A 3-component extension of the Camassa-Holm hierarchy

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(1)UNIVERSITA’ DEGLI STUDI DI BERGAMO DIPARTIMENTO DI INGEGNERIA GESTIONALE E DELL’INFORMAZIONE° † QUADERNI DEL DIPARTIMENTO. Department of Management and Information Technology Working Paper Series “Mathematics and Statistics”. n. 3/MS – 2006. A 3-component extension of the Camassa-Holm hierarchy. by. Laura Fontanelli, Paolo Lorenzoni, Marco Pedroni. ° †. Viale Marconi. 5, I – 24044 Dalmine (BG), ITALY, Tel. +39-035-2052339; Fax. +39-035-562779 Il Dipartimento ottempera agli obblighi previsti dall’art. 1 del D.L.L. 31.8.1945, n. 660 e successive modificazioni..

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(3) A 3-component extension of the Camassa-Holm hierarchy Laura Fontanelli, Paolo Lorenzoni Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca Via Roberto Cozzi 53, I-20125 Milano, Italy laura.fontanelli@unimib.it, paolo.lorenzoni@unimib.it Marco Pedroni Dipartimento di Ingegneria Gestionale e dell’Informazione Università di Bergamo Viale Marconi 5, I-24044 Dalmine (BG), Italy marco.pedroni@unibg.it. Abstract We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a 3-component extension of the CH equation.. Keywords: Integrable hierarchies of PDEs, Camassa-Holm equation, bi-Hamiltonian manifolds, Marsden-Ratiu reduction. 2000 Mathematics Subject Classification: 35Q53, 35Q58, 37K10, 53D17.. 1.

(4) 1. Introduction. One of the many remarkable facts concerning the theory of integrable PDEs is that almost all these equations can be obtained as suitable reductions from (integrable) hierarchies living on loop-algebras. The main example is the Drinfeld-Sokolov reduction [11], leading from (the loop-algebra of) a simple Lie algebra g to the so-called generalized KdV equations. It is well-known (see, e.g., [10]) that the choice g = sl(n) gives rise to the Gelfand-Dickey hierarchies. In particular, sl(2) corresponds to the KortewegdeVries (KdV) hierarchy, whose most famous member is the KdV equation ut = −3uux + uxxx ,. (1). describing shallow water waves. Another important model for such waves is the CamassaHolm (CH) equation ut − utxx = −3uux + 2ux uxx + uuxxx ,. (2). introduced in [4, 14]. Like KdV, it belongs to a hierarchy of commuting evolution equations and possesses a bi-Hamiltonian formulation. Many papers have been devoted to the CH hierarchy, and part of them [2, 3, 15, 21] have investigated the connections with the KdV hierarchy. The Whitham modulation theory has been recently discussed in [1]. In [18] it has been shown that the bi-Hamiltonian structure of CH can be seen as a reduction of a suitable bi-Hamiltonian structure on the space M = C ∞ (S 1 , sl(2)) of C ∞ maps from the unit circle to sl(2). The reduction process — called bi-Hamiltonian reduction — is a particular instance of the Marsden-Ratiu reduction [20] and can be canonically performed on every bi-Hamiltonian manifold. In [7] it has been introduced and applied to the KdV hierarchy, while in [9, 22] its equivalence with the DrinfeldSokolov reduction has been showed. In this paper we show that there exists a bi-Hamiltonian hierarchy on M — to be called the matrix CH hierarchy — giving rise to the usual (i.e., scalar) CH hierarchy after that the bi-Hamiltonian reduction is performed. This allows us to interpret also CH as a reduced hierarchy, and to define a 3-component extension of the CH equation. This should be related to the 2-component extension introduced in [12, 13, 17]. The paper is organized as follows: In Section 2 we recall the bi-Hamiltonian reduction process and its application to the CH case. In Section 3 we present the 3-component extension of the CH equation and we show that it belongs to the (nonlocal) matrix CH hierarchy, studied in Section 4. Section 5 is devoted to conclusions, including a brief discussion on the local matrix CH hierarchy. Acknowledgments. We wish to thank Gregorio Falqui, Franco Magri, and Giovanni Ortenzi for useful discussions and suggestions. 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, and by the European Community through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652). M.P. would like to thank for the hospitality the Department Matematica e Applicazioni of the Milano-Bicocca University, where most of this work was done. 2.

(5) 2. Some information about the bi-Hamiltonian reduction. In this section we review some facts on the geometry of bi-Hamiltonian manifolds (see, e.g., [19, 5]), and we recall from [18] that the bi-Hamiltonian structure of the (usual) CH hierarchy can be obtained by means of a reduction. Let (M, P1 , P2 ) be a bi-Hamiltonian manifold, i.e., a manifold M endowed with two compatible Poisson tensors P1 and P2 . Let us fix a symplectic leaf S of P1 and consider the distribution D = P2 (KerP1 ) 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. 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 [9] for the Drinfeld-Sokolov case) is very useful. Suppose Q to be a submanifold of S which is trasversal to the distribution E, in the sense that Tp Q ⊕ Ep = Tp S. for all p ∈ Q .. (3). Then Q (which is locally diffeomorphic to N ) also inherits a bi-Hamiltonian structure from M. The reduced Poisson pair on Q is given by ¡ rd ¢ Pi p α = Πp ((Pi )p α ˜) , i = 1, 2 , (4) where p ∈ Q, α ∈ Tp∗ Q, Πp : Tp S → Tp Q is the projection relative to (3), and α ˜ ∈ Tp∗ M satisfies α ˜ |D p = 0 , α ˜ |Tp Q = α . (5) Let us suppose now that {Hj }j∈Z be a bi-Hamiltonian hierarchyPon M, that is, −j P2 dHj = P1 dHj−1 for all j. This amounts to saying that H(λ) = is a j∈Z Hj λ (formal) Casimir of the Poisson pencil P1 − λP2 . The bi-Hamiltonian vector fields associated with the hierarchy can be reduced on the quotient manifold N according to Proposition 2 The restrictions of the functions Hj to S are constant along the distribution E, and therefore they give rise to functions on N . These functions form a bi-Hamiltonian hierarchy with respect to the reduced Poisson pair. The vector fields Xj = P2 dHj = P1 dHj−1 are tangent to S and project on N . Their projections are the vector fields associated with the reduced hierarchy. Now let M = C ∞ (S 1 , sl(2)) be the loop-space on the Lie algebra of traceless 2 × 2 real matrices. The tangent space TS M at S ∈ M is identified with M itself, and we will assume that TS M ' TS∗ M by the nondegenerate form Z hV1 , V2 i = tr(V1 (x)V2 (x)) dx, V1 , V 2 ∈ M , 3.

(6) where the integral is taken (here and in the rest of the paper) on S 1 . It is well-known (see, e.g., [16]) that the manifold M has a 3-parameter family of compatible Poisson tensors P(λ1 ,λ2 ,λ3 ) = λ1 ∂x + λ2 [ · , S] + λ3 [ · , A] , (6) where λ1 , λ2 , λ3 ∈ R , A is a constant matrix in sl(2), and µ ¶ p q S= ∈M. r −p In this paper we are interested in the pencil Pλ = P1 − λP2 = [ · , S] − λ(∂x + [ · , A]) ,. (7). obtained from (6) setting λ2 = 1, λ3 = λ1 − λ, and µ ¶ 1 0 1 . A= 2 1 0 In [18] the bi-Hamiltonian reduction procedure has been applied to the pair (P1 , P2 ). In this case ½µ ¶ ¾ (µp)x + 12 (q − r)µ (µq)x + 2µp ∞ 1 DS = | µ ∈ C (S , R) . (µr)x − µp −(µp)x − 21 (q − r)µ The distribution D is not tangent to the generic symplectic leaf of P1 , but it is tangent to the symplectic leaf ¶ ¾ ½µ p q 2 | p + qr = 0, (p, q, r) 6= (0, 0, 0) , (8) S= r −p so that Ep = Dp ∩ Tp S coincides with Dp for all p ∈ S. It is not difficult to prove that the submanifold ½ µ ¶ ¾ 0 q ∞ 1 1 Q = S(q) = | q ∈ C (S , R), q(x) 6= 0 ∀x ∈ S (9) 0 0 of S is transversal to the distribution E and that the projection ΠS(q) : TS(q) S → TS(q) Q is given by ΠS(q) : (p, ˙ q) ˙ 7→ (0, q˙ − 2p˙x ) . (10) The reduced bi-Hamiltonian structure (4) coincides with the bi-Hamiltonian structure of the Camassa-Holm hierarchy (see [18] for details): ¡ rd ¢ P1 q = 2(2q∂x + qx ) ¡ rd ¢ P2 q = 2(−∂x3 + ∂x ) . R√ Starting from the Casimir q dx of P1rd one constructs the local (or negative) CH R hierarchy. The Casimir q dx of P2rd gives rise to the nonlocal (or positive) CH hierarchy, whose second flow is the CH equation (2). We refer to [6] and the references cited therein for more details, and for a discussion about a “KP extension” of the local hierarchy. 4.

(7) 3. A 3-component extension of the Camassa-Holm equation. In this section we start to construct a bi-Hamiltonian hierarchy associated to the pencil (7). We consider the functional Z 1 H1 = − (r + q) dx , (11) 2 which is easily seen to be a Casimir of P2 . Applying P1 to the differential dH1 we obtain the vector field  1   p˙ = − 2 (r − q) (12) q˙ = p   r˙ = −p This vector field is Hamiltonian also with respect to P2 , i.e., it can be written as P2 dH2 , where Z H2 [p, q, r] = (p2 − γ 2 − γx2 ) dx . (13) To obtain this expression of H2 , let us start from (dH2 )x + [dH2 , A] = [dH1 , S] .. (14). and let us denote the components of dH2 in the following way: ¶ µ α β . dH2 = γ −α Equation (14) in componentwise form reads  1   αx − 2 (−γ + β + r − q) = 0 βx + p − α = 0   γx + p − α = 0 The last two equations implies (β + γ)x = 0. If we suppose that β = −γ, then we have    α = p + γx β = −γ   γxx − γ = 12 (q − r) − px ˙ The expression of H2 can be easily obtained evaluating dH2 on a tangent vector S. Indeed, we have Z Z Z d 2 ˙ hdH2 , Si = [2p(γ ˙ x + p) − γ r˙ + γ q]dx ˙ = p dx + γ(−2p˙x − r˙ + q)dx ˙ dt Z Z Z Z d d 2 2 2 = p dx + 2 γ(γ˙ xx − γ)dx ˙ (p − γ )dx − 2 γ˙ γ˙ x dx dt dt Z d = (p2 − γ 2 − γx2 )dx , dt 5.

(8) leading to (13). Hence the second vector field P1 dH2 of the hierarchy is given by    p˙ = −γ(r + q) q˙ = 2q(γx + p) + 2pγ   r˙ = 2pγ − 2r(p + γx ). (15). where γ satisfies the equation 1 γxx − γ = (q − r) − px . 2. (16). Let us show that the vector fields (12) and (15) project on the first members of the usual CH hierarchy. The reduction procedure for P1 dH1 goes as follows: - we restrict P1 dH1 on the symplectic leaf S and in particular on the points (p = r = 0) of the transversal submanifold Q; - we project the restricted vector field according to the formula (10). We obtain qt0 = q˙ − 2p˙x = −qx As far as P1 dH2 is concerned, its restriction at the points of Q is    p˙ = −γq q˙ = 2qγx   r˙ = 0. (17). where, from (16), one has that γxx − γ = 21 q. This shows that the usual change of dependent variable for the CH equation arises naturally in the reduction procedure. The projection of (17) on T Q is qt1 = q˙ − 2p˙x = 4qγx + 2qx γ , that is, γxxt1 − γt1 = 4γxx γx − 6γγx + 2γγxxx , which becomes the Camassa-Holm equation (2) after putting u = 2γ. Thus (15) is a 3-component extension of the Camassa-Holm equation. Notice that the reduced Hamiltonian Z H2 [0, q, 0] = − (γ 2 + γx2 )dx is the first Hamiltonian of the Camassa-Holm equation. Now we want to show that this procedure can be iterated, i.e., that the vector fields (12) and (15) belong to a (bi-Hamiltonian) hierarchy of (commuting) vector fields, whose reduction is the (scalar, nonlocal) CH hierarchy. To this aim, we construct a Casimir of the pencil (7) using the classical method of dressing transformations [24, 11, 8]. 6.

(9) First of all we observe that the elements µ ¶ v1 v2 V = v3 −v1 of ker Pλ satisfy the equation −λVx + [V, S − λA] = 0 ,. (18). which implies that. d tr V 2 = 0 , dx so that the spectrum of V does not depend on x: tr. V2 = F (λ) . 2. (19). Therefore, there exists a nonsingular matrix K such that V (λ) = K −1 ΛK , where. µ Λ=. 0 F (λ) 1 0. ¶ .. Proposition 3 If F (λ) does not depend on the point S, then V (λ) is an exact 1-form whose potential is given by Z H(λ) = tr (M Λ) dx , (20) where M = K −1 (S − λA)K + λK −1 Kx .. (21). Proof. From (18) it follows that −λK −1 Vx K + K −1 [V, S − λA]K − λΛx + [Λ, M ] = [Λ, M ] = 0 , R ˙ ˙ we which implies that tr ([M, K −1 K]Λ) dx = 0. Thus, for every tangent vector S, have Z Z ˙ ˙ ˙ ˙ dx + tr ([M, K −1 K]Λ) hdH, Si = tr (M Λ) dx = tr (K −1 SKΛ) Z Z −1 ˙ ˙ ) dx = hV, Si ˙ . = tr (SKΛK ) dx = tr (SV Let us compute explictly M and H. A possible choice for K is µ −1 1 ¶ v 2 v1 v − 2 K= , 1 0 v2 7.

(10) where v = v3 . From (21) and (20) we easily obtain µ ¶ 2r − λ 0 v −1 (r − λ2 )F (λ) Λ M= = λ −1 v (r − 2 ) 0 2v and. Z. F (λ)(2r − λ) dx . v The functional H(λ) can also be written as Z p H(λ) = 2λ F (λ) h dx , H(λ) =. (22). (23). where the density h is clearly defined up to a total x-derivative. Using this freedom we can choose h in such a way that it satifsies a Riccati-type equation. Indeed, from the first two equations of the system (18) written in componentwise form, 1 1 −λv1 x + v2 (2r − λ) − v (2q − λ) = 0 2 2 −λv2x + v1 (2q − λ) − 2v2 p = 0 , we can write v1 and v2 as functions of v = v3 : −λ1 vx + 2vp 2r − λ λ2 rx vx − 2λrx pv −λ2 vxx + 2λpvx + 2λvpx v(2q − λ) v2 = 4 + 2 + . (2r − λ)3 (2r − λ)2 2r − λ v1 =. (24). Substituting these expressions of v1 and v2 in (19) we obtain the following equation, (2r − λ)3¡F (λ) − λ2 (2r − λ)vx2 + 2λ2 (2r − λ)vxx v − 4rx λ2 vx v +v 2 λ3 + 2λ2¢(2px − q − 2r) + 4λ(−2px r + 2rx p + p2 + 2qr + r2 ) −8r(p2 + qr) = 0 ,. (25). whose solution gives, with (24), the differential V (λ) of the Casimir H(λ) of the Poisson pencil. It is now easy to guess the most convenient choice of the total x-derivative involved in the definition of h. Putting p F (λ)(2r − λ) vx + h= 2v 2λv we get from (25) the following Riccati equation for h: (hx + h2 )(2 r − λ)λ2 − 2rx h λ2 = − 41 λ3 + ( r + 12 q − px ) λ2 +(−r2 − 2 rx p − 2 q r + 2 px r − p2 ) λ + 2 qr2 + 2 p2 r .. (26). If h is a solution of this equation, then (23) is a Casimir of the Poisson pencil and its coefficients constitute a bi-Hamiltonian hierarchy. We remark that equation (26), evaluated at the points of Q, is the Riccati equation for the scalar CH hierarchy (see [23] and [6]). 8.

(11) 4. The matrix CH hierarchy. We want to show that a suitable choice of a solution of the Riccati equation (26) leads to a bi-Hamiltonian hierarchy which starts from the Casimir (11) of P2 and contains the 3-component extension of the CH equation presented in the previous section. We need to find a solution h(λ) of (26) as a formal series expansion in negative powers of λ, ∞ X h= hi λ−i . i=0. Substituting in (26) we obtain: ¡ ¢³ P∞ ¡ ¢ −i ´ Pi P −i 2rλ2 − λ3 h + h h − 2rx λ2 ∞ = ix i=0 j=0 i−j j λ i=0 hi λ ¡ ¢ 3 = − λ4 + 2q + r − px λ2 − (+r2 + 2rq − 2rpx + p2 + 2rx p)λ + 2r(rq + p2 ). (27). It is possible to find the coefficients hi recursively, by solving a differential equation at every step. The first step corresponds to the coefficients of λ3 in (27): h0 x + h20 =. 1 4. .. The only periodic solutions of this equation are h0 = ± 21 . We choose the positive solution, so that the next equation become 1 h1x + h1 = − (r + q) − rx + px . (28) 2 ¡ ¢ The differential operator 1 + ∂x is invertible in the space of periodic smooth functions C ∞ (S 1 , R), so the previous equation has a unique solution Z x Z 1 ¡ ¢−1 1 y−x h1 = 1 + ∂x m e m(y)dy + ey−x m(y)dy, e − 1 0 0 were m = − 12 (r + q) − rx + px . The next step is h2 x + h2 = 2r(h1 x + h1 ) − h21 − 2rx h1 + r2 + 2rq − 2rpx + p2 + 2rx p . Substituting the expression for h1 we get i ¢−1 ¡ ¢−1 ¢2 ¢−1 h ¡¡ ¡ 2rm − 1 + ∂x m − 2rx 1 + ∂x m + r2 + 2rq − 2rpx + p2 + 2rx p h2 = 1 + ∂x (29) Similarly, we obtain h3 x + h3 = 2r(h2 x + h2 + h21 ) − 2h1 h2 − 2rx h2 − 2r(qr + p2 ) ,. 9.

(12) so that ¡ ¢−1 h = 1 + ∂ × 3 x h ¡ ¡¡ ¢−1 ¢2 ¡ ¢−1 ¢ 2r 2rm − 1 + ∂x m − 2rx 1 + ∂x m + r2 + 2rq − 2rpx + p2 + 2rx p ¡¡ ¢−1 ¢2 ¡ ¡ ¢−1 ¢¡ ¡ ¡¡ ¢−1 ¡ ¢−1 ¢2 +2r 1 + ∂x m − 2rx + 1 + ∂x m 1 + ∂x 2rm − i 1 + ∂x m ¡ ¢−1 ¢¢ −2rx 1 + ∂x m + r2 + 2rq − 2rpx + p2 + 2rx p − 2r(qr + p2 ) and so on. The matrix CH hierarchy is the bi-Hamiltonian hierarchy on (M, P1 , P2 ) R given by the functionals Hj = hj dx, for j ≥ 1. This corresponds to putting H(λ) = P −j H and F (λ) = 4λ1 2 in equation (23). jλ j≥1 Integrating both sides of (28) with respect to x we obtain Z Z 1 h1 dx = − (r + q)dx = H1 . 2 Moreover, using the identity 1 m = (r − q) − (r + rx ) + px = (1 − ∂x2 )γ − (1 + ∂x )r , 2 that is, (1 + ∂x )−1 m = γ − γx − r , it is easy to show that. Z. Z h2 dx =. (p2 − γ 2 − γx2 )dx. coincides with the first Hamiltonian H2 of the vector field (15).. 5. Conclusions. In this paper we showed that the Camassa-Holm equation has a property that seems to be a common feature of integrable PDEs, namely, that it can be interpreted as a reduction of a member of a matrix hierarchy. Indeed, we constructed a bi-Hamiltonian hierarchy for the Poisson pair (P1 = [ · , S], P2 = ∂x + [ · , A]), starting from a Casimir of P2 and giving rise to the scalar CH hierarchy after a (bi-Hamiltonian) reduction. In particular, this allowed us to find a 3-component extension of the Camassa-Holm equation. We conclude with two natural developments of our study. First of all, the restriction of the matrix CH hierarchy to the symplectic leaf S, defined by the constraint p2 + qr = 0, is by construction a 2-component extension of the CH hierarchy. Is there a parametrization of S showing that this extension coincides with the (bi-Hamiltonian) one discussed in [12, 13, 17]? If the answer is affirmative, how can the bi-Hamiltonian structure on M be reduced on S? Secondly, the¡ scalar ¢ CH hierarchy has a negative (or local) counterpart, generated rd by a Casimir of P1 q = 2(2q∂x + qx ). In order to find a local matrix CH hierarchy, one 10.

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