Compatible Hamiltonian Operators for the Krichever-Novikov Equation

Transcription

1 arxiv: v [math.ap] 3 May 207 Compatible Hamiltonian Operators for the Krichever-Novikov Equation Sylvain Carpentier* Abstract It has been proved by Sokolov that Krichever-Novikov equation s hierarchy is hamiltonian for the Hamiltonian operator H 0 = u x u x and possesses two weakly non-local recursion operators of degree 4 and 6, L 4 and L 6. We show here that H 0, L 4 H 0 and L 6 H 0 are compatible Hamiltonians operators for which the Krichever-Novikov equation s hierarchy is hamiltonian. November 7, 208 * Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 0239, USA

2 In the study of finite gap solutions of KP, an integrable (+)-dimensional PDE was discovered, the Krichever-Novikov equation. One of its forms (equivalent to the original one in [KN80]) is du dt = u 3 3 u P (u), () 2 u u where u = u(t, x), u n = ( d dx )n (u), and P is a polynomial of degree at most 4. Let V = C[u, u ±, u 2,...] and K be the fraction field of V. Let us denote d dx by. The differential order d F of a function F V is the highest integer n such that F u n 0. One of the attributes of equation () is to be part of an infinite hierarchy of compatible evolution PDEs of odd differential orders du dt i = G i V, i 0, (2) where G i has differential order (2i + ). One says that F, G V are compatible, or symmetries of one another, if {F, G} := X F (G) X G (F ) = 0, (3) where X F denotes the derivation of V induced by the evolution equation u t = F, that is X F = F (n). (4) u n 0 n (3) endows V with a Lie algebra bracket, and the G i s span an infinitedimensional abelian subalgebra of (V, {.,.}), which we will denote by S. The first four equations in the hierarchy are: G 0 = u, G = u 3 3 u P (u), 2 u u G 2 = u (5) 5 u 4u 2 5 u u 2 u P u 2 u 5 P 2 u 8 u 3 u 3 u 2 2 u u P uu, 45 u u 3 3 P u u 2 6 P u2 2 u 3 (5)

3 G 3 = u 7 7 u 2u 6 7 u 5 (2P + 2u u 6 u 2 3 u 27u 2 2) 2 u u 4 (2P u 2 2 u 2 u 2) 3 7 u 4 (2P 3 u 2 u u 5u 2 u 3 ) + 49 u u 2 3 (22P 47u 2 2 u 2 2 u 2) u 4 2 u 3 8 u P u 2 u u u P u2 2 u u u 3 (2P 8 u 4 uu u 4 P 2 ) 575 u u 4 2 P 6 u 5 24 u u 3 2 P 6 u 3 u + 49 u 2 2 (6P 36 u 5 uu u 4 5P 2 ) 7 u 2 (2P 9 u 3 uuu u 4 5P P u ) + 7 P 3 54 u P P uu + 7 u 9 P uuuuu 3 7 Pu 2. 8 u (6) It is known ([IS80], [MS08]) that all integrable hierarchies admit a pseudodifferential operators L V(( )) satisfying X 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 u n n V[ ]. (8) A pseudodifferential operator satisfying (7) is called a recursion operator (for F ). In [DS08] two rational recursions operators for () were found, of order 4 and 6: L 4 = H H0, L 6 = H 2 H0, (9) where H 0 =u u, H = 2 (u u 2 ) + (2u 3 u 9 2 u P ) + (2u 3u 9 2 u P ) + G G + u G 2 + G 2 u, H 2 = 2 (u u 2 ) + (3u 3 u 9 2 u2 2 P ) (3u 3 u 9 2 u2 2 P ) + (u 5 u 9u 3 u u2 3 2 u 3 (5P 39u 2 3 u 2) + u2 2 (5P 9u 2 u 2) (u 5 u 9u 3 u u2 3 2 u 3 (5P 39u 2 3 u 2) + u2 2 (5P 9u 2 u 2) G G 2 + G 2 G + u G 3 + G 3 u. 2 P 2 u 2 P 2 u 2 + u 2 P uu ) (0) + u 2 P uu )

4 Moreover, L 4 and L 6 are both weakly non-local, i.e. of the form E( ) V[ ] + i p i δρ i, () where the ρ i s are conserved densities of (). derivative δ is defined as follows: δf = D F () = n Recall that the variational ( ) n ( F u n ). (2) In [S84], Sokolov showed that the space of symmetries of (), S, is preserved by L 4. The same argument applies to L 6, which was found later. He also establishes that the hierarchy of the Krichever-Novikov equation is hamiltonian for H 0 : there exists a sequence φ i V such that G i = H 0 ( δφ i ) for all i 0. (3) A Hamiltonian operator H = AB 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 (6.3) in [DSK3]) 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))) (4) for all F, G V. Lemma. Let L V( ) be a skewadjoint rational operator. If there exists an infinite-dimensional (over C) subspace W V such that B(W ) δ V and such that for all G W, E = A(G) satisfies X E (L) = D E L + LD E, (5) then L is a Hamiltonian operator. Conversely, if L is a hamiltonian operator and G V, then D B(G) = DB(G) if and only if A(G) satisfies equation (5) 3

5 Proof. Let us first give an equivalent form of (5) involving only differential operators. (.5) X E (A) D E A = AB (X E (B) + D EB) X E (A) D E A = B A (X E (B) + D EB) A (X E (B) + D EB) = B (D E A X E (A)) A (X E + D E)B = B (D E X E )A. A (D B(F ) (E) + D E(B(F ))) = B (D E (A(F )) D A(F ) (E)) F V (6) Comparing the last line of (6) with (4), it is clear that if H is Hamiltonian, then E = A(G) satisfies equation (5) is and only if D B(G) is self-adjoint. It is also clear that if A(G) satisfies (5) and D B(G) is self-adjoint, then (F, G) satisfies (4) for any F V. Therefore, if we consider W V infinitedimensional subspace of V such that A(W) satisfies (5) and B(W) δ V, we deduce that (4) is satisfied for any (F, G) V W. To conclude, we note that (4) can be rewritten as an identity of bidifferential operator, i.e. it amounts to say that some expression of the form m ij F (i) G (j), where m ij V is trivial, i.e. m ij = 0 for all i, j. Namely, (4) is equivalent to A (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 )), (7) where given a differential operator P, an element F V, the differential operator (D P ) F is defined by (D P ) F (G) = X G (P )(F ) G V. (8) 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 = CD be a rational operator and (F n ) n 0 a sequence spanning an infinite-dimensional subspace of K satisfying C(F n ) = D(F n+ ) V for all n 0. Assume that L is recursion for all the D(F n ) s and that the D(F n ) 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(F n ) s are hamiltonian (n ). 4

6 Proof. By Lemma, H satisfies equation (5) for all D(F n ), n 0, hence so does LH (L is recursion for D(F n ) for all n 0). To conclude using Lemma, one needs to check that D(F n ) = LH( δρn ) for some ρ n V for all n. Let P, Q V[ ] be right coprime differential operators such that LH = P Q. Let A, B be right coprime differential operators such that H = AB. D(F n ) is hamiltonian for H for all n 0, meaning that there exist two sequences in V, (φ n ) n 0 and (ρ n ) n 0, such that δρn = B(φ n) and D(F n ) = A(φ n ) for all n 0. In the language of [CDSK5], δρn and C(F n) are CD AB associated, hence (quote result) there exists ψ n such that C(F n ) = P (ψ n ) and Q(ψ n ) = δρn for all n 0. Therefore, by Lemma., LH is a Hamiltonian operator for which (C(F n )) n 0 are hamiltonian. Theorem. H 0, H and H 2 are compatible Hamiltonian operators. Proof. Let α, β, γ C and let L α,β,γ = (αh 0 + βh + γh 2 )H 0. L α,β,γ is a recursion operator for the whole Krichever-Novikov hierarchy S. Moreover, it maps S to itself as was proved in [S84], meaning that if L α,β,γ = AB 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 that H = H 2 H H 0 is a Hamiltonian operator of degree. However, it is not weakly non-local. More generally all the (H 2 H ) n H 0, for n Z are pairwise compatible Hamiltonian operators. Remark 4. Every Hamiltonian operator K = AB over V, where A and B are right coprime induces a Lie algebra bracket on the space of functionals F(K) := { f V/ V δf ImB}, (well-)defined by { f, g} = δf AB ( δg ) (see section 7.2 in [DSK3]). Note that F(H 0) = V/ V but that F(H ) and F(H 2 ) consist only of the conserved densities of the Krichever-Novikov equation. We recall that if a rational differential operator L = AB, with A, B V[ ] right coprime generates an infinite dimensional abelian subspace of (V, {.,.}), in the sense that there exist (F n ) n 0 K such that A(F n ) = B(F n+ ) for all n 0 and such that the B(F n ) 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 [C7]). The recursion operators L α,β,γ satisfy this condition. The author thanks Vladimir Sokolov for useful discussions, and Victor Kac for his interest in this work. 5

7 References [C7] S. Carpentier, A sufficient condition for a Rational Differential Operator to generate an Integrable System, Japan. J. Math. 2, (207) [CDSK5] S. Carpentier, A. De Sole, V. Kac, Singular Degree of a Rational Matrix Pseudodifferential Operator Int Math Res Notices (205) 205 (3): [DS08] D.K. Demskoi, V.V. Sokolov, On recursion operators for elliptic models, Nonlinearity, 2:6, [DSK3] A. De Sole, V.G. Kac, Non-local Hamiltonian structures and applications to the theory of integrable systems, Jpn. J. Math. 8 (203), no. 2, [IS80] N.Kh. Ibragimov, A.B. Shabat, Evolution equations with nontrivial Lie-Bäcklund group, Funkstional. Anal. i Prilozhen, 4, no., (980). [KN80] J. M. Krichever and S. P. Novikov, Holomorphic bundles over algebraic curves and non-linear equations Russian Math. Surveys, 35:6 (980), [MS08] A.V. Mikhailov and V.V. Sokolov, Symmetries of differential equations and the problem of integrability in : A.V. Mikhailov, ed., Integrability, Lect. Notes Phys. 767 ( Springer, 2008). [S84] V.V. Sokolov, On the hamiltonian property of the Krichever-Novikov equation, Soviet. Math. Dokl. Vol. 30 (984), No. 6