is L n = v , [ α ( u , + u − , ) − β ( u , + u , − )] . (88) t is obvious that the equation (85) is self-adjoint. The infinitesimal generators arerespectively extended to Y = X − v , ∂∂v , , Y = X + ( − m + n v , ∂∂v , . (89) Since the Lagrangian depends on backward shifts of u , , we may shift it forward andthen apply the same procedure we used above. However, taking Y as an example,we can reach a conservation law directly by pr ( ∞ ) Y ( L n ) = − v , [ α ( u , + u − , ) − β ( u , + u , − )]+ u , [ α ( v − , + v , ) − β ( v , − + v , )]+ α ( S − id)( u , v − , − u − , v , ) + β ( S − id)( u , − v , − u , v , − ) . (90) Hence we get a conservation law for the combined system as P = α ( u , v − , − u − , v , ) , P = β ( u , − v , − u , v , − ) . (91) Setting v = u , this becomes a conservation law for the original equation. However, itis trivial. In particular, when α = β , it is obvious that v , = 1 and v , = ( − m + n are two special solutions of the adjoint equation. Hence, respectively, we get twoconservation laws for the original equation, namely P = α ( u , − u − , ) , P = α ( u , − − u , ) (92) and P = α ( − m + n +1 ( u , + u − , ) , P = α ( − m + n ( u , − + u , ) . (93) Similarly, for Y , we can get a conservation law for the combined system ( P = α ( − m + n ( u , v − , + u − , v , ) ,P = β ( − m + n +1 ( u , − v , + u , v , − ) . (94) Let u = v , and it becomes a conservation law for the original system, namely P = 2 α ( − m + n u , u − , , P = 2 β ( − m + n +1 u , − u , . (95) When α = β , again we can follow the same procedure as that for Y to obtainconservation laws. However, those conservation laws via substitutions v , = 1 and v , = ( − m + n are equivalent to the ones we already obatined. By applying symmetries to known conservation laws, it is possible to constructnew conservation laws, which is called a symmetry method for conservation lawsin [30]. Let X be an infinitesimal generator and P be a conservation law for somedifference system. Since the prolonged symmetry generator and shift operatorscommute with each other, i.e.[ pr ( ∞ ) X, S J ] = 0 , for all J, (96)we have that pr ( ∞ ) X ( P ) is again a conservation law. Therefore, we can use onesymmetry generator again and again to obtain more conservation laws. In particular,if a difference system admits infinitely many symmetries, then we may constructinfinityly many conservation laws either by using the method developed in this paper r the symmetry method. Though no promise is given that such new conservationlaws are independent from one another. Neither are they nontrivial. For example,the two symmetry generators in Example 4.1 and the first integral (79) will lead toeither trivial or equivalent first integrals. Similar consequence happens for Example4.2 and Example 4.3. Nevertheless, higher order symmetries are usually helpful. Example 4.4.
Let us consider the equation in Example 4.3 again. It admits in-finitely many symmtries with infinitesimal generators X ij = u i,j ∂∂u , (97) and e X ij = ( − m + n u i,j ∂∂u , , (98) for any integers i and j . In this situation, both our new method and the symmetrymethod will lead to infinitely many conservation laws. Here since we already obtaineda conservation law via our method in Example 4.3, the symmetry method seems moreimmediate. Applying pr ( ∞ ) X ij to the conservation law (95), we obtain infinitelymany conservation laws ( P ij = 2 α ( − m + n ( u − , u i,j + u , u i − ,j ) ,P ij = 2 β ( − m + n +1 ( u , − u i,j + u , u i,j − ) . (99) By using X ij and the new conservation laws P kl , we can get even more conservationlaws, namely ( P ijkl = 2 α ( − m + n ( u i − ,j u k,l + u i,j u k − ,l ) ,P ijkl = 2 β ( − m + n +1 ( u i,j − u k,l + u i,j u k,l − ) . (100) We may continue this procedure, though it is possible that the newly obtained con-servation laws will be trivial or equivalent to the ones we already got. Similarly for e X ij , we get ( e P ij = 2 α ( u − , u i,j − u , u i − ,j ) , e P ij = 2 β ( u , u i,j − − u , − u i,j ) , (101) and e P ijkl = 2 α ( − m + n + k + l ( u − , u i + k,j + l + u , u i + k − ,j + l ) , e P ijkl = 2 β ( − m + n + k + l ( u , − u i + k,j + l + u , u i + k,j + l − ) . (102) For each pair of fixed k and l , we can always find e P ijkl from the set { P ij } (with amultiplication of ( − k + l ), that is, the expression of e P ijkl provides no more nonequiv-alent conservation laws. Acknowledgements
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