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solv-int/9601001 Hamiltonian Structures for the Generalized Dispersionless KdV Hierarchy 6 9 9 1 J. C. Brunelli* n a J Departamento de F´ısica Matemática 5 1 Instituto de F´ısica da Universidade de São Paulo v 1 C.P. 66318, CEP 05389-970 0 0 São Paulo, SP - BRAZIL 1 0 6 9 / t n i - v l o Abstract s : v i X We study from a Hamiltonian point of view the generalized dispersionless KdV hier- r a archy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and third Hamiltonian structures are calculated directly from the r-matrix approach. Since the third structure is not related recursively with the first two ones the generalized dispersionless KdV hierarchy can be characterized as a truly tri-Hamiltonian system. * Work supported by CNPq. E-mail address: [email protected] 1 1. Introduction The representation of nonlinear integrable equations by a system of linear equations is due to Lax and have been studied extensively in the past [1-3]. An interesting class of nonlinear equations are the so called dispersionless Lax equations which are the quasi- classical limit of ordinary Lax equations. This quasi-classical limit corresponds to the solutions which slowly depend on the variables x,t. Let us take the KdV equation 4u = u +6uu (1.1) t xxx x and after dropping out the dispersive term (or make the substitution ∂ → ǫ∂ , ∂ → ǫ∂ ∂t ∂t ∂x ∂x and ǫ → 0) we end up with the equation 3 u = uu (1.2) t x 2 This is the Riemann equation (also called inviscid Burgers’, Hopf etc.). Solutions of (1.2) can be written through the implicit form [4] u = f(x−ut) (1.3) and this dependence gives rise to the breaking of the wave shape leading to a transition from conservative to dissipative behaviour [5]. As it is well known the balancing between the dispersive and nonlinear terms in (1.1) is responsible for the soliton solutions and the integrabilityof(1.1). Whatisinterestingisthattheevolutionequation(1.2),atleastbefore the breaking of its wave solutions, is a integrable Hamiltonian system much like the KdV system(1.1). Infactthisbehaviourwasobservedforthedispersionlessequationsdescribing an incompressible nonviscous fluid with a free surface in the approximation of long waves, the Benney’s equation [6]. A Hamiltonian structure for this equation was obtained and studied by Kupershmidt and Manin [7], Manin [8], Lebedev and Manin [9], and Lebedev [10]. As point out by Olver and Nutku [11] equation (1.2) has an infinite sequence of zero order (the highest order derivative which a F(u,u ,...) depends) conserved charges x 1 1 2 3 H = dxu, H = dxu , H = dxu ,... (1.4) 1 3 5 4 8 Z Z Z 2 and has three first-order Hamiltonian structures given by D =2∂ 1 D =u∂ +∂u 2 (1.5) 2 2 D =u ∂ +∂u 3 These three Hamiltonian operators are compatible in the Magri’s sense [12,13] making (1.2) a tri-Hamiltonian system, i.e., equation (1.2) can be written in three Hamiltonian forms δH δH 3 δH 5 3 1 u = D = D = D (1.6) t 1 2 3 δu δu 4 δu However, it is important to point out that since D 6= D D−1D (1.7) 3 2 1 2 the Hamiltonian operators are not trivially related. Therefore, the Hamiltonian structures (1.5) make the Riemann equation a truly tri-Hamiltonian system. This is to be contrasted with the Kupershmidt’s equations for the dispersive water wave equations [14]. In that situation the three Hamiltonian structures are related by a unique recursion operator. It follows, for instance from Magri’s theorem [12,13], that the Hamiltonians (1.4) are in involution with respect to any of the three Poisson brackets δH δH n m {H ,H } = dx D = 0, i = 1,2,3,... (1.8) n m i i δu δu Z Whence, (1.2) is an integrable Hamiltonian system. We can associate with (1.2) a Lax representation using the Chen et al approach [15]. The Lax pair is ∂L = [L,B] (1.9) ∂t where L is the recursion operator obtained from (1.5) L = D D−1 = u+∂u∂−1 (1.10) 2 1 3 and B is the Frechét derivative obtained through a linearization of the Riemann equation (1.2) 3 d 3 v = (u+ǫv)(u+ǫv) = ∂uv = Bv (1.11) t x 2dǫ ǫ=0 2 (cid:12) Also, there is another Lax representation (1.9) w(cid:12)ith (cid:12) L =∂ +u 3 (1.12) 2 B = u 4 Although we have Lax representations for the Riemann equation we do not know how to use them for an inverse scattering problem or use the pseudo-differential operator algebra to obtain its Hamiltonian structures through the Gelfand-Dickey approach [3]. However, Lebedev [10] has notice that for the case of Benney’s equation an alternative Lax representation is possible. This Lax representation is called dispersionless Lax equation and was also considered by Krichever [16] in his studies about topological minimal models. In this paper we will use this dispersionless Lax representation and the algebraic setup behind it to derive the Hamiltonian structures of the Riemann equation and the hierarchy of equations which contains it as the first nontrivial equation. This paper is organized as follows. In section 2 we introduce the generalized dispersionless KdV hierarchy of equations and we show that the Riemann equation belongs to it. In section 3 we obtain the first Hamiltonian structure. The second and third Hamiltonian structures for the generalized dispersionless KdV hierarchy are obtained in section 4 by the r-matrix method. We present our conclusions in section 5. 2. Dispersionless KdV Hierarchy The dispersionless KdV equation (1.2) can be obtained directly, bypassing the dispersionless limit of (1.1). Let E be the polynomial of degree n in p [16] n n E = pn +u pn−1 +u pn−2 +···+u = u pi (2.1) n −1 0 n−2 n−i−2 i=0 X whereu = 1andthepolynomialcoefficients u arefunctions ofthevariablexandvarious −2 i time variables t (k = 1,2,3,...). We denote A and A the parts of the Laurent poly- k + − nomial A containing nonnegative and negative powers of p respectively. The generalized 4 dispersionless KdV hierarchy is given by the Lax equation ∂E n = {(Ek/n) ,E } = {E ,(Ek/n) } (2.2) ∂t n + n n n − k where the bracket is defined [9,10] to be {A,B} = ∂ A∂ B −∂ B∂ A (2.3) p x p x and Ek/n is the kth power of the Laurent polynomial E1/n satisfying (E1/n)n = E . n n n n First, let us denote the degree of a Laurent polynomial of the form A = a pi as i i P degA ≡ maximum i for which a 6= 0 (2.4) i and observe that the bracket reduces the degree of a polynomial by one unit. Therefore, deg{A,B} = degA+degB −1 (2.5) From (2.2) we have ∂E deg n = deg{E ,(Ek/n) } ≤ n−1−1 = n−2 (2.6) ∂t n n − k and when we compare with (2.1) we conclude that ∂u −1 = 0 (2.7) ∂t k for any k. Now, for general Laurent polynomials of the form +∞ A = a (x)pi (2.8) i i=−∞ X we define the residue as the coefficient of the p−1 term ResA = a (2.9) −1 and the Adler trace [17] as TrA = dxResA (2.10) Z 5 For general Laurent polynomials A = +∞ a pi and B = +∞ b pi it is straightfor- i=−∞ i i=−∞ i ward to show that P P +∞ Res{A,B} = i(a b )′ (2.11) i −i i=−∞ X which implies Tr{A,B} = 0 (2.12) Also, it follows easily from (2.12) the useful property Tr(A{B,C}) = Tr(B{C,A}) (2.13) Let us note from (2.2) that we can write m/n ∂E n = {(Ek/n) ,Em/n} (2.14) ∂t n + n k for an arbitrary integer m. Taking the trace of (2.14) and after using (2.12) we obtain ∂ Tr(Em/n) = 0 (2.15) ∂t n k Thus, we define the conserved charges as n H = Tr(Em/n), m = 1,2,3... (2.16) m m n Now let us show that the flows given by (2.2) commute. Let be ∂2E ∂ n = {(Ek/n) ,E } ∂t ∂t ∂t n + n ℓ k ℓ ={(∂Ek/n/∂t ) ,E }+{(Ek/n) ,{(Eℓ/n) ,E }} (2.17) n ℓ + n n + n + n ={{(Eℓ/n) ,Ek/n} ,E }+{(Ek/n) ,{(Eℓ/n) ,E }} n + n + n n + n + n where we have used (2.14). Now using the Jacobi identity ∂2E n ={{(Eℓ/n) ,Ek/n} ,E }+{{(Ek/n) ,(Eℓ/n) },E }+{(Eℓ/n) ,{(Ek/n) ,E }} ∂t ∂t n + n + n n + n + n n + n + n ℓ k ={{(Eℓ/n) ,(Ek/n) } ,E }+{(Eℓ/n) ,{(Ek/n) ,E }} n + n − + n n + n + n ={{Eℓ/n −(Eℓ/n) ,(Ek/n) } ,E }+{(Eℓ/n) ,{(Ek/n) ,E }} n n − n − + n n + n + n ={{Eℓ/n,Ek/n −(Ek/n) } ,E }+{(Eℓ/n) ,{(Ek/n) ,E }} n n n + + n n + n + n ={{(Ek/n) ,Eℓ/n} ,E }+{(Eℓ/n) ,{(Ek/n) ,E }} n + n + n n + n + n ∂2E n = ∂t ∂t k ℓ (2.18) 6 Therefore, the hierarchy of equations (2.2) has an infinite number of conserved laws (2.16) and an infinite number of commuting flows and can be formally be considered integrable. Let us illustrate these results for (2.1) with n = 2 and u = 0. We obtain for E ≡ E −1 2 and u ≡ u 0 2 E =p +u 1 1 1 5 E1/2 =p+ up−1 − u2p−3 + u3p−5 − u4p−7 +... 2 8 16 128 3 3 E3/2 =p3 + up+ u2p−1 +... 2 8 (2.19) 5 15 5 E5/2 =p5 + up3 + u2p+ u3p−1 +... 2 8 16 7 35 35 35 E7/2 =p7 + up5 + u2p3 + u3p+ u4p−1 +... 2 8 16 128 . . . From ∂E = {(Ek/2) ,E} (2.20) + ∂t k we get the hierarchy of equations ∂u =u x ∂t 1 ∂u 3 = uu x ∂t 2 3 ∂u 15 = u2u (2.21) x ∂t 8 5 ∂u 35 3 = u u x ∂t 16 7 . . . The conserved charges from (2.16) are 2 H = Tr(Em/2) (2.22) m m 7 and results in H = dxu 1 Z 1 2 H = dxu 3 4 Z 1 H = dxu3 (2.23) 5 8 Z 5 4 H = dxu 7 64 Z . . . Thus, from (2.21) we get the Riemann equation as the first nonlinear equation in the hierarchy. Also, the charges (2.23) are exactly the Hamiltonians (1.4). Then we will call (2.20) and (2.2) the dispersionless KdV (Riemann) hierarchy and the generalized dispersionless KdV (Riemann) hierarchy, respectively. 3. First Hamiltonian Structure Now we will derive in a systematic way the first Hamiltonian structure associated with the generalized dispersionless KdV hierarchy of equations (2.2). Particularly, as an example we will obtain the first Hamiltonian structure in (1.5) for the Riemann equation. Here we will follow the Drinfeld and Sokolov approach [18] as in [19] for the usual KdV hierarchy. Let us observe that n−1 (Ek/n) = Ek/n −(Ek/n) = Res Ek/npi−1 p−i +O(p−n) (3.1) n − n n + n Xi=1 (cid:16) (cid:17) k/n andeven though(E ) has aninfinitenumber oftermsthe onlyones thatwillcontribute n − to give dynamical equations are the ones up to degree p−n+1. Since δE pi = dx n (3.2) δu n−i−2 Z we rewrite (3.1) as n−1 δE (Ek/n) = Res Ek/n dx n p−i +O(p−n) n − n δu i=1 (cid:18) Z n−i−1(cid:19) X (3.3) n−1 δH = n+k p−i +O(p−n) δu i=1 n−i−1 X 8 where H is given by (2.16). n+k Using (3.3) in (2.2) we get n−1 ∂E δH n = {E , n+k p−i} (3.4) n ∂t δu k i=1 n−i−1 X since the terms of order O(p−n) in (3.3) do not contribute. Let us introduce the dual to E n n Q = q p−i (3.5) n−i−1 i=1 X where the q’s are assumed to be independent of the u’s. This yields a linear functional n−1 TrE Q = dx u q (3.6) n n−i−2 n−i−2 Z i=0 X Also, let us note that n−1 n−1 δH δH n+k p−i = dy n+k V (x,y) (3.7) i δu δu (y) i=1 n−i−1 i=1 Z n−i−1 X X where V (x,y) ≡ δ(x−y)p−i (3.8) i and which gives TrE V (x,y) = u (y) (3.9) n i n−i−1 We have thus from (3.4) using (3.9) and (2.13) n−1 ∂ δH n+k TrE Q = dyTrE {V (x,y),Q(x)} (3.10) n n i ∂t δu (y) k n−i−1 i=1 Z X If we want to write this equation in Hamiltonian form as ∂ TrE Q ={TrE Q,H } n n n+k 1 ∂t k n−1 (3.11) δH n+k = dy{TrE Q,TrE V } n n i 1 δu (y) i=1 Z n−i−1 X (where we have used (3.9)) we get after comparing it with (3.10) {TrE Q,TrE V } = TrE {V ,Q} (3.12) n n i 1 n i 9 In this way, the dispersionless Lax equation (2.2) can be written in Hamiltonian form with respect to the first Poisson bracket {TrE Q,TrE V} = TrE {V,Q} (3.13) n n 1 n for any dual Q and V relative to E . As an example let be n 2 E ≡ E = p +u p+u (3.14) 2 −1 0 with the duals V =v p−2 +v p−1 −1 0 (3.15) Q =q p−2 +q p−1 −1 0 We get that TrE{V,Q} = 2 dxq v′ (3.16) 0 0 Z and TrEQ = dx(u q +u q ) −1 −1 0 0 Z (3.17) TrEV = dx(u v +u v ) −1 −1 0 0 Z Now, (3.13) yields {u (x),u (y)} =0 −1 −1 1 {u (x),u (y)} =0 (3.18) −1 0 1 {u (x),u (y)} =2∂δ(x−y) 0 0 1 and constraining (3.14) to E = p2+u, where u ≡ u , we must set u = 0. We finally get 0 −1 from (3.18) {u(x),u(y)} = 2∂δ(x−y) = D δ(x−y) (3.19) 1 1 which is the first Hamiltonian structure for the Riemann equation (1.2). We now could try to write the dispersionless Lax equationin otherHamiltonianforms. However, in the next section we will use the algebraic structure behind the dispersionless hierarchy and apply the r-matrix formalism to obtain the other Hamiltonian structures. 10