A generalized Boite–Pempinelli–Tu (BPT) hierarchy and its bi-Hamiltonian structure

Physics Letters A 317 (2003) 280–286 www.elsevier.com/locate/pla

A generalized Boite–Pempinelli–Tu (BPT) hierarchy and its bi-Hamiltonian structure Yufeng Zhang a,b a Institute of Mathematics, Information School, Shandong University of Science and Technology, Taian 271019, PR China b Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences,

Beijing 100080, PR China Received 3 May 2003; accepted 27 August 2003 Communicated by A.R. Bishop

Abstract A subalgebra of loop algebra A˜ 1 is presented. It follows that a new Liouville integrable hierarchy with 6 potential functions, which possesses a bi-Hamiltonian structure, is obtained. Since it can reduce to Boite–Pempinelli–Tu (BPT) hierarchy, we call it a generalized BPT system. Two symplectic operators of the system obtained are directly derived from recurrence relations, not from a recurrence operator. As a reduction case, the famous KdV-mKdV equation is given.  2003 Elsevier B.V. All rights reserved. Keywords: Loop algebra; Integrable system; Hamiltonian structure

1. Introduction By making use of the method for the compatibility of Lax pairs, one has been obtained a host of integrable Hamiltonian hierarchies of evolution equations [1–3]. An efficient and straightforward method, called Tu scheme [4,5], is popular to get integrable systems by the compability of Lax pairs at present. Via use of Tu scheme, some celebrated hierarchies, such as AKNS hierarchy, BPT hierarchy, KN hierarchy, etc. were generated [4–7]. Professor Guo Fukui developed Tu scheme into the cases of high dimension subalgebras of loop algebra A˜ 1 , and found some intrinsic integrable Hamiltonian systems [8,9]. In this Letter, we construct a subalgebra of loop algebra A˜ 1 , which is different from any one in Refs. [8,9], so that a Liouville integrable system with 6 potential functions is presented. As its reduction, the well-known Boite–Pempinelli–Tu (BPT) hierarchy is presented. Therefore, we call the system a generalized BPT hierarchy. Furthermore, a reduction case of the generalized BPT hierarchy is the famous combined KdV-mKdV equation. In addition, two symplectic operators in the generalized BPT hierarchy are directly worked out by the recurrence relations obtained, not by the form recurrence K = J L, where J is a Hamiltonian operator, L is a recurrence operator. Also do we prove the generalized BPT hierarchy possesses a bi-Hamiltonian structure, and the conjugate operator of the recurrence operator L is a hereditary symmetry. E-mail address: zhang_ [email protected] (Y. Zhang). 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.08.057

Y. Zhang / Physics Letters A 317 (2003) 280–286

281

2. A subalgebra of loop algebra A˜ 1 and its application Let 1 h(n) = 2 f± (n) =




1 2

λ4n 0


0 −λ4n

0 ±λ4n+3

 , λ4n−1 0



1 ¯ h(n) = 2 ,




λ4n+2 0

0 −λ4n+2

 ,

1 e± (n) = 2




0 ±λ4n+1

λ4n−3 0

 ,

[e− (m), f− (n)] = 0,

¯ [h(m), f± (n)] = f∓ (m + n), [h(m), e± (n)] = f∓ (m + n), [h(m), e± (n)] = e∓ (m + n), ¯ ¯ [e− (m), e+ (n)] = h(m + n − 1), [e− (m), f+ (n)] = h(m + n), [h(m), f± (n)] = e∓ (m + n + 1), ¯ ¯ [f− (m), e+ (n)] = h(m + n), [f− (m), f+ (n)] = h(m + n), [h(m), h(n)] = [e+ (m), f+ (n)] = 0, deg h(n) = 4n,

¯ deg h(n) = 4n + 2,

deg e± (n) = 4n − 1,

deg f± (n) = 4n + 1

(1)

which is different from any one in Refs. [8,9]. Consider a linear isospectral problem λt = 0, ψ = (ψ1 , ψ2 )T , ψx = U ψ,
u2 +u3 u5 +u6  u4 1 u1 + λ 2 λ + λ3 + λ5 U= 6 λ3 + (u2 − u3 )λ − u5 +u −u1 − uλ42 λ ¯ = f+ (0) + u1 h(0) + u2 e+ (0) + u3 e− (0) + u4 h(−1) + u5 f− (−1) + u6 f+ (−1), rank U = rank ∂ = rank u1 = deg f+ (0) = 1,

rank u2 = rank u3 = 2,

rank u4 = 3,

rank u5 = rank u6 = rank λ = 4. Set V=

(2)

  ¯ am h(−m) + bm e+ (−m) + cm e− (−m) + dm f+ (−m) + em f− (−m) + fm h(−m) , m
0

substituting it into the stationary zero curvature equation Vx = [U, V ] yields am+1x = −cm+1 − u2 em+1 + u3 dm+1 + u5 bm − u6 cm , bmx = u1 cm − u3 am + u4 em − u5 fm , cmx = −fm+1 + u1 bm − u2 am + u4 dm − u6 fm , dm+1x = u1 em+1 − u3 fm+1 + u4 cm − u5 am , em+1x = −am+1 + u1 dm+1 − u2 fm+1 + u4 bm − u6 am , fm+1x = −u2 cm − em+1 + u3 bm + u5 dm − u6 em , a0 = c0 = e0 = d0 = f0 = 0, b0 = β, f1 = βu1 , 
β 2 1 3 e1 = β(−u1x + u3 ), a1 = β u1xx − u3x − u1 − u1 u2 + u4 , d 1 = − u1 , 2 2 
1 2 3 4 1 2 2 b1 = β −u1 u1xx + u1x − u1 u4 + u1 u2 + u1 + u3 + u1 u3x − u1x u3 , 2 8 2

(3)

282

Y. Zhang / Physics Letters A 317 (2003) 280–286


3 2 1 2 c1 = β −u1xx + u3xx + u1 u1x + (u1 u2 )x − u4x + u1x u2 − u2 u3 − u1 u3 + u5 , 2 2 rank am = 4m − 1, rank bm = rank cm = 4m, rank dm = rank em = 4m − 2,

rank fm = 4m − 3.

(4)

Note n     (n) V+ = λ4n V + = am h(n − m) + bm e+ (n − m) + cm e− (n − m) + dm f+ (n − m) m=0

(n)

 ¯ − m) , + em f− (n − m) + fm h(n

(n)

V− = λ4n V − V+ , then (3) can be written as   (n) (n)  (n) (n)  −V+x + U, V+ = V−x − U, V− .

(5)

The terms of the left-hand side in (5) are of degree
−4, while the terms of the right-hand side are of degree  −1. Therefore, the terms of both sides in (5) are of degree −4, −3, −2, −1. Thus,   (n) −V+x + U, V+(n) = fn+1 e− (0) + (dn+1 − u1 en+1 + u3 fn+1 )f+ (−1) ¯ + (en+1x + an+1 − u1 dn+1 + u2 fn+1 )f− (−1) + (fn+1x + en+1 )h(−1) + (an+1x + u2 en+1 − u3 dn+1 + cn+1 )h(−1). Taking a modified term of V+(n) as ∆n = −bn e+ (0) − cn e− (0), i.e., V (n) = V+(n) + ∆n , then the zero-curvature equation   Ut − Vx(n) + U, V (n) = 0 (6) admits the following Lax integrable system     u1 −cn u1 cn − bnx   u2       u1 bn − fn+1 − cnx   u3   ut =   =   −fn+1x − en+1 − u2 cn + u3 bn   u4       u5 −en+1x − an+1 + u1 dn+1 − u2 fn+1 + u4 bn u6 t −dn+1x + u1 en+1 − u3 fn+1 + u4 cn      an+1 an+1 0 0 0 0 1 0 0 0 0 −u1 −∂   dn+1   dn+1   0     1 0 0 −1 ∂ u1   −en+1   −e   0 =   = J1  n+1   0 0 1 −∂ u u f f      2 2 3 n+1 n+1       −1 u1 ∂ −u2 0 u4 −cn −cn 0 −∂ −u1 −u3 −u4 0 bn bn        −cn fn+1 0 0 1 0 0 0 fn+1 u1 cn − bnx 0 0   bn     bn   0 −∂ −u1 0        ∂ 0 0 0   −cn   u1 bn − fn+1 − cnx  1  −1 u1  −cn  = =   = J2  ,  0 0 0 −u6 −u5   an   u6 e n − u5 d n  2  0  an         u6 a n 0 0 0 u6 0 0 −en −en u5 a n 0 0 dn dn 0 0 0 u5 where J1 and J2 are Hamiltonian operators, which are directly derived from (4).

(7)

Y. Zhang / Physics Letters A 317 (2003) 280–286

283

Let 1 V= 2




a + f λ2 (b − c)λ + (d − e)λ3

+ d+e λ −a − f λ2 b+c λ3

 ,

a=



am λ−4m , . . . ,

m
0

then a direct calculation reads          ∂U ∂U ∂U 1 1 1 V, V, V, = a + f λ2 , = d + bλ−2 , = − e + cλ−2 , ∂u1 2 ∂u2 2 ∂u3 2    ∂U 1 V, = f + aλ−2 , ∂u4 2    ∂U 1 V, = [−2u4a + u2 (2c − b) + u3 (c − 2b)]λ−3 ∂λ 2   1 + u2 (2e − d) + u3 (e − 2d) − 2u4 f + 2c + b + (u5 + u6 )(b − c) λ−1 2    1 5 5 + d + 2e + (u5 + u6 )(d − e) λ − (u5 + u6 )(d + e)λ−7 − (u5 + u6 )(b + c)λ−9 . 2 2 2 (8) Inserting (8) into trace identity gives rise to     V , ∂U   ∂u1  ∂U     V , ∂u2         ∂U        V , ∂u3   δ ∂  γ ∂U  V, = λ−γ λ   . ∂U   δu ∂λ ∂λ   V ,      ∂u    ∂U4       V, ∂u5   ∂U  V , ∂u6

(9)

Comparing the coefficients of λ−4n−5 in (9) leads to  δ −2u4 an+2 + u2 (2cn+2 − bn+2 ) + u3 (cn+2 − 2bn+2 ) + u2 (2en+1 − dn+1 ) + u3 (en+1 − 2dn+1 ) δu  1 5 + (u5 + u6 )(bn+1 − cn+1 ) − 2u4 fn+1 + 2cn+1 + bn+1 − (u5 + u6 )(bn−1 + cn−1 ) 2 2   an+1  dn+1     −e  = (−4n − 4 + γ )  n+1  .  fn+1    −cn bn Again comparing the coefficients of λ−4n−3 yields   δ 1 5 dn+1 + 2en+1 + (u5 + u6 )(dn+1 − en+1 ) − (u5 + u6 )(dn−1 + en−1 ) δu 2 2

(10)

284

Y. Zhang / Physics Letters A 317 (2003) 280–286

 fn+1  bn     −cn  = (−4n − 2 + γ )  .  an    −en dn 

(11)

Substituting the initial values into (10) and (11) gives γ = 1. Therefore, we may present two Hamiltonian functions as follows   an+1  dn+1     −en+1  δH1 (n) ,  =  fn+1  δu   −cn bn  H1 (n) = −2u4 an+2 + u2 (2cn+2 − bn+2 ) + u3 (cn+2 − 2bn+2 ) + u2 (2en+1 − dn+1 ) + u3 (en+1 − 2dn+1 ) 1 + (u5 + u6 )(bn+1 − cn+1 ) − 2u4 fn+1 + 2cn+1 + bn+1 2  5 − (u5 + u6 )(bn−1 + cn−1 ) (4n + 3)−1 , 2  fn+1  bn     −cn  δH2 (n) , =   an  δu   −en dn   1 5 H2 (n) = dn+1 + 2en+1 + (u5 + u6 )(dn+1 − en+1 ) − (u5 + u6 )(dn−1 + en−1 ) (4n + 1)−1 . 2 2

(12)



(13)

Thus, the integrable system (7) has the following Hamiltonian structures δH1 (n) δH2 (n) = J2 . (14) δu δu From the recurrence relations (4), a recurrence operator is generated     an+1 l1 l2 l3 −u6 − u1 ∂ −1 u5 u1 ∂ −1 u1 u6 − ∂u6 u1 ∂ −1 u1 u5 − ∂u5  dn+1   −∂ −1 (u ∂ − u ) ∂ −1 u u ∂ −1 (u u − u )  −∂ −1 u5 ∂ −1 u1 u6 ∂ −1 u1 u5 1 3 1 3 1 2 4     −en+1   −∂ u u 0 u u5   3 2 6 =     1 0 0 0 0 0  fn+1      0 0 1 0 0 0 −cn 0 1 0 0 0 0 bn   fn+1  bn     −cn  = L (15) ,  an    −en dn ut = J1

Y. Zhang / Physics Letters A 317 (2003) 280–286

285

where l1 = ∂ 2 − u1 ∂ −1 u1 ∂ − u1 ∂ −1 u3 − u2 , l2 = −∂u3 + u1 ∂ −1 u1 u3 + u4 , l3 = u1 ∂ −1 u1 u2 − ∂u2 − u1 ∂ −1 u4 . It is easy to verify that J1 L = L∗ J1 = J2 holds. Hence, we conclude that (14) is Liouville integrable. Also can we prove L∗ = J2 J1−1 is a hereditary symmetry according the approach in Refs. [8,10]. In terms of (12), (13), (15), the system (7) or (14) can be written as

ut = J1 Sn = J1

δH1 (n) δH2 (n) = J1 LTn = J2 , δu δu

where Sn = (an+1 , dn+1 , −en+1 , fn+1 , −cn , bn )T , Tn = (fn+1 , bn , −cn , an , −en , dn )T . We can prove (16) is a bi-Hamiltonian structure of the system (7). In fact, let J = c1 J1 + c2 J2 , f = (f1 , f2 , f3 , f4 , f5 , f6 )T , g = (g1 , g2 , g3 , g4 , g5 , g6 )T , h = (h1 , h2 , h3 , h4 , h5 , h6 )T , then 

     f5 f3 F1 −u1 f5 − f6x    −f2x − u1 f3   F2        −f4 + f5x + u1 f6    −f + u1 f2 + f3x   F3  Jf = c1   + c2  1  =  , f3 − f4x + u2 f5 + u3 f6    −u6 f5 − u5 f6   F4        −f1 + u1 f2 + f3x − u2 f4 + u4 f6 u6 f4 F5 −f2x − u1 f3 − u3 f4 − u4 f5 u5 f4 F6     0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 −F1 0  0 0  0 0 −F1 0     0 0 0 F1  0 0 0 0  0 0  0 F1  J [Jf ] = c1   + c2  , 0 0 −F6 −F5  0 0 F2 F3  0 0 0 0     0 F1 0 −F2 0 F4 0 0 0 F6 0 0 0 0 0 F5 0 0 −F1 −F3 −F4 0 0 0

(16)

286

Y. Zhang / Physics Letters A 317 (2003) 280–286

     J [Jf ]g, h = F1 c1 (h5 g2 − g5 h2 + g6 h3 − h6 g3 ) + c2 (h3 g2 − h2 g3 ) + F2 c1 (h4 g5 − h5 g4 ) + F3 c1 (h4 g6 − g4 h6 ) + F4 c1 (h5 g6 − h6 g5 ) + F5 c2 (h6 g4 − h4 g6 ) + F6 c2 (h5 g4 − h4 g5 )  = c12 f6x h5 g4 − f6x h4 g5 + f5x h4 g6 − f5x g4 h6 + f4x h6 g5 − f4x h5 g6 + u1 (f5 h5 g4 − f5 h4 g5 + f6 h4 g6 − f6 g4 h6 ) + u2 (f5 h5 g6 − f5 h6 g5 ) + u3 (f6 h5 g6 − f6 h5 g5 ) + f5 h5 g2  − f5 g5 h2 + f5 g6 h3 − f5 h6 g3 + f4 g4 h6 − f4 h4 g6 + f3 h5 g6 − f3 h6 g5  + c1 c2 u1 (f2 h4 g6 − f2 g4 h6 + f2 h6 g4 − f2 h4 g6 ) + u2 (f4 h4 g6 − f4 h6 g4 ) + u3 (f4 h4 g5 − f4 h5 g4 ) + u4 (f5 h4 g5 − f5 h5 g4 + f6 h6 g4 − f6 h4 g6 ) + u5 (f6 h6 g5 − f6 h5 g6 ) + u6 (f5 h6 g5 − f5 h5 g6 ) + f1 g4 h6 − f1 h4 g6 + f3 h5 g2  − f3 g5 h2 + f3 g6 h3 − f3 h6 g3 + f5 h4 g5 − f5 h5 g4   + c22 f3 h3 g2 − f3 h2 g3 + u5 (f4 h5 g4 − f4 h4 g5 ) + u6 (f4 h6 g4 − f4 h4 g6 ) . Thus,        J [Jf ]g, h + J  [J g]h, f + J  [J h]f, g ∂ = c12 (f6 h5 g4 − f6 h4 g5 + f5 h4 g6 − f5 g4 h6 + f4 h6 g5 − f4 h5 g6 ) ∼ 0, ∂x which implies that {J1 , J2 } is a Hamiltonian operator pair. Therefore, (16) is a bi-Hamiltonian structure. A special case. Taking u4 = u5 = u6 = 0 in (4), we can deduce the following      u1 ∂ −u3 −u2 an δH1 (n) , ut n = u2 = u3 0 0 dn = J˜ δu u3 t n 0 0 −en u2 which is just the well-known BPT hierarchy [1]. Especially, taking n = 1, u1 = u, u2 = 1, u3 = u4 = u5 = u6 = 0 in (7), the celebrated combined KdV-mKdV equation is presented as follows
 3 2 ut = β uxxx − u ux − 2ux . 2

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

F. Magri, J. Math. Phys. 19 (1978) 1156. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transformation, SIAM, Phyladelphia, PA, 1981. C. Gu, et al., Soliton Theory and its Application, Zhejiang Publishing House of Science and Technology, 1990. G. Tu, J. Math. Phys. 30 (2) (1989) 330. W. Ma, Chinese Ann. Math. Ser. A 13 (1) (1992) 115. X. Hu, J. Phys. A: Math. Gen. 27 (1994) 2497. E. Fan, J. Math. Phys. 41 (11) (2000) 7769. F. Guo, Acta Math. Appl. Sinica 23 (2) (2000) 181. F. Guo, J. Systems Sci. Math. Sci. 22 (1) (2002) 36. W. Ma, J. Phys. A: Math. Gen. 23 (1990) 2707.