On recursion operators for elliptic models

Transcription

1 On recursion operators for elliptic models D K Demskoi and V V Sokolov 2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia demskoi@maths.unsw.edu.au 2 Landau Institute for Theoretical Physics, Kosygina 2, 9334, Moscow, Russia Abstract. New quasilocal recursion and Hamiltonian operators for the Krichever-Novikov and the Landau- Lifshitz equations are found. It is shown that the associative algebra of quasilocal recursion operators for these models is generated by a couple of operators related by an elliptic curve equation. A theoretical eplanation of this fact for the Landau-Lifshitz equation is given in terms of multiplicators of the corresponding La structure. AMS classification scheme numbers: 7B80, 7B63, 32L8, 4H70 Submitted to: Nonlinearity Introduction One of the main algebraic structures related to + -dimensional integrable PDE of the form u t = F u, u,..., u n, n 2, u i = i u, t 0. is an infinite hierarchy of commuting flows or, the same, generalized symmetries [] u ti = G i u,..., u mi. 0.2 We identify the symmetry 0.2 with its right hand side G i. The symmetry G satisfies the equation D t G F G = 0, 0.3 D t stands for the derivation in virtue of 0. and F denotes the Fréchet derivative of F : F = n i=0 i F u i D i. The dual objects for symmetries are cosymmetries which satisfy the equation D t g + F t g = 0, F t is the differential operator adjoint to F. The product g G of any cosymmetry g and symmetry G is a total -derivative. It is well known [] that for any conserved density ρ the variational derivative δρ/δu is a cosymmetry.

2 On recursion operators for elliptic models 2 The simplest symmetry for any equation 0. is u. The usual way to get other symmetries is to act to u by a recursion operator R. By definition, the recursion operator is a ratio of two differential operators that satisfies the identity [D t F, R] = R t [F, R] = It follows from 0.3 and 0.4 that for any symmetry G the epression RG is a symmetry as well. Most of known recursion operators have the following special form R = R + k i= G i D g i, 0.5 R is a differential operator, G i and g i are some fied symmetries and cosymmetries common for all members of the hierarchy. For all known eamples the cosymmetries g i are variational derivatives of conserved densities. Applying such operator to any symmetry we get a local epression, i.e. a function of finite number of variables u, u,... u i,... since the product of any symmetry and cosymmetry belongs to Im D. Moreover, a different choice of integration constants gives rise to an additional linear combination of the symmetries G,..., G k. Probably for the first time ansatz 0.5 was used in [2]. We call recursion operators 0.5 quasilocal. Most of known integrable equations 0. can be written in a Hamiltonian form δρ u t = H, δu ρ is a conserved density and H is a Hamiltonian operator. It is known that this operator satisfies the equation D t F H = HD t + F t, 0.6 which means that H takes cosymmetries to symmetries. Besides 0.6 the Hamiltonian operator should satisfy certain identities see for eample, [] equivalent to the skew-symmetricity and the Jacobi identity for the corresponding Poisson bracket. It is easy to see that the ratio H 2 H of any two Hamiltonian operators is a recursion operator. As the rule, the Hamiltonian operators are local i.e. differential or quasilocal operators. The latter means that H = H + m i= G i D Ḡi, 0.7 H is a differential operator and G i, Ḡi are fied symmetries. It is clear that acting by the operator 0.7 on any cosymmetry, we get a local symmetry. For eample, for the Korteweg-de Vries equation u t = u + 6 u u the simplest recursion operator see [3] R = D 2 + 4u + 2u D 0.8 is quasilocal with k =, G = 2u, and g =. This operator is the ratio of two local Hamiltonian operators H = D, H 2 = D 3 + 4uD + 2u. For the systematic investigation of quasilocal weakly nonlocal in terminology of [4] operators related to the Korteweg-de Vries and the nonlinear Schrödinger equations see [4, 5].

3 On recursion operators for elliptic models 3 It is possible to prove that for the Korteweg-de Vries equation the associative algebra A of all quasilocal recursion operators is generated by operator 0.8. In other words, this algebra is isomorphic to the algebra of all polynomials in one variable. Our main observation is that it is not true for such integrable models as the Krichever-Novikov and the Landau-Lifshitz equations. These equations play a role of the universal models for the classes of KdV-type and NLS-type equations. It appears that other integrable equations from these classes are limiting cases or can be linked to these models by differential substitutions [6, 7]. It turns out that for these models, known to be elliptic, the algebra A is isomorphic to the coordinate ring of the elliptic curve. Since the algebra A is defined in terms of the equation 0. only, this is an invariant way to associate a proper algebraic curve with any integrable equation and, in particular, to give a rigorous intrinsic definition of elliptic models. This paper is organized as follows. In Section, we consider the Krichever-Novikov equation [8], which is the simplest known one-field elliptic model. We show that there eist two quasilocal recursion operators of orders 4 and 6 related by the equation of elliptic curve. These recursion operators are ratios of the corresponding quasilocal Hamiltonian operators. The simplest quasilocal Hamiltonian operator of order H 0 = u D u for the Krichever-Novikov equation was found in [2]. From the results of this paper it follows that this equation has also a Hamiltonian operator of order 3. In Section we present one more quasilocal Hamiltonian operator of the fifth order for the hierarchy of the Krichever-Novikov equation. It seems to be interesting to investigate this multi-hamiltonian structure for the Krichever-Novikov equation in frames of the bi-hamiltonian approach [9]. In Section 2 we obtain similar results for the Landau-Lifshitz equation. It turns out that for this model there eist quasilocal recursion operators of orders 2 and 3 related by an elliptic curve equation. The corresponding quasilocal Hamiltonian operators are also found. In Sections,2 we have used the direct way to construct recursion and Hamiltonian operators based on the ansatzes 0.5 and 0.7. On the other hand there are several schemes that use L A-pairs for this purpose. For eample, there is a construction related to the squared eigenfunctions of the La operator L [0]-[2]. An alternative approach is based on the eplicit formulas for the A- operators see [3, 4, 5]. Some version of this approach has been suggested in [6]. In Section 3 we generalize the main idea of this work to the case of the Landau-Lifshitz equation. As a result, a deep relationship between the algebra A of quasilocal recursion operators and the algebra of multiplicators of the La structure becomes evident. In particular, the eistence of two recursion operators related by elliptic curve equation follows from a similar property for multiplicators.. The Krichever-Novikov equation. The Krichever-Novikov equation [8, 7] can be written in the form u t = u 3 u 2 + P u, P V = u u Denote by G the right hand side of.9. The fifth order symmetry of.9 is given by u 2 3 G 2 = u 5 5 u 4u u 2 u 2 u 3 u 2 2 u u 4 2 u P u 3 u P u2 2 u P u 2 5 P 2 u 8 u u P.

4 On recursion operators for elliptic models 4 The simplest three conserved densities of.9 are ρ = u u 2 ρ 3 = u2 4 u u3 3 u 3 P 3 u 2, ρ 2 = u u 2 9 u 2 3u u P u2 3 u P u3 2 u P u2 2 u 2 + P 3 27 u 6 4 P P 27 u 2 3 u u P u2 2 u 4 + P 2 8 u P, u 6 2 u u 4 2P 36 u 6 7 P 2 27 u P IV u 2. In the paper [2] the forth order quasilocal recursion operator of the form R = D 4 + a D 3 + a 2 D 2 + a 3 D + a 4 + G D δρ was found. Here the coefficients a i are given by a = 4 u 2, a 2 = 6 u2 2 u u 2 2 u 3 4 u 3 P, u 2 a 3 = 2 u u 3u 2 u u 2 6 u3 2 u 3 + 4P u 2 u 3 2 P, 3 u δu + u D δρ 2 δu, P 2 u2 2 u 6 a 4 = u 5 2 u2 3 u u u 3u 2 2 u 3 4 u 4u 2 u 2 3 u4 2 u P 2 9 u P u2 2 u P 8 3 P u 2 u 2. The following statement can be verified directly. Theorem. There eists one more quasilocal recursion operator for.9 of the form R 2 = D 6 + b D 5 + b 2 D 4 + b 3 D 3 + b 4 D 2 + b 5 D + b 6 2 u D δρ 3 δu +G D δρ 2 δu + G 2D δρ δu,

5 On recursion operators for elliptic models 5 b = 6 u 2 u, b 2 = 9 u 3 u 2 P u u2 2 u 2, b 3 = u u 3u 2 u u 2 + 4P u 2 u 3 57 u3 2 u 3 3 P, u b 4 = 4 u 5 u + 38 u 4u P 2 u 4 u 2 + 2P u 2 u 2 P, b 5 = 2 u u 4u 3 u u P u3 2 u P u 3 u P u 4 u P u 2u 3 u u2 3 u u4 2 u 4 55 u 3u 2 2 u P u 3 u 3 44P u2 2 u 4 P P u 3 04 u 2u 2 3 u 3 70 u 4u 2 2 u u3 2u 3 u 4 22P u2 2 u 3 + 2P u 2 6 u 3 P 2 u 2 u 5 08 u5 2 u 5, + 4 u 5u 2 u 2 b 6 = u 7 6 u 2u 6 u u P 2 u2 2 u 6 95 u2 3u 2 2 u 4 + 6P u2 3 u P u4 2 u P P u 2 u u 4u 3 u 2 u P u 4u 2 u 4 72 u6 2 u P u P u2 2 u P u 4 u P u3 2 u 4 0 u2 4 u u3 3 u P u 3u 2 2 u P IV u P u 3u 2 u P u 5 u P u 3 4 u 3 P 2 u 3 u 5 89 u 4u 3 2 u u 3u 4 2 u 5 3 u 5u 3 u u 5u 2 2 u 3 7 P 2 9 u 2 8 P 3 27 u 6 4 P P 9 u 2. The operators R and R 2 are related by the following elliptic curve R 2 2 = R 3 φr θ,.0 φ = 6 P 2 2P P + 2P IV P, 27 θ = P P 2 P IV + P P P + 2P IV P P P P 2. Remark. The relation.0 is understood as an identity in the non-commutative field of pseudodifferential series [8] of the form A = a m D m + a m D m + + a 0 + a D Here a i are local functions and the multiplication is defined by + a 2 D 2 +. k, m Z and ad k bd m = a bd m+k + C kd bd k+m + C 2 kd 2 bd k+m 2 +, C j n = nn n 2 n j + j!. Remark 2. It is easy to verify that φ and θ are constants for any polynomial P u, deg P 4. Under Möbius transformations of the form u αu + β γu + δ

6 On recursion operators for elliptic models 6 in equation.9 the polynomial P u changes according to the same rule as in the differential ω = du P u. The epressions φ and θ are invariants with respect to the Möbius group action. Remark 3. Of course, the ratio R 3 = R 2 R satisfies equation 0.4. It belongs to the noncommutative field of differential operator fractions [9]. Any element of this field can be written in the form P P2 for some differential operators P i. So, according to our definition, R 3 is a recursion operator of order 2. However, this operator is not quasilocal and it is unclear how to apply it even to the simplest commuting flow u. The recursion operators presented above appear to be ratios R = H H0, R 2 = H 2 H0 of the following quasilocal Hamiltonian operators H = 2 u2 D 3 + D 3 u 2 + 2u u 9 2 u2 2 3 P D + D 2u u 9 2 u2 2 3 P +G D G + u D G 2 + G 2 D u, H 2 = 2 u2 D 5 + D 5 u 2 + 3u u 9 2 u2 P D 3 + D 3 3u u 9 2 u2 P +hd + D h + G D h = u u 9u u u2 2 3 G 2 + G 2 D u u G + u D G 3 + G 3 D u, 5P 39u 2 + u2 u 2 5P 9u P 2 3 u 2 + u 2 P, and G 3 = R G = R 2 u is the seventh order symmetry of.9: u 5 G 3 = u 7 7 u 2u 6 7 u 6 u 2 2P + 2u 3 u 27u u 2 u 3 u 2 2P u u 4 3 u 2 2P u 5u 2 u u u u u 3 22P 47u u u 4 u P u 2 u 2 u P u2 2 u 4 u 3 35 u 3 8 u 4 2P u 4 P u u u u 5 P 203 u u 3 P + 49 u u 5 6P u 4 5P 2 7 u 2 9 u 3 2P u 4 5P P + 7 P 3 54 u P P + 7 u 9 P u 3 7 P 2. 8 u u 2 4 u 4 2. The Landau-Lifshitz equation. In this section we consider the Landau-Lifshitz equation written in the form u t2 = u + 2 u 2 P u + 2 P u v t2 = v + 2 v 2 P v 2 P v, 2. = u v, and P is an arbitrary fourth degree polynomial. The usual vectorial form of Landau-Lifshitz equation gives rise to a system of the form 2. after the stereographic projection see Section 3 for details.

7 On recursion operators for elliptic models 7 The third order symmetry of 2. is given by u t3 v t3 = u 6u u + 6u u P v 3u P v 6u 2 P v, = v + 6v v + 6v v P v 3v P v 6v 2 P v. In case of the system of evolution equation 2. the symmetries and cosymmetries can be treated as twodimensional vectors. We introduce the following notation for symmetries: G = u, v t, G 2 = u t2, v t2 t, G 3 = u t3, v t3 t. 2.2 Thus G ij is j-th component of the symmetry G i. The simplest cosymmetries are given by g = 2 v, u t, g 2 = 2 v t2, u t2 t, g 3 = 2 v t3, u t3 t. 2.3 These cosymmetries are variational derivatives δρ δu, δρ δv t of the following conserved densities ρ = 2 u + v, ρ 2 = 2 u v P v 2 P v 2 P v, ρ 3 = 2 2 u v u v + 3 u v u + v 2 u 4 2 P u + P u 3P u. The coefficients of operators F, R, and H from 0.4, 0.6 are matrices. For the quasilocal recursion operators the non-local terms can be written as AD B t, A and B are k 2 matrices whose columns are symmetries and cosymmetries, correspondingly. For quasilocal Hamiltonian operators the columns are symmetries for both A and B. Theorem 2. Equation 2. possesses the following quasilocal recursion operators: R 0 G G R = 2 2 D 0 R22 G 2 G g2 g g g and R 2 = R 2 R 2 2 R 2 2 R 2 22 G G G 3 G 2 G 22 G 32 D g 3 g 32 g 2 g 22 g g 2, 2.5 G ij and g ij are defined by 2.2, 2.3, and R = D 2 4u D u v + 3u 2u 3 P v 4P v 2 2P v, R22 = D 2 + 4v D v u + 3v + 2v 3 P v 4P v 2 2P v, R 2 = D 3 6u D u 2 P v 6u 2 P v 3P v D + P vu 2u +4 3 u 6P v 6u 2 v 2 u v + 2 2v u 2u v + 9P vu + 8u u, R2 2 = 2 2 P u u 2 D 4 3 P uu v + u 2 v u u P u 2u, R2 2 = 2 2 P v vd P vu v vu 2 + v 3 2 v P v 2v, R22 2 = D 3 6v D P v 3v 2 6v + 3P v + 2 P vd 4 3 v 6v 2 + v u + u v P v 9v v u v + v u 3 2 v P u +2 3 v P v + P u + v P v 2v. Operators R and R 2 are related by the following elliptic curve equation R 2 2 R 3 ϕr ϑe = 0, 2.6

8 On recursion operators for elliptic models 8 E stands for the unity matri, and ϕ = 2 4 P v P u P v P vp u P vp u 3P vp v 2 2P vp v + P vp u + 3P v 2 P vp v 2 P v 2, ϑ = 6 6 P v P v P u P v P u P vp u + 3P vp v + 4 P vp u 2 3P v 2 P u + 6P v P up v + 2P v 2 + 4P vp v P v P u + 2 P v P v P vp v + 6 P vp u + 6 P v 2 P v + 08 P v P v 2 P u P v P vp u P vp u + 4P vp vp v + 2P v 3. It is easy to verify that ϕ and ϑ are constants for any polynomial P, if deg P 4. They are invariants with respect to the Möbius group action just as in the case of the Krichever-Novikov equation see Remark 2. Remark 4. Possibly the odd recursion operator found in [20] is a rational function of recursion operators from Theorem 2. It turns out that the above recursion operators are ratios R = H H0, R 2 = H 2 H0 of the following quasilocal Hamiltonian operators 0 H 0 = 2, 0 2 H = 0 H 2 H 2 0 H 2 = H 2 H 2 2 H 2 2 H 2 22 G G 2 2 D G 2 G G2 G 22, 22 G G 2 G G G 3 G 2 G 22 G 32 D G 3 G 32 G 2 G 22 G G 2 H2 = 2 D2 4 v D 2 v + 2v2 + 6u v P v P v 4P v, H2 = 2 D2 4 u D 2 u 6u v 2u 2 + P v P v + 4P v, H 2 = 2u 2 P ud + 2u u P uu, H 2 22 = 2v 2 P vd + 2v v P vv, H2 2 = 2 D3 6 v D2 + 2u v + 6 v 2 v P v P v P v P v D + v 2 v + 6v v + 8v u + 4u v + 4u + v 3P v 4u v + P v3u + 6v, H2 2 = 2 D3 + 6 u D2 + 2u v + 6 u 2 + u P v P v P v D +3v P u + 2 u u P v + 6u u + 8u v + 4v u 6P vu +6u v u + v 2P vu + P uv.,

9 On recursion operators for elliptic models 9 3. Recursion operators and multiplicators. The original vector form of Landau-Lifshitz reads as follows U t = U U + U JU. 3.7 Here U = u, u 2, u 3, U =, symbol stands for the vector product, and J = diagp, q, r is an arbitrary constant diagonal matri. The usual way to represent equation 3.7 as a two-component system is to make use of the stereographic projection. The transformation U = uv, i + iuv, u + v, i 2 =, 3.8 coincides with the stereographic projection if we set v = /ū. This transformation takes equation 3.7 into the system of the form 2.: u τ = u + 2u 2 P u + 2 P u, v τ = v + 2v 2 P v 2 P v, 3.9 t = iτ, P u = 4 p qu4 2 p + q 2ru2 + p q. 4 Here and below the epression A, B denotes the standard Euclidean scalar product of the vectors A and B. Note that for equation 3.9 curve 2.6 has the form R 2 2 = R + 2p q r 3 E R + 2q p r 3 E R + 2r p q 3 Recall the algebraic structure lying behind the elliptic La pair [2] see also [22] for the Landau-Lifshitz equation. Let e = , e 2 = , e 3 = be the standard basis in the Lie algebra so3. The La operator L for 3.7 is given by L = D E. u i E i, 3.20 j= E = λ e pλ2, E 2 = λ e 2 qλ2, E 3 = λ e 3 rλ2. The operators A i defining the La representations for the commuting flows L ti = [A i, L] 3.2 U ti = H i U, U,... from the Landau-Lifshitz hierarchy belong to the Lie algebra G generated by E, E 2, E 3. Lemma. The algebra G is spanned by λ 2i E, λ 2i E 2, λ 2i E 3, Ē, λ2i Ē2, λ2i λ 2i Ē3, i = 0,, 2,..., 3.22

10 On recursion operators for elliptic models 0 Ē = λ 2 e qλ 2 rλ 2, Ē 2 = λ 2 e 2 pλ 2 rλ 2, Ē 3 = λ 2 e 3 pλ 2 qλ 2. Our main observation is that the recursion operators are in one-to one correspondence with the multiplicators see [23, 24] of the algebra G. Definition. A scalar function µλ is called the multiplicator for the algebra G if µλ G G. The order of pole of µλ at λ = 0 is called the order of the multiplicator. It is easy to prove the following Lemma 2. The set of all multiplicators for G coincides with the polynomial ring generated by, µ λ = pλ λ 2, µ 2 qλ 2λ = 2 rλ 2 λ 3. It is clear that µ 2 2 = µ pµ qµ r. Thus the ring of multiplicators is isomorphic to the coordinate ring of an elliptic curve. The following construction establishes a correspondence between multiplicators and recursion operators. Let µ be a multiplicator of order k > 0. To find a relation between operators A n and A n+k we use the following anzats A n+k = µa n + R n, R n G, ord R n < k. Substituting this into 3.2 and taking into account 3.20, we get j= H n+k j E j = µ j= H n j E j + [ j= u j E j, R n ] dr n d Both sides of this relation belong to G. Equating in 3.23 the coefficients of basis elements 3.22, we find step by step unknown coefficients of R n and eventually an epression for H n+k j in terms of H n j i.e. a recursion operator of order k. For the simplest multiplicator µ = λ 2, we have R n = M j E j + j= F j Ē j. It is easy to verify that 3.23 is equivalent to the following identities: j=. H n F U = 0, 2. F + U M = 0, H n+2 = M + F JU, M = M, M 2, M 3, F = F, F 2, F 3. It follows from the first identity and the condition U = that F = U H n + f U To find f we note that the second identity implies U, F = 0. Substituting the epression 3.25 to this relation, we get f = D U, H n U.

11 On recursion operators for elliptic models The second identity can be rewritten as M = U F + mu = UU, H n H n + fu U + mu for some function m. To find m, we substitute into the third identity the latter epression for M and take the scalar product by U of both sides of the relation thus obtained. Taking into account that H n+2, U = 0, we get m = D JU, H n U, H n. Now the third identity produces the following recursion operator for equation 3.7: H n+2 = U, H n U H n + U, H n U U + U JUD U U, U H n U U + U, JUH n U, H n JU, H n. U, U H n U D 3.26 The recursion operator 3.26, along with the bi-hamiltonian structure of the Landau-Lifshitz equation, was originally discovered in [25], as eplicit formulas for higher symmetries were given earlier by Fuchssteiner [26]. For the second multiplicator µ 2 we set R n = K j E j + j= M j Ē j + λ 2 j= Upon the substitution of 3.27 to 3.23 we get the following identities F j E j j=. H n = F U, 2. F = M U, 3. M JH n = K U, 4. H n+3 = K + M JU The first two relations in 3.28 are analogous to 3.24, and therefore F = U H n + fu, f = D U, H n U, It follows from that M = U, H n U H n + fu U + mu. K = U M JH n + κu = U, H n U U U JH n U H n U U, H n U U, U U + U f + mu U + κu. Functions m and κ can be found from the conditions which yield U, M JH n = 0, U, H n+3 = 0, m = D U, JH n U, H n, κ = D U, U JH n U, H n U 2 U, Hn U U, JU + U, U +U, H n JU 2 U, U U, JUD U, H n U.

12 On recursion operators for elliptic models 2 Thus the second recursion operator reads as H n+3 = H n U + H n U + H n, UU U U, H n U U +H n U, H n, UU JU + H n, UU U H n JU + U, H n JUU U U 2U, H n U U, U U U JH n U JH n +U, U JH n U + JH n, UU U U, H n U U 2U, H n U U U U + U JUD H n, U JH n, U U UU, U 3 2 U U, JU D U, H n U +U D U, U JH n U, H n U 2 U, Hn U U, JU 2 U, Hn U U, U + H n JU, U One can verify that under transformation 3.8 the operators defined by formulas 3.26 and 3.29 become 3 p + q + re R and R 2 correspondingly. Acknowledgments The authors are grateful to Professors I.M. Krichever and W.K. Schief for useful discussions. The research was partially supported by: RFBR grant , NSh grants and References [] P J Olver, Applications of Lie groups to differential equations, Second ed., Springer-Verlag, New York, 993. [2] V V Sokolov, Hamiltonian property of the Krichever-Novikov equation, Sov. Math. Dokl., 30, 44 47, 984. [3] P J Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 8, 22-25, 977. [4] A Ya Maltsev, S P Novikov, On the local systems Hamiltonian in the weakly non-local Poisson brackets, Physica D, 56, 53 80, 200. [5] B Enriquez, A Orlov, V Rubtsov, Higher Hamiltonian structures sl2 case, Sov. JETP Lett., 588, , 993. [6] A V Mikhailov, V V Sokolov and A B Shabat, The symmetry approach to classification of integrable equations, in What is Integrability?, Ed. V E Zakharov, Springer series in Nonlinear Dynamics, 5 84, 99. [7] V E Adler, A B Shabat and R I Yamilov, Symmetry approach to the integrability problem, Theor. Math. Phys., 253, , [8] I M Krichever, S P Novikov, Holomorphic bundles over algebraic curves and non-linear equations, Rus. Math. Surveys, 35, 53 79, 980. [9] F Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 9, 56 62, 978. [0] A S Fokas and R L Anderson, On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems, J. Math. Phys., 23, , 982. [] A S Fokas, Symmetries and integrability, Stud. Appl. Math., 77, , 987. [2] A P Fordy and J Gibbons, Factorization of operators. II., J. Math. Phys, 22, 70 75, 98. [3] W Symes, Relations among generalized Korteweg-de Vries systems, J. Math. Phys., 20, , 979. [4] M Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equation, Inventiones Math., 50, , 979. [5] M Antonowicz and A P Fordy, in Nonlinear Evolution Equations and Dynamical Systems NEEDS 87, 45 60, edited by J. Leon, World Scientific, Singapore, 988. [6] M Gurses, A Karasu, and V V Sokolov, On construction of recursion operator from La representation, J. Math. Phys., 40, , 999. [7] S I Svinolupov, V V Sokolov, Evolution equations with nontrivial conservation laws, Funct. Anal. Appl., 64, 86 87, 982. [8] I M Gel fand, L A Dikii, Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl., 0, 3 29, 976. [9] O Ore, Theory of non-commutative polynomials, Ann. of Math., 34, , 933. [20] A B Yanovski, Recursion operators and bi-hamiltonian formulations of the Landau Lifshitz equation hierarchies, J. Phys. A., 39, , [2] E K Sklyanin, On complete integrability of the Landau-Lifshitz equation. LOMI preprint, E-3, 979.

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