On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators
aa r X i v : . [ n li n . S I] A ug Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2011), 081, 8 pages On a Recently Introduced Fifth-OrderBi-Hamiltonian Equationand Trivially Related Hamiltonian Operators
Daryoush TALATI and Refik TURHANDepartment of Engineering Physics, Ankara University 06100 Tando˘gan Ankara, Turkey
E-mail: [email protected], [email protected]
Received April 25, 2011, in final form August 18, 2011; Published online August 20, 2011http://dx.doi.org/10.3842/SIGMA.2011.081
Abstract.
We show that a recently introduced fifth-order bi-Hamiltonian equation witha differentially constrained arbitrary function by A. de Sole, V.G. Kac and M. Wakimoto isnot a new one but a higher symmetry of a third-order equation. We give an exhaustive listof cases of the arbitrary function in this equation, in each of which the associated equationis inequivalent to the equations in the remaining cases. The equations in each of the casesare linked to equations known in the literature by invertible transformations. It is shownthat the new Hamiltonian operator of order seven, using which the introduced equationis obtained, is trivially related to a known pair of fifth-order and third-order compatibleHamiltonian operators. Using the so-called trivial compositions of lower-order Hamiltonianoperators, we give nonlocal generalizations of some higher-order Hamiltonian operators.
Key words: bi-Hamiltonian structure; Hamiltonian operators
A hierarchy of evolution equations u t i = F i [ u ] are called bi-Hamiltonian integrable if by twocompatible Hamiltonian operators (HO’s) J and K there exist a Magri scheme u t i = F i [ u ] = Kδ u Z h i [ u ]d x = J δ u Z h i +1 [ u ]d x, i = 0 , , , , . . . with conserved densities h i [ u ] which are assumed to exist for all i , where δ u is the variationalderivative. HO’s are skew-adjoint operators satisfying Jacobi identity. Two HO’s are said tobe compatible if their arbitrary linear combinations are also HO. A direct byproduct of a bi-Hamiltonian structure determined by compatible pair of HO’s K and formally invertible J is therecursion operator R = KJ − which maps an equation F i to the next equation F i +1 = RF i inthe symmetry hierarchy u t i = F i , i = 0 , , , . . . . Square brackets, like F [ u ] denote differentialfunctions of x , u and x -derivatives of u up to some finite order.Because of their central role in the integrability theory, various operator classes are classifiedfor their HO content. Local and scalar HO’s of order 1 and 3 are classified by Gel’fand–Dorfman,Astashov–Vinogradov, Mokhov, Olver in [1, 2, 3, 4], and the 5th-order ones in [5] by Cooke.Recently in [6], A. de Sole, V.G. Kac and M. Wakimoto (deSKW) classified HO’s of order 7 upto 13 under equivalence up to contact transformations. They also gave a conjecture for HO’s oforder grater than 13.Contact transformations, i.e. the invertible transformations x = P ( y, v, v ′ ) and u = Q ( y, v, v ′ )leaving the equation w = 0 of the contact form w = d u − u ′ d x invariant, provide a natural classof equivalence transformations for HO’s because they preserve locality of HO’s and, moreover, D. Talati and R. Turhanthe order of the transformed HO remains equal to that of the original one [3]. In what followsthe HO’s and the other relevant objects like symmetries, conserved densities etc. which can betransformed into each other by contact transformations will be called equivalent .Here we consider an integrable 5th-order equation with an arbitrary function subject toa differential constraint, obtained by a pair of compatible newly obtained 7th-order and 3rd-order HO’s in [6] which is announced to be new. We show that the obtained equation is nota new one but a higher symmetry of a 3rd-order equation which is equivalent to one of threeparticular equations given in Mikhailov–Shabat–Sokolov (MSS) classification [7] according tothe arbitrary function it contains. We give a lower-order recursion operator and show that thenewly obtained 7th-order HO is trivially related to a pair of known 5th-order and 3rd-orderHO’s. By using successive trivial compositions of the compatible 5th- and 3rd-order HO’s weprovide some nonlocal generalizations of the HO’s of order grater than 7 obtained in deSKWclassification. The result of the deSKW classification of 7th-order HO’s in [6] is the following theorem.
Theorem 2.1 ([6]) . Any HO of order is equivalent either to a quasiconstant coefficient skew-adjoint differential operator or to the operator H (7 ,c ( x )) + b D , where H (7 ,c ( x )) = − B ∗ (3 ,c ( x )) · D · B (3 ,c ( x )) , B (3 ,c ( x )) = 1 u D u D + c ( x ) D − c ′ ( x ) , and c ′′′ ( x ) = 0 , b = const . These two types of HO’s are not equivalent. The HO’s H (7 ,c ( x )) + b D and H (7 ,c ( x )) + b D are equivalent if and only if α c ( x ) = c ( α x + β ) and α b = ± b for someconstants α = 0 and β . Such a HO is equivalent to a linear combination of the operators H ( j, if and only if c ( x ) = c = const , and one has H (7 ,c ) = H (7 , + 2 cH (5 , + c H (3 , . In the above theorem, quasiconstant refers to arbitrary functions of x only and ( ∗ ) denotes(formal) adjoint. H ( N, are HO’s defined for N = 2 n + 3 ≥ H ( N, = D · (cid:18) u D (cid:19) n · D, D n = d n d x n . (2.1)Using the compatible pair of HO’s H (7 ,c ( x )) with c ′′′ ( x ) = 0 and D , the following equation u t = H (7 ,c ( x )) δ u Z h d x = u (4) u − u ′ u ′′′ u − u ′′ u + 105 u ′′ u ′ u − u ′ u + (cid:18) c ( x ) − c ′′ ( x ) x + 12 c ′ ( x ) x (cid:19) u ′′ u − (cid:18) c ′ ( x ) x + 6 c ( x ) − c ′′ ( x ) x (cid:19) u ′ u + 5 c ′ ( x ) u ′ u − c ′′ ( x ) u − c ( x ) + 916 c ( x ) c ′′ ( x ) x − c ( x ) c ′ ( x ) x + 364 c ′′ ( x ) x − c ′ ( x ) c ′′ ( x ) x + 316 c ′ ( x ) x (cid:19) ′ = D δ u Z h d x (2.2)is introduced as the first member of a new integrable symmetry hierarchy. The associated initialconserved density which is a Casimir functional for the HO D , is h = − x u, n a Recently Introduced Fifth-Order Bi-Hamiltonian Equation 3and the second conserved density is given as h = a ( x ) u + 1 u (cid:18) c ( x ) − c ′′ ( x ) x + 14 c ′ ( x ) x (cid:19) − u ′ u , with a ( x ) = x (cid:0) − c ( x ) − c ( x ) c ′′ ( x ) x + 20 c ( x ) c ′ ( x ) x − c ′′ ( x ) x + 5 c ′ ( x ) c ′′ ( x ) x − c ′ ( x ) x (cid:1) . Discarding the trivial symmetry u t − = 0 = D δ u R h d x , the hierarchy constructed by theHO’s H (7 ,c ( x )) with c ′′′ ( x ) = 0 and D starts from the 5th-order equation (2.2). The nextsymmetry u t = H (7 ,c ( x )) δ u R h d x is of order 9. The fourth-order recursion operator R (4 ,c ( x )) = H (7 ,c ( x )) D − (2.3)gives the symmetries each being 4 orders higher than the one it succeeds. Interestingly, for none of the functions c ′′′ ( x ) = 0, the 5th-order equation (2.2) is equivalent toany of the 5th-order equations given in MSS classification [7] where the scalar equations of orderup to 5 are extensively classified with respect to existence of sufficiently many higher conserveddensities for the existence of a formal symmetry.From the recursion operator point of view, the order of the recursion operator R (4 ,c ( x )) in (2.3)does not match those of the 5th-order equations of Sawada–Kotera and Kaup–Kupershmidtboth of which have recursion operators of order 6. Note that the order of a recursion operatoris preserved by plenty of transformations.All these facts lead us to reconsider the equation (2.2) for its lower-order symmetries at first.The result is the following proposition. Proposition 3.1.
The th-order equation (2.2) with c ′′′ ( x ) = 0 is a higher symmetry of the rd-order equation u t = u ′′′ u − u ′′ u ′ u + 12 u ′ u − c ′ ( x ) = (cid:18) u − u ′′ − u − u ′ − c ( x ) (cid:19) ′ . (3.1)If we further search for symmetries of order higher than 5 we obtain a 7th-order symmetrybefore the one at order 9. All these intermediate symmetries suggest existence of a 2nd-orderrecursion operator. Indeed, it can be shown (but not needed in view of the results of nextsection) that Proposition 3.2.
The nd-order operator R (2 ,c ( x )) = (cid:0) H (7 ,c ( x )) D − (cid:1) = D u D u D − + c ( x ) + 32 c ′ ( x ) D − (3.2) is a hereditary recursion operator for equation (3.1) . Since the inverse of D is not uniquely defined, by applying R (2 ,c ( x )) on the r.h.s. of (3.1) weobtain the following linear combination u t = F + kF , (3.3) D. Talati and R. Turhanwhere F = u (4) u − u ′ u ′′′ u − u ′′ u + 105 u ′′ u ′ u − u ′ u + 52 c ( x ) u ′′ u − c ( x ) u ′ u + 5 c ′ ( x ) u ′ u − c ′′ ( x ) u − c ( x ) (cid:19) ′ (3.4)and F is the r.h.s. of equation (3.1). If the arbitrary constant k in (3.3) is chosen to be theconstant k = − c ′′ ( x ) x + 12 c ′ ( x ) x − c ( x ) , c ′′′ ( x ) = 0 , (3.5)then the linear combination of symmetries (3.3) becomes (upon taking all the derivatives) exactlythe equation (2.2). So, the F part in equation (2.2) is redundant. Rather than equation (2.2)(with or without the F part in it) we can concentrate on its 3rd-order symmetry (3.1) since itis a lower-order symmetry in the hierarchy which is not of Lie-point type.Having stepped down to order 3, in the following proposition, we divide the functions c ( x )which are constrained to satisfy c ′′′ ( x ) = 0, into three subclasses. For each subclass, equa-tions (3.1) are mutually inequivalent under contact transformations. We further relate equationsfrom each subclass with the ones given in literature. Proposition 3.3.
The equation (3.1) for i ) c ( x ) = c ( x ) where c ′ ( x ) = 0 , is a special case of the equation (4.1.34) in [7] , whosepotentiation is a symmetry in a Riemann hierarchy [8] , which by a further hodographtransformation becomes a linear equation with only a rd-order term; ii ) c ( x ) = c ( x ) where c ′′ ( x ) = 0 and c ′ ( x ) = 0 , is equivalent to u t = (cid:0) u − u ′′ − u − u ′ − x (cid:1) ′ (3.6) which is the equation (4.1.23) in [7] , through x (cid:18) c ′ ( x ) (cid:19) x, t (cid:18) c ′ ( x ) (cid:19) t ; iii ) c ( x ) = c ( x ) where c ′′′ ( x ) = 0 and c ′′ ( x ) = 0 , is equivalent to u t = (cid:18) u − u ′′ − u − u ′ + 32 x (cid:19) ′ (3.7) which is the equation (4.1.24) in [7] , through x (cid:18) − c ′′ ( x ) (cid:19) x − c ′ (0) c ′′ ( x ) , t (cid:18) − c ′′ ( x ) (cid:19) t. Let us note that under the contact transformations, each of the cases in the above propositionis an equivalence class. And the three cases exhaust all functions c ( x ) such that c ′′′ ( x ) = 0. Thesecases are the cases by which the HO’s of form H (7 ,c ( x )) in Theorem 2.1 are exhaustively dividedinto three equivalence class of HO’s too.n a Recently Introduced Fifth-Order Bi-Hamiltonian Equation 5 The recursion operator R (4 ,c ( x )) is a consequence of the compatible HO’s H (7 ,c ( x )) with c ′′′ ( x ) = 0and D as the ratio given in (2.3). Taking its square root we obtained the 2nd-order recursionoperator R (2 ,c ( x )) . Also considering the restored missing members of the symmetry hierarchyas above, it is natural to ask whether R (2 ,c ( x )) has a bi-Hamiltonian factorization too. Uponassuming the second HO to be again D , the answer reads R (2 ,c ( x )) = H (5 ,c ( x )) D − , where H (5 ,c ( x )) = D u D u D + c ( x ) D + 32 c ′ ( x ) D , c ′′′ ( x ) = 0 , (4.1)is the HO given in Remark 3.8 in [6] (denoted by H (5 , ,c ( x )) there). It is equivalent to a HOobtained by Cooke in [5].So, as a summary the hierarchy obtained by H (7 ,c ( x )) with c ′′′ ( x ) = 0 and D is nothing buta subset of the hierarchy that the compatible pair of HO’s H (5 ,c ( x )) with c ′′′ ( x ) = 0 and D givesrise to. Let us note, for the sake of completeness that the 3th-order equation (3.1) is obtainedby u t = (cid:18) u − u ′′ − u − u ′ − c ( x ) (cid:19) ′ = H (5 ,c ( x )) δ u Z (cid:18) − x u (cid:19) d x = D δ u Z (cid:18) u − (cid:18) D − ( c ( x )) + f ( x ) (cid:19) u (cid:19) d x with any function f ( x ) such that f ′′′ ( x ) = 0. The next symmetry in the hierarchy u t = H (5 ,c ( x )) δ u Z (cid:18) u − (cid:18) D − ( c ( x )) + f ( x ) (cid:19) u (cid:19) d x, is either the equation (3.4) for the choice of the constant f ′′ ( x ) = 0, or the linear combina-tion (2.2) for the choice f ′′ ( x ) = k where k is the constant given in (3.5). A HO K which is obtainable from other compatible HO’s K and J as K = K J − K , iscalled trivially related in [8] since, as seen here, the pair ( K , J ) gives a subset of the structureof symmetries/conserved quantities that the pair ( K , J ) gives. The sequence of operators K n obtained by trivially composing compatible HO’s K and J as K n = ( K J − ) n K , n = 1 , , . . . are proved to be HO in [9]. It can be further shown by constructing trivial compositions J + m X n =0 λ n (cid:0) K J − (cid:1) n K ! J − J + m X n =0 ¯ λ n (cid:0) K J − (cid:1) n K ! , (cid:0) K J − (cid:1) = 1 , of the successive partial sums m = 0 , , , . . . of linear combinations J + m P n =0 λ n ( K J − ) n K with arbitrary constants λ n , ¯ λ n and by induction on m that all K n , n = 0 , , , . . . are mutuallycompatible HO’s if so is the HO’s K and (formally) invertible J . D. Talati and R. TurhanAs an example, consider the HO’s H ( N, and H ( M, with N = 2 n + 3 ≥ M = 2 m + 3 ≥ n, m = 0 , , , , . . . , of type (2.1) whose Darboux form was obtained in [10]. Their trivialcompositions H (2 N − M, = H ( N, (cid:0) H ( M, (cid:1) − H ( N, = D · (cid:18) u D (cid:19) s · D, where s = 2 n − m = 0 , ± , ± , ± , . . . are all mutually compatible HO’s.Trivially related HO’s were first singled out in [8] where the considered equations were ofso-called hydrodynamic type. Equations of this type may possess plenty of Hamiltonian repre-sentations with more than two compatible HO’s. The 1st-order scalar equations in general [11],and the Riemann equation as a particular representative thereof [8], is an extreme case inthe number of compatible Hamiltonian formulations admitted. They possess infinitely manyHamiltonian structures [12] on the same set of variables not only with 1st-order HO’s, even ifthe trivially related HO’s are isolated [13].In deSKW classification [6], all the obtained higher-order HO’s as well as the 5th-order onesobtained by Cooke in [5] are given in a particular symmetric form which is not only beautiful butalso very suitable to observe possible trivial lower-order decompositions. Taking into accountthat D B (3 ,c ( x )) = H (5 ,c ( x )) , c ′′′ ( x ) = 0 , where the operator B (3 ,c ( x )) is given in Theorem 2.1 and H (5 ,c ( x )) in (4.1), we observe H (7 ,c ( x )) = H (5 ,c ( x )) D − H (5 ,c ( x )) , which is a trivial composition of the HO’s H (5 ,c ( x )) with c ′′′ ( x ) = 0 and D . A pair of which isknown to be compatible from the classification of 5th-order HO’s.The fact that H (7 ,c ( x )) is trivially related to the HO’s H (5 ,c ( x )) and D has the followingconsequence: Proposition 5.1.
The operator of order J (7 ,c ( x )) = H (7 ,c ( x )) + aH (5 ,c ( x )) + bD , where H (7 ,c ( x )) is given in Theorem and H (5 ,c ( x )) in (4.1) with arbitrary constants a and b isHO iff c ′′′ ( x ) = 0 . Note that the HO H (7 ,c ( x )) , being a trivial composition of HO’s H (5 ,c ( x )) and D , formsa compatible triple with the latter two. In general, compatibility of one of the HO’s, say J ofa compatible pair ( J , J ) with a third one J (not proportional to J or J ) does not implycompatibility of the pair ( J , J ). But this property holds automatically for trivially relatedHO’s.The HO H (7 ,c ( x )) is a strict trivial composition of the the pair H (5 ,c ( x )) and D in the followingsense: Proposition 5.2.
Let the functions c ( x ) and f ( x ) and a constant a be such that c ′′′ ( x ) = 0 , f ′′′ ( x ) = 0 and a = 0 . Then H (7 ,f ( x )) + aH (5 ,c ( x )) + bD is HO only if f ( x ) = c ( x ) , for any value of constant b . n a Recently Introduced Fifth-Order Bi-Hamiltonian Equation 7Let us proceed with the compatible pair H (7 ,c ( x )) and H (5 ,c ( x )) with c ′′′ ( x ) = 0 to obtain theHO of order 9 J (9 ,c ( x )) = H (7 ,c ( x )) (cid:0) H (5 ,c ( x )) (cid:1) − H (7 ,c ( x )) = (cid:0) H (5 ,c ( x )) D − (cid:1) H (5 ,c ( x )) = B ∗ (3 ,c ( x )) (cid:18) D u D u D + c ( x ) D + 12 c ′ ( x ) − c ′′ ( x ) D − (cid:19) B (3 ,c ( x )) , c ′′′ ( x ) = 0 , where B (3 ,c ( x )) is given in Theorem 2.1. For c ′′ ( x ) = 0 the HO J (9 ,c ( x )) is an operator witha nonlocal term D − and thus outside of the deSKW classification.As it was the case with H (7 ,c ( x )) , J (9 ,c ( x )) cannot give any new structure of symmetries orconservation laws other than the ones that the pair H (5 ,c ( x )) with c ′′′ ( x ) = 0 and D givesbecause eventually it is a trivial composition of the latter two which implies the following: Proposition 5.3.
The operator of order J (9 ,c ( x )) + k H (7 ,c ( x )) + k H (5 ,c ( x )) + k D with arbitrary constants k , k and k is HO iff c ′′′ ( x ) = 0 . The c ′′ ( x ) = 0 special case of Proposition 5.3 gives a local HO which agrees with the Theo-rem 3.11 of deSKW classification of HO’s of order 9 in [6].At order 11 there is the following trivial composition J (11 ,c ( x )) = H (9 ,c ( x )) (cid:0) H (7 ,c ( x )) (cid:1) − H (9 ,c ( x )) = (cid:0) H (5 ,c ( x )) D − (cid:1) H (5 ,c ( x )) = − P ∗ (11 ,c ( x )) DP (11 ,c ( x )) ,P (11 ,c ( x )) = (cid:18) u D u D + c ( x ) − c ′ ( x ) D − (cid:19)(cid:18) u D u D + c ( x ) D − c ′ ( x ) (cid:19) , c ′′′ ( x ) = 0 , which is a local operator only for the case c ′ ( x )=0. Trivial compositions (cid:0) H (5 ,c ( x )) D − (cid:1) n H (5 ,c ( x )) , n = 4 , , . . . , i.e. at orders 13 and higher are HO’s with nonlocal tail unless c ′ ( x ) = 0.Trivial compositions in the opposite direction (cid:0) D ( H (5 ,c ( x )) ) − (cid:1) m D , m = 1 , , , . . . arepseudodifferential operators of order 1 and lower which are intractable. Therefore, trivial com-positions of the local pair ( H (5 ,c ( x )) , D ) shows us conversely that in those cases with intractableHO pairs, there may exist an associated tractable, i.e. at least weakly nonlocal [14] or betterlocal, pair of HO’s which are trivial (de)compositions of the original intractable ones. We have shown that a recently introduced 5th-order integrable equation (2.2) is a linear combi-nation of a 5th-order equation with its 3rd-order symmetry and related through the recursionoperator R (2 ,c ( x )) in (3.2), to the 3rd-order equation (3.1) as a higher symmetry. The equa-tion (3.1), depending on the form of arbitrary function c ( x ) in it, is equivalent to three wellknown equations. In other words, equation (2.2) is a higher symmetry of equation (3.1) whichis, in a sense, a compact representation of three well known equations in a single expression.The fact that none of the equations (2.2) is in the MSS list of 5th-order integrable equationsis understandable since, as noted, those 5th-order equations which are higher symmetries oflower-order equations are omitted in the MSS list of 5th-order equations. Correspondence ofeach case in Proposition 3.3 with only one equation in MSS list is in agreement with the factthat the cases in Proposition 3.3 and the equations in the MSS classification are both dividedaccording to the equivalence under contact transformations.Integrable third-order scalar evolution equations were extensively classified using variousdefinitions of integrability, see e.g. [7, 15] and the references therein. As also noted in [7], equa-tions (3.6) and (3.7) are related to the KdV and pKdV equations respectively via potentiation D. Talati and R. Turhanand hodograph transformation which is not uniquely invertible and thus does not belong to theclass of contact transformations. But if the class of contact transformations is extended, theultimate 3rd-order equation is the KdV equation from which all the equations considered herecan be derived [16].On the side of HO’s, we have explained by using trivial (de)compositions of HO’s that the5th-order equation (2.2) obtained by the compatible pair ( H (7 ,c ( x )) , D ) in deSKW classificationis a higher symmetry in a finer hierarchy constructed by the compatible pair ( H (5 ,c ( x )) , D ),a trivial composition of which is the HO H (7 ,c ( x )) . Moreover, as a consequence of using trivialcompositions of lower-order HO’s we have obtained nonlocal generalizations of some of thehigher-order HO’s obtained in deSKW classification. Acknowledgements
R.T. thanks Professors M. G¨urses and A. Karasu for stimulating discussions; and ProfessorV.G. Kac for comments and clarification of the parametric nature of their results, removinga misinterpretation with incorrect consequences. D.T. is supported by T ¨UB˙ITAK PhD Fellow-ship for Foreign Citizens.
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