Variational Poisson-Nijenhuis structures for partial differential equations

V ARIA TIONAL POISSON–NIJENHUIS STR UCTURES F OR P AR TIAL DIFFERENTIAL EQUA TIONS V. GOLO VKO, I. KRASIL ′ SHCHIK, AND A. VERBOVETSKY Abstract. W e explore v ariational Poisson–Nijenh uis structures on nonlinear PDEs and establish relations bet ween Schout en and Nijenhuis brac kets on the initial equation with the Lie brack et of symmetries on i ts natural extensions (co ve ri ngs). This approac h allows to construct a f ramework f or the theo ry of nonlocal structures. Introduction Poisson–Nijenh uis structures [1 6] play an imp ortant role b oth in classica l dif- ferential geometry (se e , for e x ample [1 , 10]) a nd in geometry of par tial differential equations, se e [11, 1 4]. In the la tter case existence of a Poisson–Nijenh uis s tructure virtually amo un ts to complete integrabilit y of the equation under co nsideration. Infinite-dimensional P ois son–Nijenhuis structures are well descr ib ed for the cas e of jets and for evolutionary different ial equatio ns rega rded as flows on the jet spac e. As fo r general differential equatio n, the corres p o nding theory was not introduced for a lo ng time. In our r elatively r ecent w or ks [7, 6] we outlined a n appr oach to the theory in a pplication to evolution equation in geometric a l setting. This appr o ach is based on the notion of ∆- c ov erings and reduces co ns truction of bo th r ecursion op erator s a nd Hamiltonian structures to solution of the linearised equation ℓ E (Φ) = 0 (1) on sp ecia l extensio ns of the initial equation E . W e call thes e extensio ns the ℓ - and ℓ ∗ -c overings and they play the ro le of tangent and cotangent bundles in the category of differ ent ial e quations. The ab ov e men tioned appro ach seems to work for general equations as well and we exp o s e its g eneralisatio n b elow. In Section 1 we define Poisson–Nijenh uis structur es in the “absolute case”, i.e., for the manifold of infinite jets. T o this end, w e r edefine the Schouten and F r¨ olicher– Nijenh uis bra ck ets (cf. with [7, 12], see als o [17]). W e als o expres s the compatibility condition betw een a Poisson bi-v ector and a Nijenhuis o per ator in ter ms of a sp e - cial bracket closely r elated to Vinog radov’s unifie d br acket , see [3]. The main r esult of this section is Theorem 2 that states the existence of infinite families of pair - wise compatible Hamiltonian structures related to the initial Poisson–Nijenhuis structure. In Section 2 Poisson–Nijenhuis str uctures on evolution equations are int ro duce d. W e show that an inv aria n t (with resp ect to the flow deter mined by the eq ua tion) Nijenhuis tensors are r e cu rsion op er ators for the symmetries, while inv ar iant Poisson bi- vectors amount to Hamiltonian structur e s. W e also define the ℓ - and ℓ ∗ -cov erings and r educe constructio n of rec ur sion op erato r s and Hamiltonian 2000 Mathematics Subje ct Classific ation. 37K05, 35Q53. Key wor ds and phr ases. Poisson–Nijenh uis structures, symmetries, conserv ation law, cov eri ngs, nonlocal structures. This work w as suppor ted in part b y the NW O–RFBR gran t 047.017.015 and RFBR– Consortium E.I.N.S.T. E.I.N grant 06-01-92060. 1 structures to solution of equatio n (1). The Schouten a nd F r¨ olicher–Nijenh uis brack- ets as well as the co mpa tibilit y conditions are reformulated in terms o f the Jaco bi brack ets of the corresp onding solutions and explicit for m ulas for these brackets ar e obtained. Section 3 genera lises the results to arbitrar y nonlinear partial differen- tial equatio n. Finally , in Section 4 we outline an approa ch to deal with nonlo c al Poisson–Nijenh uis structures. 1. V aria tional Poisson –Nijenhuis structures on J ∞ ( π ) 1.1. Geometrical structures. Let us r ecall definitions and res ults we sha ll use. F or details w e r efer to [2]. Let π : E → M be a vector bundle o ver an n -dimensional ma nifold M and π ∞ : J ∞ ( π ) → M be the infinite jet bundle of lo cal sections o f the bundle π . If x 1 , . . . , x n are lo cal co ordina tes in the bas e a nd u 1 , . . . , u m are co or dinates alo ng the fib er of π the c anonic al c o or dinates u j σ arise in J ∞ ( π ) defined by j ∞ ( s ) ∗ ( u j σ ) = ∂ | σ | s j ∂ x σ , where s = ( s 1 , . . . , s m ) is a loca l sec tio n of π , j ∞ ( s ) is its infinite jet and σ = i 1 i 2 . . . i | σ | , i α = 1 , . . . , n , is a multi-index. Denote by F ( π ) the algebra of smo oth functions on J ∞ ( π ). The ba sic geometrica l structure o n J ∞ ( π ) is the Cartan distribution C tha t is spanned by total derivatives D i = ∂ ∂ x i + X j,σ u j σi ∂ ∂ u j σ . Different ial o p erators on J ∞ ( π ) in total deriv a tives will be ca lled C - differ en t ial op er ators . In lo cal c o ordinates, they hav e the form k P σ a σ ij D σ k , where a σ ij ∈ F ( π ). Let P and Q be F ( π )-mo dules of sectio ns of some v ector bundles over J ∞ ( π ). All C - differential op erators fro m a P to Q for m an F ( π )-mo dule denoted by C Diff ( P , Q ). The adjoint op er ator to a C -differential opera tor ∆ : P → Q is denoted by ∆ ∗ : ˆ Q → ˆ P , where ˆ P = Hom F ( π ) ( P, ¯ Λ n ( π )) and ¯ Λ n ( π ) is the F ( π )-mo dule of horizo nt al n - form on J ∞ ( π ), i.e., forms ω = a dx 1 ∧ · · · ∧ dx n . Denote by C Diff sk ew ( k ) ( P, Q ) the mo dule o f k -linear skew-symmetric C -differential op erator s P × · · · × P → Q and by C Diff sk-ad ( k ) ( P, ˆ P ) ⊂ C Diff sk ew ( k ) ( P, ˆ P ) the subset of op erator s skew-adjoin t in each argument. A π ∞ -vertical vector field on J ∞ ( π ) is called evolutionary if it preser ves the Car - tan distribution. There is a o ne-to-one cor resp ondence b etw een e volutionary vector fields and sectio ns o f the bundle π ∗ ∞ ( π ). Denote b y κ ( π ) the corr esp onding mo dule of sections. In lo cal co o r dinates, the evolutionary vector field that co rresp onds to a section (the gener ating s ection, or function) ϕ = ( ϕ 1 , . . . , ϕ m ) is of the form З ϕ = X j,σ D σ ( ϕ j ) ∂ ∂ u j σ . The comm utator of evolutionary vector fields induces in κ ( π ) a Lie algebr a struc- ture that is given b y the higher Jac obi br acket defined a s a unique se c tio n { ϕ, ψ } satisfying [ З ϕ , З ψ ] = З { ϕ, ψ } . The bracket { ϕ, ψ } is express ed b y { ϕ, ψ } = З ϕ ( ψ ) − З ψ ( ϕ ) . F ollowing [7, 8], we s hall call elements of κ ( π ) variational ve ctors , elements of the module C Diff sk-ad ( k − 1) ( ˆ κ , κ ) will be called variatio nal k -ve ctors , while elemen ts of ˆ κ will b e called variational 1 -forms and elements o f the mo dule C Diff sk-ad ( k − 1) ( κ , ˆ κ ) 2 v aria tional k - forms , r esp ectively . The Lie deriv ative on v ariationa l vectors L ϕ : κ → κ takes the form L ϕ = З ϕ − ℓ ϕ , (2) where the line arization op er ator ℓ ϕ is defined by the equalit y ℓ ϕ ( α ) = З α ( ϕ ), α ∈ κ . Un lo cal c o ordinates, it has the form ℓ ϕ ( α ) = X j,σ ∂ ϕ ∂ u j σ D σ ( α j ) , α = ( α 1 , . . . , α m ) . The Lie der iv ative on v ar iational forms L ϕ : ˆ κ → ˆ κ is of the form L ϕ = З ϕ + ℓ ∗ ϕ . (3) 1.2. V ariational P o isson–Ni jenh uis structures. Recall (see [7]) that the vari- ational Schouten br acket of t wo ope rators A , B ∈ C Diff sk-ad ( ˆ κ , κ ) is defined by [ [ A, B ] ]( ψ 1 , ψ 2 ) = − ℓ A, ψ 1 ( B ψ 2 ) + ℓ A, ψ 2 ( B ψ 1 ) − A ( ℓ ∗ B , ψ 1 ( ψ 2 )) − ℓ B , ψ 1 ( Aψ 2 ) + ℓ B , ψ 2 ( Aψ 1 ) − B ( ℓ ∗ A, ψ 1 ( ψ 2 )) , ψ 1 , ψ 2 ∈ ˆ κ . (4) An o pe rator A is called H amiltonian if the [ [ A, A ] ] = 0 and tw o Ha miltonian oper - ators A a nd B are c omp atible if their Schouten brack et v anishes. Remark 1. Here and b elow the notation ℓ ∆ ,p 1 ,...,p n ( ϕ ), ϕ ∈ κ , for a C - differential op erator ∆ : P × · · · × P → Q means ℓ ∆ ,p 1 ,...,p n ( ϕ ) = З ϕ (∆)( p 1 , . . . , p n ) , p 1 , . . . , p n ∈ P. F or tw o op er ators R , S ∈ C Diff ( κ , κ ) their F r¨ olic her–Nijenhuis br acket (cf. with [12], see a lso [17]) is defined by [ R, S ] F N ( ϕ 1 , ϕ 2 ) = { R ϕ 1 , S ϕ 2 } + { S ϕ 1 , R ϕ 2 } − R ( { S ϕ 1 , ϕ 2 } + { ϕ 1 , S ϕ 2 } − S { ϕ 1 , ϕ 2 } ) − S ( { Rϕ 1 , ϕ 2 } + { ϕ 1 , R ϕ 2 } − R { ϕ 1 , ϕ 2 } ) , ϕ 1 , ϕ 2 ∈ κ . (5 ) If [ R, R ] F N = 0 we shall refer to R as a Nijenhuis op er ator . F o r particula r co mpu- tations it is conv enient to use the equality [ R, S ] F N ( ϕ 1 , ϕ 2 ) = − ℓ R, ϕ 1 ( S ϕ 2 ) − ℓ S, ϕ 1 ( Rϕ 2 ) + ℓ R, ϕ 2 ( S ϕ 1 ) + ℓ S, ϕ 2 ( Rϕ 1 ) + R ( ℓ S, ϕ 1 ( ϕ 2 ) − ℓ S, ϕ 2 ( ϕ 1 )) + S ( ℓ R, ϕ 1 ( ϕ 2 ) − ℓ R, ϕ 2 ( ϕ 1 )) . (6) Definition 1 (cf. with [11]) . A Hamiltonia n op erato r A ∈ C Diff sk-ad ( ˆ κ , κ ) and a Nijenh uis op er ator R ∈ C Diff ( κ , κ ) constitute a variational Poisson–Nijenhuis structur e ( A, R ) on J ∞ ( π ) if the following compatibilit y co nditions ho ld (i) R ◦ A = A ◦ R ∗ , (7) (ii) C ( A, R )( ψ 1 , ψ 2 ) = L Aψ 1 ( R ∗ ψ 2 ) − L Aψ 2 ( R ∗ ψ 1 ) + R ∗ L Aψ 2 ( ψ 1 ) − R ∗ L Aψ 1 ( ψ 2 ) + E h ψ 1 , AR ψ 2 i − R ∗ E h ψ 1 , Aψ 2 i = 0 , (8) where E : ¯ H n ( π ) → ˆ κ is the Euler op er ator and ¯ H n ( π ) is the n th ho rizontal de Rham co homology gr oup, while h . , . i : ˆ κ × κ → ¯ H n ( π ) is the natural pa iring. In terms o f linearizatio n oper ators condition (ii) has the form C ( A, R )( ψ 1 , ψ 2 ) = − ℓ R ∗ , ψ 1 ( Aψ 2 ) + ℓ R ∗ , ψ 2 ( Aψ 1 ) + ℓ ∗ A, ψ 1 ( R ∗ ψ 2 ) + ℓ ∗ R ∗ , ψ 1 ( Aψ 2 ) − R ∗ ( ℓ ∗ A, ψ 1 ( ψ 2 )) = 0 . Similarly to the finite-dimensional c a se, we ha ve the following Prop ositio n 1. L et a Hamiltonian op er ator A ∈ C Diff sk-ad ( ˆ κ , κ ) and a Nijenhuis op er ator R ∈ C Diff ( κ , κ ) define a Poisson–Nijenhuis structur e on J ∞ ( π ) . Then the c omp osition R ◦ A is a Hamiltonian op er ator c omp atible with A . 3 Pr o of. By straightforward computations one can prov e that [ [ RA, RA ] ]( ψ 1 , ψ 2 ) − 2 R [ [ A, R A ] ]( ψ 1 , ψ 2 ) + R 2 [ [ A, A ] ]( ψ 1 , ψ 2 ) − [ R, R ] F N ( Aψ 1 , Aψ 2 ) = 0 (9 ) and 2[ [ A, RA ] ]( ψ 1 , ψ 2 ) − [ [ A, A ] ]( R ∗ ψ 1 , ψ 2 ) − [ [ A, A ] ]( ψ 1 , R ∗ ψ 2 ) − 2 A ( C ( A, R )( ψ 1 , ψ 2 )) = 0 , (10) from where the statement follo ws immediately .  Theorem 2 . L et a Hamiltonian op er ator A ∈ C Diff sk-ad ( ˆ κ , κ ) and a Nijenhuis op er ator R ∈ C Diff ( κ , κ ) define Poisson–Nijenhuis structur e on J ∞ ( π ) . Then on J ∞ ( π ) ther e is a hier ar chy of iter ate d Hamiltonian op er ators , that is a se- quenc e of Hamiltonian op er ators R i A , i ≥ 0, which ar e p air-wise c omp atible , i.e. , [ [ R i A, R j A ] ] = 0, i, j ≥ 0 . Pr o of. The pro of is by inductio n on n = max( i, j ). F or n = 1 the statement follows from the pr op osition ab ov e. Assume now that [ [ R i A, R j A ] ] = 0, i, j = 0 , . . . , n , and A ( C ( R i A, R j )) = 0, i + j ≤ n , and let us prov e that [ [ R i A, R n +1 A ] ] = 0, i = 0 , . . . , n + 1, and A ( C ( R i A, R j )) = 0 , i + j ≤ n + 1. First, note that b y direct computations one ca n prove the follo wing formulas [ [ RA, RB ] ]( ψ 1 , ψ 2 ) − R [ [ RA, B ] ]( ψ 1 , ψ 2 ) − R [ [ A, R B ] ]( ψ 1 , ψ 2 ) + R 2 [ [ A, B ] ]( ψ 1 , ψ 2 ) − [ R, R ] F N ( Aψ 1 , B ψ 2 ) − [ R, R ] F N ( B ψ 1 , Aψ 2 ) = 0 , (11) [ [ RA, B ] ]( ψ 1 , ψ 2 ) + [ [ A, RB ] ]( ψ 1 , ψ 2 ) − [ [ A, B ] ]( R ∗ ψ 1 , ψ 2 ) − [ [ A, B ] ]( ψ 1 , R ∗ ψ 2 ) − A ( C ( B , R )( ψ 1 , ψ 2 )) − B ( C ( A, R )( ψ 1 , ψ 2 )) = 0 , (12) C ( RA, R )( ψ 1 , ψ 2 ) + C ( A, R 2 )( ψ 1 , ψ 2 ) − C ( A, R )( R ∗ ψ 1 , ψ 2 ) − C ( A, R )( ψ 1 , R ∗ ψ 2 ) − R ∗ ( C ( A, R )( ψ 1 , ψ 2 )) = 0 . (13) Let us substitute R n A a nd R n − l A , l = 1 , . . . , n , for A and B in (12), r esp ectively . Then we get [ [ R n +1 A, R n − l A ] ] − R n − l A ( C ( R n A, R )) = 0 . (14) Now let us take R n − 1 A , R n − l A , l = 1 , . . . , n , and R 2 for A , B and R in (12) , resp ectively . Then w e have [ [ R n +1 A, R n − l A ] ] − R n − l A  C ( R n − 1 A, R 2 )  = 0 . (15) If we ha ve in (13) R n − 1 A for A w e ge t C ( R n A, R ) + C ( R n − 1 A, R 2 ) = 0 . (16) Therefore, ta king the sum o f (14) and (15) we obtain tha t [ [ R n +1 A, R n − l A ] ] = 0 for l = 1 , . . . , n . Let us substitute now R n − 1 A a nd R n A for A and B in (11), res pe c tively , and then R n A for A in (9). Thus we get [ [ R n +1 A, R n A ] ] = 0 a nd [ [ R n +1 A, R n +1 A ] ] = 0. In order to pr ove that A ( C ( R i A, R j )) = 0 for i + j ≤ n + 1 , one has to put B = R n − l A , l = 0 , . . . , n and ta ke R l +1 for R in (12).  F or subsequen t constr uctions we need the op era tor C ∗ ( A, R )( ψ , ϕ ) = − ℓ A, ψ ( Rϕ ) + ℓ R, ϕ ( Aψ ) + R ( ℓ A, ψ ( ϕ )) + A ( ℓ ∗ R, ϕ ( ψ ) − ℓ R ∗ , ψ ( ϕ )) defined by h C ( A, R )( ψ 1 , ψ 2 ) , ϕ i = h ψ 2 , C ∗ ( A, R )( ψ 1 , ϕ ) i . (17) 4 2. V aria tional P oisson–Nijenhuis structure s on evolution equa tions 2.1. Symmetries and cosymmetries. Cons ider a system of e volution equatio ns E = { F = u t − f ( x, t, u, u 1 , . . . , u k ) = 0 } , (18) where b oth u = ( u 1 , . . . , u m ) and f = ( f 1 , . . . , f m ) ar e vectors a nd u t = ∂ u/ ∂ t , u k = ∂ k u/∂ x k . F or simplicity , w e consider the case of one space v ariable x , though everything works in ge ne r al situa tio n a s well. Denote by F ( E ) the algebra o f smo o th functions on E . Recall tha t equation (18) E can be unders to o d as the space J ∞ ( π ) × R with the Cartan distribution generated by the fields D x and D t = ∂ /∂ t + З f . H ere t the co ordinate along R . In lo cal co ordinates , these fields are of the for m D x = ∂ ∂ x + m X j =1 X k ≥ 0 u j k +1 ∂ ∂ u j k , D t = ∂ ∂ t + m X j =1 X k ≥ 0 D k x ( f j ) ∂ ∂ u j k . A symmetry of the equation E is a π ∞ -vertical vector field on E that preserves the Cartan distr ibution. The set o f a ll symmetr ie s for ms a Lie algebra over R denoted b y sym( E ) and there is a one-to- one corr esp ondence b etw een sym( E ) a nd smo oth sections ϕ ∈ Γ( π ∗ ∞ ( π )) = κ ( E ) satisfying the equa tion ℓ E ( ϕ ) = 0 , (19) where ℓ E = D t − ℓ f is the linea rization op er ator of E . A c onservation law fo r the equation (18) is a horizontal 1-form η = X dx + T dt closed with resp e ct to ho rizontal de Rham differen tial ¯ d : ¯ Λ 1 ( E ) → ¯ Λ 2 ( E ), i.e ., such that D t ( X ) = D x ( T ) , where X , T ∈ F ( E ). A co nserv a tion law η is trivial if it is of the for m η = ¯ dh , h ∈ F ( E ). The space of equiv alence classes of conser v ation laws coincides with the first horizontal de Rham co homology group and is denoted b y ¯ H 1 ( E ). T o any conserv ation law η = X dx + T dt there co r resp ond its g enerating function ψ η = E ( η ) that satis fies the equation ℓ ∗ E ( ψ η ) = 0 . (20) Solutions of the last equa tion are called c osymmetries of e q uation E and the space of cosymmetries of E will b e denoted by sym ∗ ( E ). 2.2. In v arian t P oiss o n–Nijenhuis structures. Consider the mo dules κ and ˆ κ on the space J ∞ ( π ) × R of e x tended jets, i.e., we admit explicit dep endence of their elements on t . Since v ector fields on E act on κ and ˆ κ by Lie deriv atives, w e can give the following Definition 2. An op erator O acting from κ to κ (or from κ to ˆ κ , etc.) is called inv ar iant if L D t ◦ O = O ◦ L D t . Prop ositio n 3. A n op er ator A : ˆ κ → κ is invariant iff ℓ F ◦ A + A ◦ ℓ ∗ F = 0 . (21) An op er ator R : κ → κ is invariant iff ℓ F ◦ R − R ◦ ℓ F = 0 . (22) Pr o of. Indeed, from (2 ) one has for the action on κ L D t = L ∂ ∂ t + З f = ∂ ∂ t + З f − ℓ f = ℓ F , while from (3) for the action on ˆ κ we obtain L D t = L ∂ ∂ t + З f = ∂ ∂ t + З f + ℓ ∗ f = − ℓ ∗ F 5 and this pr ov es bo th (21) a nd (22).  Remark 2. F rom (22) we see that an in v ariant C -differential oper atos takes sym- metries of equation E to s y mmetries, i.e., is a r e cursion op er ator for symmetries o f E . On the other hand, a C - differential op erator A that e njoys (21) tak es co symmetries to symmetries. As it was shown in [7], if A is skew-adjoint and satisfies [ [ A, A ] ] = 0 it is a Hamiltonian structu re for E and a ll Hamiltonian structures can b e found in such a w ay . Definition 3. Let ( A, R ) b e a Poisson–Nijenhuis structure on J ∞ ( π ) × R . W e say that it is a Poisson–Nijenhuis structure on E if b oth A and R ar e inv a riant op erator s . Thu s, a Poisson–Nijenhuis o n E c o nsists of a Hamiltonian structur e A and a recursion op era tor which is a Nijenhuis ope rator compatible with A . 2.3. The ℓ ∗ - and ℓ -co v erings. W e shall now lo ok a t Poisson–Nijenhuis str uctures from a differen t p o in t of view. T o this end, consider the following extension of the equation E . Let us add to E new o dd v ar iable p = ( p 1 , . . . , p m ) that satisfies p t = − ℓ ∗ f ( p ) . (23) The system consisting of the initial equation E a nd equation (23) is ca lled the ℓ ∗ -c overing of E and is denoted by L ∗ E . The extended total deriv a tives on L ∗ E are ˜ D x = D x + m X j =1 X k ≥ 0 p j k +1 ∂ ∂ p j k , ˜ D t = D t − m X j =1 X k ≥ 0 ˜ D k x ( ˜ ℓ ∗ f ( p j )) ∂ ∂ p j k . Note that a ny C -differential op erato r O o n E can b e lifted to an o p erator ˜ O on L ∗ E by sup er s cribing tildes o ver co rresp onding total de r iv atives. Let A : ˆ κ → κ be a C -differential op erator o f the form k P i ≥ 0 a l ij D i x k . Consider a p -linear vector function H A = ( H 1 A , . . . , H m A ), H l A = X i,j a l ij p j i , a l ij ∈ F ( E ) . (24) Theorem 4 (see [7]) . A skew-adjoint op er ator A satisfies (21) iff ˜ ℓ E ( H A ) = 0 . (25 ) In the terminolo gy o f the cov ering theory [1 3], a vector function satisfying (25) is nothing but a shadow of s y mmetry o f E in the ℓ ∗ -cov ering or, mor e pr ecisely , H A is genera ting section of the shadow ˜ З H A = X k ≥ 0 ˜ D k x ( H j A ) ∂ ∂ u j k . In co or dinate-free terms, a s ha dow is a der iv ation of F ( E ) with v alues in the function algebra on L ∗ E that pr eserves the Cartan distribution. Lemma 5. L et A ∈ C Diff sk-ad ( ˆ κ , κ ) and ˜ З H A b e the c orr esp onding shadow. Then ther e exists a symmet ry of the e quation L ∗ E su ch that its r estriction to F ( E ) c oin- cides with ˜ З H A . Pr o of. Consider the v ector function α = ( α 1 , . . . , α m ) defined b y α = − 1 2 ˜ ℓ ∗ H A ( p ) and set ˜ З H A , ¯ H A = ˜ З H A + X k,j ˜ D k x ( α j ) ∂ ∂ p j k . (26) 6 It is ea sily chec ked that the field ˜ З H A , ¯ H A is the desired symmetry .  Consider now t wo oper ators A , B ∈ C Diff sk-ad ( ˆ κ , κ ) and using Lemma 5 define the Jac obi br acket of H A and H B by {H A , H B } = ˜ З H A , ¯ H A ( H B ) + ˜ З H B , ¯ H B ( H A ) (27) Remark 3. The plus sign in the r ight-hand side of equation (27) is due to the fact that b oth the fields ˜ З H A , ¯ H A , ˜ З H B , ¯ H B and the functions H B , H A are o dd. Remark 4. In more explicit ter ms the Jaco bi bracket ca n b e rewritten in the for m {H A , H B } = − ˜ ℓ H A ( H B ) − ˜ ℓ H B ( H A ) − A ( ˜ ℓ ∗ H B ( p )) + B ( ˜ ℓ ∗ H A ( p )) 2 . (28) Note that ther e exists a one-to-one cor r esp ondence b etw een C -differential op era- tors from the mo dule C Diff sk ew ( k ) ( ˆ κ , κ ) and functions on L ∗ E tha t are ( k − 1)-linear with resp ect to the o dd v aria bles. F or ∆ ∈ C Diff sk ew ( k ) ( ˆ κ , κ ) denote b y H ∆ the corres p o nding function. Then b y dire ct computations one can prove the following Theorem 6. L et H A and H B b e shadows of symmetries in the ℓ ∗ -c overing to which ther e c orr esp ond op er ators A, B ∈ C Diff sk-ad ( ˆ κ , κ ) . Then {H A , H B } = H [ [ A,B ] ] . (29) If ψ 1 and ψ 2 are sections of the ℓ ∗ E -cov ering such that ( dψ i )( C ) ⊂ ˜ C , i = 1 , 2, where C and ˜ C are the Cartan distributions o n E and L ∗ E , resp ectively , then equation (29) can b e r ewritten as follows ψ ∗ 1 ψ ∗ 2 {H A , H B } = [ [ A, B ] ]( ψ 1 , ψ 2 ) . W e now intro duce the ob ject dual to the ℓ ∗ -cov ering. Namely , c o nsider the extension of E b y new o dd v ar iables q = ( q 1 , . . . , q m ) that satisfy the equatio n q t = ℓ f ( q ) . (30) The system consisting o f the initial equation and equation (30) will b e called the ℓ -c overing o f E a nd denoted by LE . The extended total deriv a tives on LE are ˜ D x = D x + m X l =1 X k ≥ 0 q l k +1 ∂ ∂ q l k , ˜ D t = D t + m X l =1 X k ≥ 0 ˜ D k x ( ˜ ℓ f ( q l )) ∂ ∂ q l k . T o any C -differential op erato r R : κ → κ of the for m P i ≥ 0 a l ij D i x let us put into corres p o ndence a q -linear vector function N R = ( N 1 R , . . . , N m R ), where N l R = X i,j a l ij q j i , a l ij ∈ F ( E ) . Similar to Theorem 4 we hav e the following Theorem 7 (see [7, 6]) . An op er ator R satisfies r elation (22), i.e , is a r e cursion op er ator for the e quation E iff ˜ ℓ E ( N R ) = 0 , (31) i.e. , N R is a shado w of a symmetry in the ℓ - c overing. The corre sp onding deriv atio n is of the fo r m ˜ З N R = X k ≥ 0 ˜ D k x ( N j R ) ∂ ∂ u j k . In par a llel to Lemma 5 w e have the followin g auxilia ry result. 7 Lemma 8. L et R ∈ C Diff ( κ , κ ) and ˜ З N R b e the c orr esp onding shadow. Then t her e exists a symmet ry of the e quation LE su ch t hat its r estriction t o F ( E ) c oincides with ˜ З N R . Pr o of. Consider the v ector function β = ( β 1 , . . . , β m ) with β j = X k,l ∂ N j R ∂ u l k q l k and set ˜ З N R , ¯ N R = ˜ З N R + X k,j ˜ D k x ( β j ) ∂ ∂ q j k . (32) This is the symmetry we are lo oking for .  Using Lemma 8 we define the Jac o bi brack et o f vector functions N R and N S by {N R , N S } = ˜ З N R , ¯ N R ( N S ) + ˜ З N S , ¯ N S ( N R ) . (33) In explicit terms the Ja cobi brack et has the for m {N R , N S } = − ˜ ℓ N R ( N S ) − ˜ ℓ N S ( N R ) + R ( ˜ ℓ N S ( q )) + S ( ˜ ℓ N R ( q )) . (34) Denote b y N [ R,S ] F N the function o n LE bilinear with resp ect to the v a riables q and corresp onding to the op era tor [ R, S ] F N ∈ C Diff sk ew (2) ( κ , κ ). Then b y the dir e c t computations we obtain Theorem 9. L et N R and N S b e shadows of symmetries in the ℓ -c overing to which ther e c orr esp ond op er ators R and S ∈ C Diff ( κ , κ ) . Then {N R , N S } = N [ R,S ] F N . If ϕ 1 , ϕ 2 are sectio ns of the ℓ -cov ering preserv ing the Ca rtan distributions then ϕ ∗ 1 ϕ ∗ 2 {N R , N S } = [ R, S ] F N ( ϕ 1 , ϕ 2 ) . 2.4. Compatibility condition. W e shall now e xpress the compatibility conditio n of a Hamiltonian structure A and a recur sion oper ator R on equa tion (18) in simila r geometric terms. T o this end, r ecall that b oth LE and L ∗ E are fib ere d over the equation E and denote the corr esp onding fiber bundles b y τ : LE → E and τ ∗ : LE → E , r esp ectively . Consider the Whitney pro duct τ ⊗ τ ∗ : LE × E L ∗ E → E o f these bundles. One can extend the total deriv atives to the s pace LE × E L ∗ E by setting ˜ D x = D x + m X l =1 X k ≥ 0  q l k +1 ∂ ∂ q l k + p l k +1 ∂ ∂ p l k  , ˜ D t = D t + m X l =1 X k ≥ 0  ˜ D k x ( ˜ ℓ f ( q l )) ∂ ∂ q l k − ˜ D k x ( ˜ ℓ ∗ f ( p l )) ∂ ∂ p l k  . Thu s the equation LE × E L ∗ E amounts to E ex tended bo th by (23) and (30). Due to the natural pr o jections LE × E L ∗ E → L ∗ E and LE × E L ∗ E → LE , the vector fields ˜ З H A , ¯ H A and ˜ З N R , ¯ N R (see equalities (26) and (32)) may b e c onsidered as deriv ations fro m F ( L ∗ E ) and F ( LE ) to F ( LE × E L ∗ E ), res p ectively . Lemma 10 . Ther e exist symmetries ˜ З H A , ¯ H A ,α and ˜ З N R , ¯ N R ,ρ of e quation LE × E L ∗ E such that their r estrictions to the function algebr as F ( L ∗ E ) and F ( LE ) c oincide with ˜ З H A , ¯ H A and ˜ З N R , ¯ N R , r esp e ctively. 8 Pr o of. Let us set ˜ З H A , ¯ H A ,α = ˜ З H A , ¯ H A + X k,j ˜ D k x ( α j ) ∂ ∂ q j k , (35) where the vector function α = ( α 1 , . . . , α m ) is defined by α = ˜ ℓ H A ( q ). Let also consider the field ˜ З N R , ¯ N R ,ρ = ˜ З N R , ¯ N R + X k,j ˜ D k x ( ρ j ) ∂ ∂ p j k , (36) where ρ = ( ρ 1 , . . . , ρ m ) is defined b y ρ = − ˜ ℓ ∗ N R ( p ) − ˜ З q ( R ∗ )( p ) . By direct computations one ca n chec k that the fields (35) and (36) p ossess the needed pro pe rties.  Using Lemma 10 let us define the brack et {H A , N R } = ˜ З H A , ¯ H A ,α ( N R ) + ˜ З N R , ¯ N R ,ρ ( H A ) , or, in explicit terms, {H A , N R } = − ˜ ℓ N R ( H A ) − ˜ ℓ H A ( N R ) − A  ˜ ℓ ∗ N R ( p ) + ˜ З q ( R ∗ )( p )  + R ( ˜ ℓ H A ( q )) . (37) Denote by C ∗ A,R the bilinear with resp ect to the v aria bles p and q function o n LE × E L ∗ E c orresp onding to the C -differ e n tial op era tor C ∗ ( A, R ) : ˆ κ × κ → κ (see equality (17)). Then the following res ult holds: Theorem 11. L et H A b e a shadow in the ℓ ∗ -c overing to which ther e c orr esp onds an op er ator A ∈ C Diff sk-ad ( ˆ κ , κ ) and N R b e a shadow in the ℓ -c overing to which ther e c orr esp onds an op er ator R ∈ C Diff ( κ , κ ) . Then {H A , N R } = C ∗ A,R . F rom the ab ov e said we get the following Theorem 12 . L et a Hamiltonian op er ator A ∈ C Diff sk-ad ( ˆ κ , κ ) and a r e cursion op er ator R ∈ C Diff ( κ , κ ) define Poisson–Nijenhuis stru ct ur e on evolution e qua- tion E , while H A and N R b e t he c orr esp onding shadows in the ℓ ∗ - and ℓ -c overing over E , r esp e ctively. Then (i) {H A , H A } = 0 , (38) (ii) {N R , N R } = 0 , (39 ) (iii) {H A , N R } = 0 . (4 0) 3. V aria tional P oisson–Nijenhuis structures in general case Consider now the infinite pr olonga tio n E ⊂ J ∞ ( π ) of a genera l differe ntial equa - tion as a submanifold in the space J ∞ ( π ) of infinite jets o f some lo cally triv ia l bundle π : E → M . In loc a l co ordina tes E is given b y the system F j ( x 1 , . . . , x n , u 1 , . . . , u m , . . . , u j σ , . . . ) = 0 , j = 1 , . . . , r . W e assume that F = ( F 1 , . . . , F r ) ∈ P , wher e P is the mo dule of sections of some vector bundle ov er J ∞ ( π ). Co nsider the linear ization op er ator ℓ E : κ → P and its a djoint ℓ ∗ E : ˆ P → ˆ κ . Similar to the evolutionary case, we can construct ℓ - a nd ℓ ∗ -cov erings by ex tending E with ℓ E ( q ) = 0 and ℓ ∗ E ( p ) = 0. 9 F ollowing the scheme exp os ed in Section 2, we are lo o king for C -differential op erator s such that the diagrams (i) κ ℓ E / / R   P ¯ R   κ ℓ E / / P , (ii) ˆ P ℓ ∗ E / / A   ˆ κ ¯ A   κ ℓ E / / P (41) are comm utative. Literally cop ying the co nstruction o f Section 2, we can put into corres p o ndence to an op era tor R from the firs t diag r am in (41) a q -linea r func- tion N R = ( N 1 R , . . . , N m R ) on LE , while for A from the second diagr am we construct a p -linea r function H A = ( H 1 A , . . . , H m A ) o n L ∗ E . W e ag ain use the gener al result prov ed in [7]: Theorem 13. An op er at or R fits diagr am (i) in (41) iff ˜ ℓ E ( N R ) = 0 , (42) while A fits dia gr am (ii) iff ˜ ℓ E ( H A ) = 0 . (43 ) Among the o p erators A let us distinguish those ones that enjoy the prop er ty similar to s kew-adjoin tness. Namely , we shall cons ide r oper ators such that A ∗ = − ¯ A. (44) Remark 5. This pro p erty means that the op era tor ( A, ¯ A ) : ˆ P ⊕ ˆ κ → κ ⊕ P is skew-adjoin t. Note now that explicit expressio ns (28), (34) and (37) for Jaco bi br ack ets do not r ely on the fact that they were given for evolutionary eq uations. Using this observ ation we g ive the following Definition 4. Let E ⊂ J ∞ ( π ) b e a different ial equation. (1) A C -differential op era tor A : ˆ P → κ is called a Hamiltonian structur e on E if it fits the left diagram (41), equation (44 ) holds and {H A , H A } = 0, where the brack et is given by (28). Two Hamilto nia n structures A and B are said to be c omp atible if {H A , H B } = 0. (2) A C -differential oper ator R : κ → κ is calle d a Nijenhuis op er ator for the equation E if it fits the right diagram (41) a nd {N R , N R } = 0, where the brack et is given b y (34). (3) A pair of C -differential op erator s ( A, R ) is called a Poisson–Nij enhuis st r u c- tur e o n E if R is a Nijenhuis op erator, A is a Ha milto nian structur e such that A ∗ ◦ R ∗ = ¯ R ◦ A ∗ and {N R , H A } = 0, where the bra ck et is given by (37). Remark 6 . If ( A, R ) is a P oiss on–Nijenhuis s tructure on E then R is a recursion op erator for symmetries of E . Mor eov er, a Hamiltonian structure A , similar to the evolutionary case, deter mines a Poisson brack et on the gro up of conserv atio n laws of E . Namely , if ω 1 , ω 2 are conserv ation laws then we set { ω 1 , ω 2 } A = L d 0 ,n − 1 1 ω 1 ( ω 2 ) , where d 0 ,n − 1 1 : E 0 ,n − 1 1 → E 1 ,n − 1 1 is the differential in Vinog r adov’s C - sp ectral s e- quence, se e [17]. F or evolutionary equatio ns this differential coinc ide s with the Euler op er ator. The following result genera lises Theorem 2: Theorem 14 . I f ( A, R ) is a Poisson–Nijenhuis stru ctur e on E then R i ◦ A , i = 0 , 1 , 2 , . . . , is a family of p air-wise c omp atible Hamiltonian structu r es on E . 10 4. Concl uding remarks: nonlocal P oisson–Nijenhuis structures Strictly sp ea king all co nstructions exp os e d ab ove a re v a lid for lo c al Poisson– Nijenh uis structures. In rea lit y , the o per ators A or R o r b o th ar e n onlo c al , i.e ., contain terms like D − 1 x . F or example, reca ll the recur sion op er ator for the KdV equation. It seems that our approach can b e extended to structur es of this type. The gener al scheme is as follows. Let E b e a differential equation and τ : f LE → LE b e a general co vering over LE in the sense of [13]. Then s olutions of the equation ˜ ℓ E ( N ) = 0 linear with resp ect to o dd v a riables g ive rise to nonlo cal C -differe ntial recurs ion op erator s R N with nonlo calities corresp onding to no nlo cal v ariables defined b y τ . These solutions are shadows of symmetries in this cov ering. The hardest problem lies in definition of the J acobi bra ck et for suc h shadows. Nontriviality o f this prob- lem is illustrated b y observ ation g iven in [15]. A wa y to commute shadows can b e derived from the results of [9] but the constructions g iven there are amb iguo us. In [5] we describ ed a cano nic a l co nstruction for the J acobi bra ck et { . , . } of shad- ows of a general nature. Given this construction a nd taking into account the ab ov e exp osed results, we can define nonlo cal Nijenh uis op erator s R N as the ones satis - fying {N , N } = 0 . In a similar ma nner, we can consider cov erings over τ ∗ : g L ∗ E → L ∗ E and, s olving the equatio n ˜ ℓ E ( H ) = 0 , lo ok for nonloc a l Hamiltonian op er ators A H corres p o nding to shadows H . The Hamiltonianity condition is given by {H , H} = 0 . Finally , the compatibility condition for R N and A H are expr essed by {H , N } = 0 , where the brack et is considered in the Whitney pro duct o f τ and τ ∗ . A detailed theo r y of nonlo cal Poisson–Nijenhuis structures will b e given else- where. Remark 7. As it was demonstrated in [7] and [6], a very efficient wa y to co ns truct nonlo cal recursio n op er ators and Hamiltonian structures for evolution equations is the use o f nonlo cal vectors (for the ℓ ∗ -cov ering) and cov ectors (for the ℓ -cov ering). In par ticula r, this method, b y its nature, leads to the so-called we akly nonlo c al op er ators . When this pap er was almost finished, Maria Clara Nucci indicated to us the work [4] whe r e the c onstruction of nonlo ca l vectors was reinv ented. The author of [4] exploits the Lagrang ian structur e of the ℓ ∗ -cov ering, thoug h the reason for existence of nonlo ca l vectors is mo re general (it suffices to compare the cons truction with the one for nonlo cal c ovectors on the ℓ -covering). References [1] J.V. Beltr´ an and J. Mont erde, Poisson–Nijenhuis structur es and t he Vino gr adov br ack e t , Ann. Global A nal. Geom., 12 (1994), no. 1, 65–78. [2] A.V. Bo charo v, V.N. Chetv eriko v, S.V. Duzhin, et al. Sy mmetries and Conservation L aws for Differ ential Equations of Mathematic al Physics . Amer. Math. So c., Providence, RI, 1999. Edited and with a preface by I. Kr asil ′ shc hik and A. Vinogradov. [3] A. Cabras and A.M. 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M orosi, A ge ometrica l char acterization of inte gra ble Hamiltonian sy stems thr ough the the ory of Poisson–Nijenhuis manifolds , Uni v ersity of Mi lan, Quaderno, S 19 (1984), 20 p. [15] P .J. Olv er, J. Sanders, and J.P . W ang, Ghost symmetrie s , J. Nonli near Math. Phys. 9 (200 2) Suppl. 1, 164–172. [16] I. V aisman, A le ctur e on Poisson–Nijenhuis structur es, inte gr able syst e ms and foliations, F euil let ages et syst` emes int´ egr ables , (Mon tp ellier, 1995) , Progr. Math., Bir kh¨ auser Boston, Boston, M A, 145 (1997), 169–185. [17] A.M. Vinogradov , Cohomolo gic al Analysis of Partial Differ ential Equations and Se c ondary Calculus , AM S T ranslation of Mathematical Monographs series, 204 , 247 pp., 2001. V alentina Golovko, Lomonosov MSU, F acul ty of Physics, Dep ar tment of Ma thema t- ics, Vorob’evy Hills, Moscow 11990 2 Russia. E-mail addr ess : golovko@mccme. ru Iosif Krasil ′ shchik, Independent University of Mosco w, B. Vlasevsky 11, 119002 Moscow, Russ ia E-mail addr ess : josephk@diffie ty.ac.ru Alexander Verbovetsky, Independent University of Moscow, B. Vla sevsky 1 1, 1190 02 Moscow, Russ ia E-mail addr ess : verbovet@mccme .ru 12