Journal of Nonlinear MathematicalPhysics 1999, V.6, N 1, 5–12. Letter Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy Zhimin JIANG 9 9 Department of Mathematics, Shangqiu Teachers College, Shangqiu 476000, China 9 1 Received October 16, 1998; Accepted December 03, 1998 n a Abstract J 1 An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the ] I hierarchy is established by using the trace identity. The isospectral problem is non- S linearized as to be finite-dimensional completely integrable systems in Liouville sense . n under Neumann and Bargmann constraints. i l n [ 1 Introduction 1 v A major difficulty in theory of integrable systems is that there is to date no completely 1 0 systematic method for choosing properly an isospectral problem ψx = Mψ so that the 2 zero-curvature representation M −N +[M,N]= 0 is nontrivial. By inserting a reference t x 1 function into AKNS and WKI isospectral problems, we have obtained successfully two 0 9 new hierarchies [1, 2]. 9 The coupled KdV hierarchy associated with the isospectral problem / n 1 1 i l − λ+ u −v n 2 2 : ψx = Mψ, M =   (1.1) v 1 1 i  1 λ− u  X  2 2    r a is discussed by D. Levi, A. Sym and S. Wojciechowsk [3]. The isospectral problem (1.1) has been nonlinearized as finite-dimensional completely integrable systems in Liouville sense [4]. In this paper, we introduce the eigenvalue problem 1 1 − λ+ u −v 2 2 ψx = Mψ, M =  , (1.2) 1 1 f(v) λ− u   2 2     Copyright (cid:13)c 1999 by Z. Jiang 6 Z. Jiang where u and v are two scalar potentials, λ is a constant spectral parameter and f(v) called reference function is an arbitrary smooth function. The bi-Hamiltonian structure of the corresponding hierarchy is established by using the trace identity [5, 6]. Since the reference function f(v) in (1.2) can be chosen arbitrarily, many new hierarchies and their Hamiltonian forms are obtained.When f = (−v)β (β ≥ 0), the isospectral problem (1.2) is nonlinearizedasfinite-dimensionalcompletely integrable sysstemsinLiouvillesenseunder Neumann and Bargmann constraints between the potentials and eigenfunctions. 2 Preliminaries Consider the adjoint representation of (1.2) ∞ a b a b N = MN −NM, N = = i i λ−j (2.1) x c −a c −a i i (cid:18) (cid:19) j=0(cid:18) (cid:19) X which leads to 1 c = b = 0, a = − α (constant), (2.2) 0 0 0 2 c = αf(v), b = −αv, a = 0, (2.3) 1 1 1 c = α(f′(v)v +uf(v)), b = α(v −uv), a = −αvf(v), (2.4) 2 x 2 x 2 a = −∂−1(vc +f(v)b ), (2.5) j j j c f(v) c c 1 = α , j+1 = L j , j = 1,2,..., (2.6) b −v b b 1 j+1 j (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) d where ∂ = , ∂∂−1 = ∂−1∂ =1, dx ∂+u+2f∂−1v 2f∂−1f L = . −2v∂−1v −∂+u−2v∂−1f ! It is easy from (1.2) and (2.1) to calculate that ∂M ∂M ∂M tr N = −a, tr N = a, tr N = −c+f′(v)b. ∂λ ∂u ∂v (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Noticing the trace identity [5, 6] δ δ ∂ , (−a)= (a,−c+f′(v)b), δu δv ∂λ (cid:18) (cid:19) hence we deduce that δ δ a (1) (2) j+1 , H = G ,G , H = , (2.7) δu δv j j−2 j−2 j (cid:18) (cid:19) (cid:16) (cid:17) where G(1) = a , G(2) = −c +f′(v)b . (2.8) j−2 j j−2 j j Neumann and Bargmann Systems and the Coupled KdV Hierarchy 7 3 The hierarchy and its Hamiltonian structure Let ψ satisfy the isospectral problem (1.2) and the auxiliary problem A B ψ = Nψ, N = , (3.1) t C −A (cid:18) (cid:19) where m−1 m m A = A + a λm−j, B = b λm−j, C = c λm−j. m j j j j=0 j=1 j=1 X X X The compatible condition ψ = ψ between (1.1) and (3.1) gives the zero-curvature xt tx representation M −N +[M,N]= 0, from which we have t x A = w(∂ +u)c +wf′(v)(∂ −u)b , m m m u c c t = θ L m = θ m+1 , (3.2) v 0 b 0 b t m m+1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1 where w = (vf′(v)+f)−1, 2 2∂w −2∂wf′(v) θ = . (3.3) 0 2wv 2wf (cid:18) (cid:19) By (2.6) we know that Eqs.(3.2) are equivalent to the hierarchy of nonlinear evolution equations u αf(v) t = θ Lm , m = 1,2,.... (3.4) v 0 −αv t (cid:18) (cid:19) (cid:18) (cid:19) Let the potentials u and v in (1.2) belong to the Schwartz space S(−∞,+∞) over (−∞,+∞). Noticing (2.5) and (2.8) we get (1) cj = θ Gj−2 , θ = −2wf′(v)∂ −2wf . (3.5) (cid:18) bj (cid:19) 1 G(j2−)2  1 (cid:18) −2w∂ 2wv (cid:19)   Then the recursion relations (2.5), (2.6) and the hierarchy (3.2) can be written as 1 G = − α(1,0)T, G = −α(0,vf′(v)+f)T, G = −α(vf,uf +uvf′(v))T, −2 −1 0 2 KG = JG , (3.6) j−1 j (u ,v )T = JG = KG , (3.7) t t m−1 m−2 where J = θ θ and K =θ Lθ are two skew-symmetric operators, 0 1 0 1 0 −2∂w K K J = , K = 11 12 , −2w∂ 0 K K 21 22 (cid:18) (cid:19) (cid:18) (cid:19) 8 Z. Jiang in which K = −2∂−4∂w(∂f′(v)+f′(v)∂)w∂, 11 K = −2∂wu+4∂w(f′(v)∂v−∂f)w,  12  K21 = −2wu∂ +4w(f∂ −v∂f′(v))w∂,  K = −4w(v∂f +f∂v)w. 22    From (2.7) we obtain the desired bi-Hamiltonian form of (3.7) δ δ ut = J δu H = K δu H . (3.8) v  δ  m+1  δ  m t (cid:18) (cid:19)  δv   δv      4 Nonlinearization of the isospectral problem Let λ and ψ(x) = (q (x),p (x))T be eigenvalue and the associated eigenfunction of (1.2). j j j δλ δλ j j Through direct verification we know that the functional gradient ∇ λ = , (u,v) j δu δv (cid:18) (cid:19) satisfies ∇ λ = q p ,−p2−f′(v)q2 , (4.1) (u,v) j j j j j (cid:0) (cid:1) p2 p2 p2 θ ∇λ = j , L j = λ j (4.2) 1 j −q2 −q2 j −q2 (cid:18) j (cid:19) (cid:18) j (cid:19) (cid:18) j (cid:19) in view of (1.2). Substituting the first expression of (4.2) into the second expression and acting with θ upon once, we have 0 K∇λ = λ J∇λ . (4.3) j j j So, the Lenard operator pair K, J and their gradient series G satisfy the basic conditions j (3.6) and(4.3) given inRefs.[7, 8]forthenonlinearization oftheeigenvalue problem(1.2). Proposition 4.1. When f(v) = (−v)β (β ≥ 0), the isospectral problem (1.2) can be nonlinearized as to be a Neumann system. N In fact, the Neumann constraint G | = ∇λ gives −1 α=1 j j=1 P hq,pi = 0,hp,pi = (β +1)(−v)β +β(−1)β−1hq,qi. (4.4) By differentiating (4.4) with respect to x and using (1.2), we have 1 hΛp,pi hΛq,qi u= +β , β+1 hp,pi hq,qi (4.5)  (cid:18) (cid:19)   v = hq,qi.   Neumann and Bargmann Systems and the Coupled KdV Hierarchy 9 Substituting (4.5) into the equations for the eigenfunctions 1 1 − λ + u −v qjx = 2 j 2 qj , j = 1,...,N, (4.6) p  1 1  p (cid:18) jx (cid:19) (−v)β λ − u (cid:18) j (cid:19) j  2 2    we obtain the Neumann system 1 1 hΛp,pi hΛq,qi q = − Λq−hq,qip+ +β q, x 2 2(β +1) hp,pi hq,qi  (cid:18) (cid:19)  1 1 hΛp,pi hΛq,qi (4.7)  px = Λp+hp,piq− +β p,  2 2(β +1) hp,pi hq,qi  (cid:18) (cid:19)  hp,pi = (−1)βhq,qiβ, hq,pi = 0.     wherep = (p ,...,p )T, q = (q ,...,q )T, Λ = diag(λ ,...,λ ), and h,i stands for the 1 N 1 N 1 N canonical inner product in RN. Proposition 4.2. When f(v) = (−v)β (β ≥ 0), the isospectral problem (1.2) can be nonlinearized as to be a Bargmann system. N In fact, the Bargmann constraint G | = ∇λ gives 0 α=1 j j=1 P 1 − β β − 1 u= hp,pihq,pi β+1 − hq,qihq,pi β+1, β+1 β +1 (4.8)   1  v = −hq,piβ+1. Substituting (4.8) into (4.6), we obtain the finite-dimensional Hamiltonian system 1 1 1 − β qx = − Λq+hq,piβ+1p+ hp,pihq,pi β+1q 2 2(β +1)   − β hq,qihq,pi−β+11q = ∂H,  2(β +1) ∂p   (4.9)   px = 1Λp− 1 hp,pihq,pi−ββ+1p+hq,piββ+1 2 2(β +1)   + β hq,qihq,pi−β+11p = −∂H.  2(β +1) ∂q    The Hamiltonian is  1 1 1 1 β H = − hΛq,pi+ hp,pihq,piβ+1 − hq,qihq,piβ+1. 2 2 2 5 Integrability of the Neumann system The Poisson brackets of two functions in symplectic space (R2N,dp∧dq) are defined as N ∂F ∂G ∂F ∂G (F,G) = − = hF ,G i−hF ,G i. q p p q ∂q ∂p ∂p ∂q j j j j j=1(cid:18) (cid:19) X 10 Z. Jiang The functions defined by (m = 0,1,2,...) 1 1 hΛiq,qi hΛiq,pi F = − hΛm+1q,pi− m 2 2 (cid:12) hΛjp,qi hΛjp,pi (cid:12) i+Xj=m(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) are in involution in pairs (see, [9]). (cid:12) (cid:12) Consider the Moser constraint on the tangent bundle 1 TSN−1 = (p,q) ∈ R2N|F = hq,pi = 0, G = (hp,pi−(−1)βhq,qiβ)= 0 . 2(β +1) (cid:26) (cid:27) Through direct calculations we have (F,F ) = 0, (F,G) = hp,pi, m 1 (F ,G) = − hΛm+1p,pi+(−1)ββhq,qiβ−1hΛm+1q,qi . m 2(β +1) (cid:16) (cid:17) Thus the Lagrangian multipliers are (F ,G) 1 hΛm+1p,pi hq,qiβ−1 µ = m = − +(−1)ββ hΛm+1q,qi . m (F,G) (β+1) hp,pi hp,pi (cid:18) (cid:19) Since F = 0 on the tangent bundle TSN−1, the restriction of the canonical equation of H∗ = F −µ F on TSN−1 is 0 0 qx = F0,p−µ0Fp|TSN−1, ( px = −F0,q +µ0Fq|TSN−1 which is exactly the Neumann system (4.7). Theorem 5.1. The Neumann system (4.7) (TSN−1,dp∧dq|TSN−1,H∗ = F0 −µ0F) is completely integrable in Liouville sense. Proof. Let F∗ = F −µ F, m = 1,...,N −1, then it is easy to verify (F∗,F∗) = 0 on m m m k l TSN−1. Hence {F∗} is an involutive system. m 6 Integrability of the Bargmann system Let N B2 kj Γ = , (6.1) k λ −λ k j jX=1 j6=k where B = p q −p q , we have (see Refs. [9, 10]) kj k j j k Lemma 6.1. hq,pi,p2 = 2p2, hq,pi,q2 = −2q2, (6.2) l l l l (cid:0) (cid:1) (cid:0) (cid:1) Neumann and Bargmann Systems and the Coupled KdV Hierarchy 11 −4B −4B p2,Γ = lk p p , q2,Γ = lk q q , k l λ −λ k l k l λ −λ k l l k l k (6.3) (cid:0) (cid:1) (cid:0) (cid:1) −2B lk (q p ,Γ )= (p q +q p ). k k l k l k l λ −λ l k Lemma 6.2. (Γ ,Γ ) = (hq,pi,Γ ) = (hq,pi,q p )= 0, (6.4) k l l l l p2,p2 = q2,q2 =(q p ,q p ) = 0, (6.5) k l k l k k l l (cid:0)q p ,p(cid:1)2 =(cid:0) 2p p(cid:1)δ , q2,p2 = 4q p δ , q2,p q =2q q δ . (6.6) k k l k l kl k l k l kl k l l k l kl (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Proposition 6.1. Let 1 1 1 β 1 1 Ek = 2hq,piβ+1p2k − 2hq,piβ+1qk2− 2λkqkpk − 2Γk, the E ,...,E constitute an N-involutive system. 1 N Proof. Obviously (E ,E ) = 0 for k = l. Suppose k 6= l, in virtue of (6.4)–(6.6) and the k l property of Poisson bracket in (R2N,dp∧dq), we have 1 1−β 1 1−β 4(Ek,El) = β+1p2khq,piβ+1 hq,pi,p2l + β +1p2lhq,piβ+1 p2k,hq,pi (cid:0) (cid:1) (cid:0) (cid:1) 1 β 1 − β+1p2k hq,pi,ql2 − β+1ql2 p2k,hq,pi −hq,piβ+1 p2k,Γl (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 1 β 1 −hq,piβ+1 Γk,p2l − β+1qk2 hq,pi,p2l − β +1p2l qk2,hq,pi (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) β β−1 β β−1 + q2hq,piβ+1 hq,pi,q2 + q2hq,piβ+1 q2,hq,pi β+1 k l β +1 l k (cid:0) (cid:1) (cid:0) (cid:1) β β +hq,piβ+1 qk2,Γl +hq,piβ+1 Γk,ql2 +λk(qkpk,Γl)+λl(Γk,qlpl). Substituting (6.2) and (6.3) i(cid:0)nto th(cid:1)e above equa(cid:0)tion yi(cid:1)elds (Ek,El)= 0. Consider a bilinear function Q (ξ,η) on RN: z N ∞ ξ η Q (ξ,η) = h(z−Λ)−1ξ,ηi = k k = z−m−1hΛmξ,ηi. z z−λ k k=1 m=0 X X The generating function of Γ is (see, [9, 10]) k N Qz(q,q) Qz(q,p) Γk = . (cid:12) Qz(p,q) Qz(p,p) (cid:12) z−λk (cid:12) (cid:12) Xk=1 (cid:12) (cid:12) Hence(cid:12)the generating funct(cid:12)ion of Ek is (cid:12) (cid:12) 1 1 1 β 1 hq,piβ+1Qz(p,p)− hq,piβ+1Qz(q,q)− Qz(Λq,p) 2 2 2 (6.7) N 1 Qz(q,q) Qz(q,p) Ek − = . 2(cid:12) Qz(p,q) Qz(p,p) (cid:12) z−λk (cid:12) (cid:12) Xk=1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 12 Z. Jiang Substituting the Laurent expansion of Q and z ∞ (z−λ )−1 = z−m−1λm k k m=0 X in to both sides of (6.7) respectively, we have Proposition 6.2. Let N F = λmE , m = 0,1,2,... m k k k=1 X then 1 1 1 β 1 F0 = hq,piβ+1hp,pi− hq,piβ+1hq,qi− hΛq,pi, 2 2 2 1 1 1 β Fm = hq,piβ+1hΛmp,pi− hq,piβ+1hΛmq,qi 2 2 1 1 m hΛj−1q,qi hΛj−1q,pi − hΛm+1q,pi− . 2 2 (cid:12) hΛm−jp,qi hΛm−jp,pi (cid:12) Xj=1(cid:12) (cid:12) (cid:12) (cid:12) Moreover, (F ,F ) = 0. (cid:12) (cid:12) k l (cid:12) (cid:12) Hence we arrive at the following theorem. Theorem 6.1. The Bargmann system defined by (4.9) is completely integrable in Liouville sense in the symplectic manifold (R2N,dp∧dq). Acknowledgement I am very grateful to Professor Cao Cewen for his guidance. This project is supported by the Natural Science Fundation of China. References [1] Jiang Z.M., Physics Letters A, 1997, V.228, 275–278. [2] Jiang Z.M., Physica A, 1998, V.253, 154–160. [3] LeviD., Sym. A.and Wojciechowsk S., Phys. A: Math. Gen., 1983, V.16, 2423–2432. [4] Cao C.W. and Geng X.G., J. Phys. A: Math. Gen., 1990, V.23, 4117–4125. [5] Tu G.Z., J. Math. Phys., 1989, V.30, 330–338. 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