It has been proved by V. Sokolov that the Krichever-Novikov equation's
hierarchy is hamiltonian for the non-local Hamiltonian operator H_0=u_x D^{-1}
u_x and possesses twi weakly non-local recursion operatos of degree 4 and 6,
L_4 and L_6. We show here that H_0, L_4H_0 and L_6H_0 are compatible
Hamiltonian operators for which the Krichever-Novikov equation's hierarchy is
hamiltonian

Carpentier, Sylvain

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Sylvain Carpentier

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Compatible Hamiltonian operators for the Krichever–Novikov equation

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C. R. Acad. Sci. Paris, Ser. I 355 (2017) 744–747Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comLie algebras/Partial differential equationsCompatible Hamiltonian operators for the Krichever–Novikov equationOpérateurs hamiltoniens compatibles pour l’équation de Krichever–NovikovSylvain CarpentierDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 12 May 2017Accepted after revision 30 May 2017Available online 12 June 2017Presented by Michèle VergneIt has been proved by Sokolov that Krichever–Novikov equation’s hierarchy is hamiltonian for the Hamiltonian operator H0 = ux∂−1ux and possesses two weakly non-local recursion operators of degrees 4 and 6, L4 and L6. We show here that H0, L4 H0 and L6 H0 are compatible Hamiltonians operators for which the Krichever–Novikov equation’s hierarchy is hamiltonian.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éIl a été démontré par Sokolov que la hiérarchie de l’équation de Krichever–Novikov est hamiltonienne pour l’opérateur hamiltonien H0 = ux∂−1ux et possède deux opérateurs de récursion faiblement non locaux de degrés 4 et 6, L4 et L6. Nous montrons ici que H0, L4 H0 et L6 H0 sont des opérateurs hamiltoniens compatibles pour lesquels la hiérarchie de l’équation de Krichever–Novikov est hamiltonienne.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.In the study of finite-gap solutions of KP, an integrable (1 + 1)-dimensional PDE was discovered, the Krichever–Novikov equation. One of its forms (equivalent to the original one in [7]) isdudt= u3 − 32u22u1+ P (u)u1, (1)where u = u(t, x), un = ( ddx )n(u), and P is a polynomial of degree at most 4. Let V = C[u, u±1 , u2, ...] and K be the fraction field of V . Let us denote ddx by ∂ . The differential order dF of a function F ∈ V is the highest integer n such that ∂ F∂un = 0.One of the attributes of equation (1) is to be part of an infinite hierarchy of compatible evolution PDEs of odd differential ordersE-mail address: syl_car@mit.edu.http://dx.doi.org/10.1016/j.crma.2017.05.0091631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.S. Carpentier / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 744–747 745dudti= Gi ∈ V, i ≥ 0, (2)where Gi has differential order (2i + 1). One says that F , G ∈ V are compatible, or symmetries of one another, if{F , G} := X F (G) − XG(F ) = 0, (3)where X F denotes the derivation of V induced by the evolution equation ut = F , that isX F =∑n≥0F (n)∂∂un. (4)(3) endows V with a Lie algebra bracket, and the Gi ’s span an infinite-dimensional abelian subalgebra of (V, {., .}), which we will denote by S . The first four equations in the hierarchy are:G0 = u1,G1 = u3 − 32u22u1+ P (u)u1,G2 = u(5) − 5 u4u2u1− 52u23u1+ 252u3u22u21− 458u42u31− 53Pu3u21+ 256Pu22u31− 53Puu2u1− 518P 2u31+ 59u1 Puu,(5)G3 = u7 − 7 u2u6u1− 76u5u21(2P + 12u3u1 − 27u22) −212u24u1+ 212u4u31(2P − 11u22)− 73u4u21(2Puu1 − 51u2u3) + 492u33u21+ 712u23u31(22P − 417u22) +24998u42u41u3913Puu2u21u3 − 5956Pu22u41u3 − 3518u3u41(2Puuu41 − P 2) −157516u62u51+ 181324u42u51P− 2036u32u31Pu + 4936u22u51(6Puuu41 − 5P 2) −79u2u31(2Puuuu41 − 5P Pu) +754P 3u51− 79PuuPu1+ 79Puuuuu31 −718P 2uu1.(6)It is known ([6,10]) that all integrable hierarchies admit a pseudodifferential operator L ∈ V((∂−1)) satisfyingX F (L) = [D F , L] (7)for all F in the hierarchy, where D F denotes the Fréchet derivative of F :D F =∑n∂ F∂un∂n ∈ V[∂]. (8)A pseudodifferential operator satisfying (7) is called a recursion operator (for F ). In [3], two rational recursions operators for (1) were found, of orders 4 and 6:L4 = H1 H−10 , L6 = H2 H−10 , (9)whereH0 =u1∂−1u1,H1 =12(u21∂3 + ∂3u21) + (2u3u1 −92u22 −23P )∂ + ∂(2u3u1 − 92u22 −23P )+ G1∂−1G1 + u1∂−1G2 + G2∂−1u1,H2 =12(u21∂5 + ∂5u21) + (3u3u1 −192u22 − P )∂3 + ∂3(3u3u1 −192u22 − P )+ (u5u1 − 9u3u2 + 192u23 −23u3u1(5P − 39u22) +u22u21(5P − 9u22) +23P 2u21+ u21 Puu)∂+ ∂(u5u1 − 9u3u2 + 192u23 −23u3u1(5P − 39u22) +u22u21(5P − 9u22) +23P 2u21+ u21 Puu)+ G ∂−1G + G ∂−1G + u ∂−1G + G ∂−1u .(10)1 2 2 1 1 3 3 1746 S. Carpentier / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 744–747Moreover, L4 and L6 are both weakly non-local, i.e. of the formE(∂) ∈ V[∂] +∑ipi∂−1 δρiδu, (11)where the ρi ’s are conserved densities of (1). Recall that the variational derivative δδu is defined as follows:δFδu= D∗F (1) =∑n(−∂)n( ∂ F∂un). (12)In [11], Sokolov showed that the space of symmetries of (1), S , is preserved by L4. The same argument applies to L6, which was found later. He also establishes that the hierarchy of the Krichever–Novikov equation is hamiltonian for H0: there exists a sequence φi ∈ V such thatGi = H0( δφiδu) for all i ≥ 0. (13)A Hamiltonian operator H = AB−1 ∈ V(∂) with A and B right coprime is a skewadjoint rational differential operator inducing a non-local Poisson lambda bracket, which is equivalent to the following identity (see equation (6.13) in [4])A∗(D B(F )(A(G)) + D∗A(G)(B(F )) − D B(G)(A(F )) + D∗B(G)(A(F )))= B∗(D A(G)(A(F )) − D A(F )(A(G)))(14)for all F , G ∈ V .Lemma 1. Let L ∈ V(∂) be a skewadjoint rational operator. If there exists an infinite-dimensional (over C) subspace W ⊂ V such that B(W ) ⊂ δδuV and such that for all G ∈W , E = A(G) satisfiesXE(L) = D E L + LD∗E , (15)then L is a Hamiltonian operator. Conversely, if L is a Hamiltonian operator and G ∈ V , then D B(G) = D∗B(G) if and only if A(G) satisfies equation (15).Proof. Let us first give an equivalent form of (15) involving only differential operators.(1.15) ⇐⇒ XE(A) − D E A = AB−1(XE(B) + D∗E B)⇐⇒ XE(A) − D E A = −B∗−1 A∗(XE(B) + D∗E B)⇐⇒ A∗(XE(B) + D∗E B) = B∗(D E A − XE(A))⇐⇒ A∗(XE + D∗E)B = B∗(D E − XE)A.⇐⇒ A∗(D B(F )(E) + D∗E(B(F ))) = B∗(D E(A(F )) − D A(F )(E)) ∀F ∈ V.(16)Comparing the last line of (16) with (14), it is clear that if H is Hamiltonian, then E = A(G) satisfies equation (15) is and only if D B(G) is self-adjoint. It is also clear that if A(G) satisfies (15) and D B(G) is self-adjoint, then (F , G) satisfies (14) for any F ∈ V . Therefore, if we consider W ⊂ V infinite-dimensional subspace of V such that A(W) satisfies (15) and B(W) ⊂ δδuV , we deduce that (14) is satisfied for any (F , G) ∈ V × W . To conclude, we note that (14) can be rewritten as an identity of bidifferential operator, i.e. it amounts to say that some expression of the form ∑mij F (i)G( j) , where mij ∈ Vis trivial, i.e. mij = 0 for all i, j. Namely, (14) is equivalent toA∗(X A(G)(B)(F ) − X A(F )(B)(G) + (D A)∗G(B(F )) + (D B)∗G(A(F )))= B∗(X A(F )(A)(G) − X A(G)(A)(F )),(17)where given a differential operator P , an element F ∈ V , the differential operator (D P )F is defined by(D P )F (G) = XG(P )(F ) ∀G ∈ V. (18)If a bidifferential operator vanishes on V × W , it must be identically 0, since W is infinite dimensional. Hence, L is an Hamiltonian operator. Lemma 2. Let L = C D−1 be a rational operator and (Fn)n≥0 a sequence spanning an infinite-dimensional subspace of K satisfying C(Fn) = D(Fn+1) ∈ V for all n ≥ 0. Assume that L is recursion for all the D(Fn)’s and that the D(Fn)’s are hamiltonian for some Hamiltonian operator H ∈ V(∂). Then, provided LH is skew-adjoint, LH is a Hamiltonian operator for which all the D(Fn)’s are hamiltonian (n ≥ 1).S. Carpentier / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 744–747 747Proof. By Lemma 1, H satisfies equation (15) for all D(Fn), n ≥ 0, hence so does LH (L is recursion for D(Fn) for all n ≥ 0). To conclude using Lemma 1, one needs to check that D(Fn) = LH( δρnδu ) for some ρn ∈ V for all n ≥ 1. Let P , Q ∈ V[∂]be right coprime differential operators such that LH = P Q −1. Let A, B be right coprime differential operators such that H = AB−1. D(Fn) is hamiltonian for H for all n ≥ 0, meaning that there exist two sequences in V , (φn)n≥0 and (ρn)n≥0, such that δρnδu = B(φn) and D(Fn) = A(φn) for all n ≥ 0. In the language of [2], δρnδu and C(Fn) are C D−1 AB−1 associated, hence (quote result) there exists ψn such that C(Fn) = P (ψn) and Q (ψn) = δρnδu for all n ≥ 0. Therefore, by Lemma 1.1, LHis a Hamiltonian operator for which (C(Fn))n≥0 are hamiltonian. Theorem. H0 , H1 and H2 are compatible Hamiltonian operators.Proof. Let α, β, γ ∈ C and let Lα,β,γ = (αH0 + βH1 + γ H2)H−10 . Lα,β,γ is a recursion operator for the whole Krichever–Novikov hierarchy S . Moreover, it maps S to itself as was proved in [11], meaning that if Lα,β,γ = AB−1 with A, B right coprime and G ∈ S , then G = B(F ) for some F ∈K and A(F ) ∈ S . The theorem follows from Lemma 2. Remark 3. It follows from Lemma 1 that H = H2 H−11 H0 is a Hamiltonian operator of degree 1. However, it is not weakly non-local. More generally, all the (H2 H−11 )n H0, for n ∈ Z are pairwise compatible Hamiltonian operators. It is known since the work of Magri ([8], see also [5]) that from a pair of compatible Hamiltonian operators, one can construct infinitely many.Remark 4. Every Hamiltonian operator K = AB−1 over V , where A and B are right coprime induces a Lie algebra bracket on the space of functionals F(K ) := {∫ f ∈ V/∂V| δ fδu ∈ ImB}, (well-)defined by {∫f , ∫g} = ∫ δ fδu AB−1( δgδu ) (see section 7.2 in [4]). Note that F(H0) = V/∂V but that F(H1) and F(H2) consist only of the conserved densities of the Krichever–Novikov equation.We recall that if a rational differential operator L = AB−1, with A, B ∈ V[∂] right coprime generates an infinite dimen-sional abelian subspace of (V, {., .}), in the sense that there exist (Fn)n≥0 ∈ K such that A(Fn) = B(Fn+1) for all n ≥ 0 and such that the B(Fn)’s span an infinite-dimensional abelian subspace of (V, {., .}), then for all λ ∈ C, Im(A + λB) must be a sub Lie algebra of (V, {., .}) (see [1]). The recursion operators Lα,β,γ satisfy this condition.Note that weakly non-local Hamiltonian operators were introduced in [9], where the authors study the complete set of weakly non-local Hamiltonian operators for both the KdV and the NLS hierarchies.The author thanks Vladimir Sokolov for useful discussions, and Victor Kac for his interest in this work.References[1] S. Carpentier, A sufficient condition for a rational differential operator to generate an integrable system, Jpn. J. Math. 12 (2017) 33–89.[2] S. Carpentier, A. De Sole, V. Kac, Singular degree of a rational matrix pseudodifferential operator, Int. Math. Res. Not. 2015 (13) (2015) 5162–5195.[3] D.K. Demskoi, V.V. Sokolov, On recursion operators for elliptic models, Nonlinearity 21 (6) (2008) 1253–1264.[4] A. De Sole, V.G. Kac, Non-local Hamiltonian structures and applications to the theory of integrable systems, Jpn. J. Math. 8 (2) (2013) 233–347.[5] B. Enriquez, A. Orlov, V. Rubtsov, Higher Hamiltonian structures (the sl2 case), JETP Lett. 58 (8) (1993) 658–664.[6] N.Kh. Ibragimov, A.B. Shabat, Evolution equations with nontrivial Lie–Bäcklund group, Funkc. Anal. Prilozh. 14 (1) (1980) 25–36.[7] J.M. Krichever, S.P. Novikov, Holomorphic bundles over algebraic curves and non-linear equations, Russ. Math. Surv. 35 (6) (1980) 53–79.[8] F. Magri, Lecture Notes in Physics, vol. 120, Springer-Verlag, Berlin, 1980, p. 233.[9] A.Ya. Maltsev, S.P. Novikov, On the local systems Hamiltonian in the weakly non-local Poisson brackets, Physica D 156 (2001) 53–80.[10] A.V. Mikhailov, V.V. Sokolov, Symmetries of differential equations and the problem of integrability, in: A.V. Mikhailov (Ed.), Integrability, in: Lecture Notes in Physics, vol. 767, Springer, 2008.[11] V.V. Sokolov, On the hamiltonian property of the Krichever–Novikov equation, Sov. Math. Dokl. 30 (1) (1984).

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Compatible Hamiltonian Operators for the Krichever-Novikov Equation

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