On the bi-hamiltonian structures of the Camassa-Holm and Harry Dym equations

Transcription

1 arxiv:nlin/ v1 [nlin.si] 6 Jul 004 On the bi-hamiltonian structures of the Camassa-Holm and Harry Dym equations aolo Lorenzoni Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca Via Roberto Cozzi 53, I-015 Milano, Italy lorenzoni@matapp.unimib.it Marco edroni Dipartimento di Matematica Università di Genova Via Dodecaneso 35, I Genova, Italy pedroni@dima.unige.it March 6, 008 Abstract We show that the bi-hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple oisson pair defined on the loop algebra of sl(, R). The reduction process is a bi-hamiltonian reduction, that can be canonically performed on every bi-hamiltonian manifold. 1

2 1 Introduction In recent years a lot of papers have been devoted to the Camassa-Holm equation (CH) or, putting m = u u xx, u t u txx = 3uu x + u x u xx + uu xxx (1) m t = mu x m x u, () introduced in [3] as a model of shallow water waves. art of them [1,, 11, 0] have investigated the connections with the Korteweg-de Vries (KdV) equation u t = 3uu x + u xxx, (3) the Hunter-Saxton (HS) equation [9] and the Harry Dym (HD) equation u txx = u x u xx uu xxx, (4) u t = u 1 xxx, (5) attributed in [1] to Harry Dym. In particular, Khesin and Misiolek [11], motivated by [1] and [], have explained the connections between KdV, CH and HS in terms of their common symmetry group, the Virasoro group. Indeed, these equations can be interpreted as Euler equations descibing the geodesic flow (with respect to different metrics): on the Virasoro group in the KdV and CH case [, 1], and on suitable homogeneous space in the HS case [11]. Moreover, to any (codimension ) coadjoint orbit there corresponds a bi-hamiltonian structure: the first oisson bracket is just the Lie-oisson bracket, while the second one is a constant bracket depending on the choice of a point in the dual of the Virasoro algebra. oints on the same orbit give rise to equivalent choices. There are three types of orbit and three different associated bi-hamiltonian structures: the KdV, the CH and the HS bi-hamiltonian structures. As regas the connections between CH and HD, they are clear in the framework of the inverse scattering techniques [1]. Indeed, both the equations are associated to the scattering problems for the family of operators L k = x + k ρ q.

3 A different choice of the boundary conditions for the function ρ(x) and of the value of the constant q selects the associated equation: the case q = 0 and ρ 1 at infinity corresponds to HD, while the case q = 1 and ρ 0 at 4 infinity corresponds to CH. In this paper we investigate the connections between CH and HD from a different point of view. More precisely, we show that the CH and HD bi-hamiltonian structures can be obtained by a bi-hamiltonian reduction procedure from the oisson pencil (λ) = + λ 1 = x + [, A] + λ[, S], defined on the space M = C (S 1, sl(, R)) of C maps from the unit circle to the Lie algebra of traceless matrices. S is a point of M and A is an arbitrary constant traceless matrix. The reduction procedure depends on the choice of the conjugacy class of A. It turns out that there are three different reduced bi-hamiltonian structures: one is the CH bi-hamiltonian structure (rank(a) = ), one is the HD bi-hamiltonian structure (rank(a) = 1) and one, up to a change of cooinates, is still the HD bi-hamiltonian structure (A = 0). Since the HD bi-hamiltonian structure can be obtained from HS bi- Hamiltonian structure just by a change of variables [10], and taking into account the correspondence between coadjoint Virasoro orbits and Joan normal forms in SL(, R), we observe that this result seems to be strictly related to those of Khesin and Misiolek. The paper is organized as follows: In section we summarize some useful techniques in the theory of the bi-hamiltonian reduction. In section 3 we formalize and prove the above mentioned results. Acknowledgments. We wish to thank aolo Casati, Gregorio Falqui, and Franco Magri for useful discussions. This work has been partially supported by INdAM-GNFM under the research project Onde nonlineari, struttura tau e geometria delle varietà invarianti: il caso della gerarchia di Camassa-Holm. The bi-hamiltonian reduction In this section we recall a reduction process of the Marsden-Ratiu type [19], that can be performed on every bi-hamiltonian manifold. It has been pre- 3

4 sented in [4] and then applied to the Drinfeld-Sokolov hierarchies [7] in [5, 3] and to the stationary reductions of KdV in [8]. Let (M, 1, ) be a bi-hamiltonian manifold, i.e., a manifold M endowed with two oisson tensors 1 and that are compatible, in the sense that their sum (and hence any linear combination) is still a oisson tensor (see, e.g., [17, 16]). Let us fix a symplectic leaf S of 1 and consider the distribution D = (Ker 1 ) on M. Theorem 1 The distribution D is integrable. If E = D T S is the distribution induced by D on S and the quotient space N = S/E is a manifold, then it is a bi-hamiltonian manifold. The reduced oisson tensors 1 red and red on N are constructed as follows. For any point p S and any covector α Tπ(p) N, where π : S N is the canonical projection, there is a covector α Tp M such that α Dp = 0, α TpS = π p α, where πp : Tπ(p) N T p S is the codifferential of π at p. Then red i α = π π(p) p (( i ) p α), i = 1,. Whenever an explicit description of the quotient manifold N is not available, the following technique to compute the reduced bi-hamiltonian structure (already employed in [5] for the Drinfeld-Sokolov case) is very useful. Theorem Suppose Q to be a submanifold of S which is trasversal the distribution E, in the sense that T p Q E p = T p S for all p Q. (6) Then Q (which is locally diffeomorphic to N) also inherits a bi-hamiltonian structure from M. The reduced oisson pair on Q is given by i α = Π p p (( i ) p α), i = 1,, (7) where p Q, α T p Q, Π p : T p S T p Q is the projection relative to (6), and α T p M satisfies α Dp = 0, α TpQ = α. (8) In the next section we will follow this procedure to construct the bi-hamiltonian structures of Camassa-Holm and Harry Dym as suitable bi-hamiltonian reduced structures. 4

5 3 The bi-hamiltonian structure of CH and HD The aim of this section is to obtain the oisson pair of Camassa-Holm and that of Harry Dym by applying the reduction procedure we have just described to a simple class of bi-hamiltonian structures on the loop algebra of sl(, R). Let M = C (S 1, sl(, R)) be the space of C -maps from the unit circle to the Lie algebra of traceless real matrices. The tangent space T S M at S M is obviously identified with M itself. As far as the cotangent space is concerned, we will assume that TS M T SM by means of the nondegenerate form V 1, V = tr (V 1 (x), V (x)) dx, V 1, V M. S 1 As well-known (see, e.g., [13]), the manifold M admits a 3-parameter family of compatible oisson tensors given by V ( (a,b,c) )S V = a xv +b[v, S]+c[V, A], S M, V T S M, (9) where a, b, c R and A is any matrix in sl(, R). We have the following theorems. Theorem 3 The bi-hamiltonian reduction process applied to the pair ( 1 = (1,1,0), = (0,0,1) ) gives rise: to the oisson pair of the KdV hierarchy if A =. 1 ( 0 ) to the oisson pair of the AKNS hierarchy if A =. 0 1 roof: See [4] and [18]. The reduction used in the latter paper is not the one presented in Theorem 1, but it is easily shown to be equivalent. Theorem 4 The bi-hamiltonian reduction process applied to the pair ( 1 = (0,1,0), = (1,0,1) ) gives rise: - to the oisson pair of the CH hierarchy if A = ( 1 ) to the oisson pair of the HD hierarchy if A =

6 roof: Let S = p q r p, A = Q R with p, q, r C (S 1, R) and, Q, R R. Since Ker ( 1 ) S is spanned by S, the symplectic leaves of 1 are the level submanifolds of det S = p qr. Moreover, we have that D S = ( ) S (Ker( 1 ) S ) = {(µs) x + [µs, A] µ C (S 1, R). Explicitly, { (µp)x + (Rq Qr)µ (µq) D S = x + (Qp q)µ (µr) x + (r Rp)µ (µp) x (Rq Qr)µ, µ C (S 1, R). The distribution D is not tangent to the generic symplectic leaf of 1, but it is easily shown to be tangent to the symplectic leaf { p q S = p + qr = 0, (p, q, r) (0, 0, 0), (10) r p so that E p = D p T p S coincides with D p for all p S. In oer to determine the reduced bi-hamiltonian structure we first show that, under the assumption that R 0, the submanifold { 0 q Q = q C (S 1, R), q(x) 0 x S 1 (11) q of S is transversal to the distribution E. Indeed, if S(q) =, then 0 0 {(ṗ ) q T S(q) S = 0 ṗ ṗ, q C (S 1, R) C (S 1, R) C (S 1, R), and every tangent vector in T S(q) S admits the unique decomposition (ṗ, q) = (ṗ, 1 R (ṗ x ṗ)) + (0, q 1 R (ṗ x ṗ)), where the first summand belongs to E S(q) and the second one to T S(q) Q. This also shows that Π S(q) : T S(q) S T S(q) Q is given by Π S(q) : (ṗ, q) (0, q 1 R (ṗ x ṗ)). (1) 6

7 At this point we can compute the reduced oisson pair on Q. For the sake of simplicity we will deal simultaneously with the oisson pencil (λ) = +λ 1. Given α T S(q) Q C (S 1, R), we look for a covector α = α1 α TS(q) α 3 α M 1 such that α DS(q) = 0 and α TS(q) Q = α. We easily find that α = ( 1 (α ) R x + α) α α 1 (α, R x + α) where α is arbitrary. Then we have that ( (λ) )S(q) α T SQ is given by ṗ = 1 R (α xx + α x ) + Rα α λαq q = α x + Q R (α x + α) α + λ q R (α x + α). Thus the reduced oisson pencil is ( (λ) α = Π ((λ) ) q S(q) )S(q) α [ = 1 ( + Q)R + R 3 x + R x + R (Q ) + λ ] R (q x + q x ) α, that is to say 1 = 1 q R (q x + q x ) ( )q = 1 ( + Q)R + R 3 x + R x + (Q ). R The case = 0, Q = R = 1, i.e., A = 0 1 1, corresponds to the 1 0 oisson pair 1 = q q x + q x = q 3 x + x. 7

8 It is well-known that it is the CH bi-hamiltonian structure [3]. Indeed, if we put q = m = u u xx : where m t = mu x m x u = 1 δh 1 δm = δh δm, (13) H 1 = 1 (u + u x)dx, H = 1 (u 3 + uu x )dx. 0 0 The case = Q = 0, R = 1, i.e., A =, gives rise to the oisson 1 0 pair 1 = q q x + q x = 1 q 3 x. It is well-known that it is the HD bi-hamiltonian structure (see [6, 15, 4]). Indeed, if we put q = u: u t = u 1 = δh 1 1 xxx δu = δh δu, (14) where H 1 = 1 u 5 u 8 x dx, H = 4 u 1 dx. Theorem 5 The bi-hamiltonian( reduction ) process applied to the pair ( 1 = 0 0 (1,1,0), = (0,0,1) ) with A =, in the space C0 0 0 (R, R) of rapidly decreasing C -maps from the real line to sl(, R), gives rise to the oisson pair ( ( x 1 p ) x + p x x 1, x ), (15) 8

9 where p(x) is a smooth function vanishing at infinity and x 1 = 1 ( x + ). roof: The distribution { (µp)x (µq) D S = x (µr) x (µp) x x µ C0 (R, R) is still tangent to S. In this case the transversal submanifold is { p 1 Q = p C0 0 p (R, R), p(x) 0 x R (16) and the projection Π S(q) : T S(q) S T S(q) Q is given by Π S(q) : (ṗ, q) (ṗ p q p x 1 x Following the same procedure used above it is easy to see that 1 = ( 1 p x p ) x + p x x 1 = p x. q, 0). (17) Moreover, taking into account that, after a change of cooinates u = u (u), a bivector transforms as u = u (s) s x ( x ) t u, (18) u (t) s 0 t 0 we obtain that, in the variable u = p x, the oisson pair ( 1, ) coincides with the Harry Dym bi-hamiltonian structure. 4 Conclusions In this paper we have shown that the bi-hamiltonian structures of CH and HD can be seen as reductions of suitable structures on C (S 1, sl(, R)). In the KdV case this is a well-known result, that can be interpreted both 9

10 from the Drinfeld-Sokolov point of view and in the bi-hamiltonian reduction scheme [3]. The results of this paper could be used to construct -field extensions of the CH and HD hierarchies, in the same way as Drinfeld and Sokolov construct a -component generalization of the KdV hierarchy. This extended hierarchy lives on a symplectic leaf of (0,0,1) and projects on the usual scalar KdV hierarchy (see [7] and [16]). Analogously, we plan to define a -field hierarchy on the symplectic leaf S of CH, whose projection on the transversal submanifold Q is the CH hierarchy. The unprojected CH equation should be compared with the -component generalization of CH recently introduced by Liu and Zhang [14]. References [1] R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Koteweg de Vries hierarchy, Adv. in Math., 140 (1998), [] R. Beals, D. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter- Saxton equations, Appl. Anal. 78 (001), [3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, hys. Lett. Rev. 71 (1993), [4]. Casati, F. Magri, M. edroni, Bi-Hamiltonian Manifolds and τ function, in: Mathematical Aspects of Classical Field Theory 1991 (M. J. Gotay et al. eds.), Contemporary Mathematics vol. 13, American Mathematical Society, rovidence, R.I., 199, pp [5]. Casati, M. edroni, Drinfeld Sokolov Reduction on a Simple Lie Algebra from the Bi-Hamiltonian oint of View, Lett. Math. hys. 5 (199), [6] I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, John Wiley and Sons Ltd, Chichester, [7] V. G. Drinfeld, V. V. Sokolov, Lie Algebras and Equations of Korteweg de Vries Type, J. Sov. Math. 30 (1985), [8] G. Falqui, F. Magri, M. edroni, J.. Zubelli, A Bi-Hamiltonian Theory for Stationary KdV Flows and their Separability, Regul. Chaotic Dyn. 5 (000), [9] J. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991),

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