A finite-dimensional integrable system related to a new coupled KdV hierarchy

Physics Letters A 355 (2006) 452–459 www.elsevier.com/locate/pla

A finite-dimensional integrable system related to a new coupled KdV hierarchy Zhenyun Qin Institute of Mathematics, Fudan University, Shanghai 200433, PR China Received 8 August 2005; received in revised form 13 September 2005; accepted 19 September 2005 Available online 9 February 2006 Communicated by A.R. Bishop

Abstract A new 4×4 isospectral problem with three potentials and the corresponding hierarchy of nonlinear evolution equations are presented. Especially, a new coupled KdV equation is produced. Their generalized bi-Hamiltonian structures are also investigated by using the trace identity. Moreover, a new finite-dimensional Hamiltonian system is given through the nonlinearization of the corresponding Lax pair. Enough conserved integrals, which are in involution and functionally independent, are created by the Lax operator to guarantee Liouville integrability of the Hamiltonian system. © 2006 Elsevier B.V. All rights reserved.

1. Introduction It is an important task to find new integrable systems in mathematical physics. In recent years, the technique of the nonlinearization of Lax pairs [1–3] is a powerful tool for obtaining finite-dimensional integrable systems from infinite-dimensional ones. There are two main forms. The first one is binary nonlinearization which is powerful for the case of general matrix spectral problems by taking into account adjoint spectral problems whether (1 + 1)-dimensional or (2 + 1)-dimensional nonlinear evolution equations [10–18]. The second one is mono-nonlinearization which has successfully been applied to various soliton hierarchies associated with 2 × 2 zero-trace matrix spectral problems [1–9]. Furthermore, it paves an effective way to get the explicit expressions of algebro-geometric solutions by means of the Riemann theta functions with the help of the elliptic variables and the Abel–Jacobi coordinates [19–28]. However, with the tedious calculations, there is much difficulty with mono-nonlinearization methods in extending the theory of the case of the higher even-order matrix spectral problems. Although some papers have dealt with the case of 4 × 4 matrix spectral problem [29–31], they have seldom concerned with the t -flows and the functional independence of the conserved integrals. In this Letter we will present a similar extension of the coupled KdV hierarchy. It is worthwhile to mention that the detailed proof of the functional independence is given according to the method [12,13] because functional independence is very important but usually difficult to verify. The Letter is organized as follows. In Section 2, a hierarchy of new coupled KdV equation is derived. In Section 3, the generalized bi-Hamiltonian structures are investigated by using the trace identity. In Section 4, a finite-dimensional Hamiltonian system is generated with the aid of the mono-nonlinearization. In Section 5, the complete integrability of the system in Liouville sense is proved. 2. A hierarchy of new cKdV equations The new 4 × 4 spectral problem is first introduced as follows E-mail address: [email protected] (Z. Qin). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.09.089

Z. Qin / Physics Letters A 355 (2006) 452–459

⎛

φx = U (u, v, w, λ)φ,

⎞ 1 0 0 0 w 0⎟ ⎠. 0 0 1 0 u−λ 0

0 ⎜u − λ U (u, v, w, λ) = ⎝ 0 v

453

(2.1)

To obtain the hierarchy of nonlinear evolution equations associated with the matrix spectral problem (2.1), we first consider the stationary zero-curvature equation Vx − [U, V ] = 0,

V = (Vij )4×4 .

(2.2)

Let V12 + V34 = 2A, V14 = B, V32 = C, V34 = 2D,    −j −j Aj −1 λ , B = Bj −1 λ , C = Cj −1 λ−j . A= j 0

j 0

(2.3)

j 0

Substituting (2.3) into (2.2), we have 2V11 = ∂ −1 (wC − vB) − 2(A − D)x , V12 = 2A − 2D,

−1 V14 = B, w(4D − 2A) − Bx , 2V13 = ∂ 2V21 = wC + vB − 2(A − D)xx + 4(u − λ)(A − D), 2V22 = ∂ −1 (wC − vB) + 2(A − D)x , 2V23 = 2wA + 2(u − λ)B − Bxx ,

2V31 = ∂ −1 v(2A − 4D) − Cx , 2V33 = ∂

−1

(vB − wC) − 2Dx ,

2V41 = 2vA − Cxx + 2(u − λ)C,



2V24 = ∂ −1 w(4D − 2A) + Bx ,

V32 = C, V34 = 2D,



2V42 = ∂ −1 v(2A − 4D) + Cx ,

2V43 = vB + wC − 2Dxx + 4(u − λ)D,

2V44 = ∂ −1 (vB − wC) + 2Dx .

Then substituting the above equations into the following ones 2V21x = −(u − λ)2V12x + w2V31 − v2V24 ,

2V23x = −(u − λ)2V14x + w(2V33 − v2V22 ),

2V41x = −(u − λ)2V32x + v(2V11 − 2V44 ),

2V43x = −(u − λ)2V34x + v2V13 − w2V42 .

Eq. (2.2) becomes

−2∂ 3 + 4ux + 8u∂ − 2w∂ −1 v − 2v∂ −1 w A + (v∂ + ∂v)B + (w∂ + ∂w)C

+ 2∂ 3 − 4ux − 8u∂ + 4w∂ −1 v + 4v∂ −1 w D = 8λ(A − D)x ,

2(v∂ + ∂v)A + 2v∂ −1 vB + 2ux + 4u∂ − 2v∂ −1 w − ∂ 3 C = 4λCx ,

2(w∂ + ∂w)A + 2ux + 4u∂ − 2w∂ −1 v − ∂ 3 B + 2w∂ −1 wC = 4λBx ,



2 v∂ −1 w + w∂ −1 v A + (v∂ + ∂v)B + (w∂ + ∂w)C + 4ux + 8u∂ − 4v∂ −1 w − 4w∂ −1 v − 2∂ 3 D = 8λDx . Eqs. (2.4) and (2.7) imply A = 2D. From this they can be rewritten as 3

−∂ + 2u∂ + 2∂u A + (v∂ + ∂v)B + (w∂ + ∂w)C = 4λAx .

(2.4) (2.5) (2.6) (2.7)

(2.8)

Thus from Eqs. (2.8), (2.5) and (2.7) we obtain the Lenard gradient sequence K(2A, B, C)T = λJ (2A, B, C)T , where

⎛

u∂ + ∂u − 12 ∂ 3 K =⎝ v∂ + ∂v w∂ + ∂w  2∂ 0 0 J = 0 4∂ 0 , 0 0 4∂

(2.9)

v∂ + ∂v 2v∂ −1 v 2u∂ + 2∂u − 2w∂ −1 v − ∂ 3 Gj = (2Aj , Bj , Cj )T ,

⎞ w∂ + ∂w 2u∂ + 2∂u − 2v∂ −1 w − ∂ 3 ⎠ , 2w∂ −1 w j  0.

Substituting (2.2) into (2.9) and comparing coefficients for the same power of λ give rise to the recursion relation J G0 = 0,

KGj = J Gj +1 ,

j  0.

(2.10)

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We take initial value G0 = (2, 0, 0)T , then we work out Gj , which is uniquely determined by the recursion relation (2.10) up to a constant term. Especially, the first few terms are as follows ⎛ 1 ⎞ ⎛ ⎞ − 4 uxx + 34 u2 + 34 vw u ⎜ ⎟ G2 = ⎝ − 18 wxx + 34 uw ⎠ . G1 = ⎝ 12 w ⎠ , 1 − 18 vxx + 34 uv 2v Suppose the auxiliary problem

φtn = V (n) φ, V (n) = λn+1 V + ,

n  0,

(2.11)

where the symbol + denotes the choice of non-negative power of λ. Then the compatibility condition of (2.1) and (2.11) gives rise to the zero-curvature representation 

Utn − Vx(n) + U, V (n) = 0, which is equivalent to  u = Xn = J Gn = KGn−1 , v w tn

(2.12)

where K and J are given by Eq. (2.9). When n = 1, ⎛ − ux 4 ⎜ −λ2 + u λ − uxx + u2 ⎜ 2 4 2 V (1) = ⎜ ⎝ − v4x vxx v 2 λ − 4 + uv

λ+ +

vw 2

ux 4 v 2 vx 4

− w4x

u 2

wxx w 2 λ − 4 + uw − u4x 2 −λ2 + u2 λ − u4xx + u2 + vw 2

w 2 wx 4

⎞

⎟ ⎟ ⎟. λ + u2 ⎠

(2.13)

ux 4

Thus, an interesting equation, a new coupled KdV equation associated with (2.1) and (2.13), is presented 1 3 3 1 3 ut = − uxxx + uux + (vw)x , vt = − vxxx + (uv)x , 4 2 4 4 2 When u = v = w, Eq. (2.14) is reduced to the KdV equation

1 3 wt = − wxxx + (uw)x . 4 2

1 ut = − uxxx + 3uux . 4

(2.14)

(2.15)

3. Generalized Hamiltonian structures To investigate the generalized Hamiltonian structures of the hierarchy (2.12), we take the King–Cartan form A, B as tr(AB). Then by direct calculations, we conclude that         ∂U ∂U ∂U ∂U V, (3.1) V, V, V, = −V12 − V34 , = V12 + V34 , = V14 , = V32 . ∂λ ∂u ∂v ∂w According to the trace identity [32]             δ δ δ ∂U ∂U ∂U ∂ ∂U −γ γ , , V, = λ λ , V, , V, . V, δu δv δw ∂λ ∂λ ∂u ∂v ∂w Substituting (2.3) and (3.1) into it, we have   δ δ δ , , (−2Aj +1 ) = (γ − j − 1)(2Aj , Bj , Cj ), δu δv δw

j  0.

To fix the constant γ , we compare the coefficient of j = 0 in the above equation and get γ = 12 . Substituting it into (3.3), we arrive at   4Aj +1 δ δ δ , , Hj = GTj , Hj = . δu δv δw 2j + 1

(3.2)

(3.3)

(3.4)

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Therefore, we obtain the desired generalized Hamiltonian structures of the hierarchy (2.12) ⎛ δHn ⎞ ⎛ δHn−1 ⎞  δu δu u ⎟ ⎜ n⎟ ⎜ = K ⎝ δHδvn−1 ⎠ = J ⎝ δH w δv ⎠ , δHn−1 δHn v tn

455

(3.5)

δw

δw

where K and J are given by (2.9). 4. Nonlinearization Now we take N distinct eigenvalues λ1 , λ2 , . . . , λN of (2.1) and suppose the corresponding solution of the Lax pair for λ = λj is (p1j , q2j , p2j , q1j )T , that is ⎛ ⎞ ⎞ p1j p1j ⎜ q2j ⎟ ⎜ q2j ⎟ ⎝ ⎠ = U (u, v, w, λj ) ⎝ ⎠, p2j p2j q1j x q1j ⎛

j = 1, 2, . . . , N,

(4.1)

where ⎛

0 ⎜ u − λj U (u, v, w, λj ) = ⎝ 0 v

1 0 0 0

0 w 0 u − λj

⎞ 0 0⎟ ⎠. 1 0

Then we define Λ = diag(λ1 , . . . , λN ), ·, · is the standard inner-product in R N and qi = (qi1 , . . . , qiN )T ,

pi = (qi1 , . . . , qiN )T ,

i = 1, 2.

Similar to the method [1–3], we carry out the variation of λ and get  ∇λj =

δλj δλj δλj , , δu δv δw

T = (2p1j p2j , −p1j p1j , −p2j p2j )T .

(4.2)

A direct calculation shows that the variational derivative satisfies the following equation K∇λj = λj J ∇λj ,

j = 1, 2, . . . , N.

(4.3)

Now consider the Bargmann constraint G1 =

N 

∇λj ,

j =1

that is u = −2p1 , p2 ,

v = −2p2 , p2 ,

w = −2p1 , p1 .

(4.4)

Substituting (4.4) into (4.1), a finite-dimensional Hamiltonian system is given qix =

∂H1 , ∂pi

pix = −

∂H1 , ∂qi

qit =

∂H2 , ∂pi

pit = −

∂H2 , ∂qi

i = 1, 2,

where H1 = −q1 , q2  − Λp1 , p2  − p1 , p2 2 − p1 , p2 p2 , p2 ,  1  1 H2 = −Λq1 , q2  − Λ2 p1 , p2 − p1 , q1 p2 , q2  + p1 , p1 q1 , q1  2 2 1 1 1 + p2 , p2 q2 , q2  − Λp1 , p1 p2 , p2  − p1 , p1 Λp2 , p2  − p1 , p2 Λp1 , p2  2 2 2 1 1 − p1 , q2 p2 , q1  + p1 , p2 q1 , q2  − p1 , q1 2 − p2 , q2 2 . 4 4

(4.5)

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5. Integrability To prove Liouville integrability of the Hamiltonian system (4.5), it is necessary to find 2N conserved integrals with involution each other and functional independence. A direct calculation can first get the following proposition. Proposition 1. The Lax equation is

 Lt = V (1) , L Lx = [U, L],

(5.1)

if and only if the constraints (4.4) hold. Where

⎛

⎞ 0 1 0 0 τj −p1 , p1  0⎟ ⎜ −p1 , p2  − λ 0 , L1 (λ) = ⎝ L(λ) = L1 (λ) + ⎠, 0 0 0 1 λ − λj j =1 0 −p1 , p2  − λ 0 −p2 , p2  ⎛p q ⎞ 2 −p1j 1j 1j −p1j p2j p1j q2j 2 ⎜ q1j q2j −p2j q2j q2j −p1j q2j ⎟ ⎟. τj = ⎜ 2 ⎝ p2j q1j −p2j p2j q2j −p1j p2j ⎠ 2 q1j −p2j q1j q1j q2j −p1j q1j N 

Its characteristic polynomial is given

F (ζ, λ) = det ζ I − L(λ) = ζ 4 + F1λ ζ 2 + F2λ ,

(5.2)

where F1λ = −L211 − L233 − L12 L21 − L34 L43 − 2L14 L23 + 2L13 L24 , Let F1λ =



F1m λ−m−1 ,

F2λ =

m0





F2λ = det L(λ) .

F2m λ−m−1

m0

by direct and tedious computation, we have 1 1 H2 = (F11 + F20 ). H1 = F10 , 2 4 , F } will be proved to commute each other. The Poisson bracket of two smooth functions in the symplectic In what follows, {F 1λ 2λ  space (R 4N , 2i=1 dpi ∧ dqi ) is defined as {f, g} =

 2  N   ∂f ∂g ∂f ∂g − . ∂qij ∂pij ∂pij ∂qij

(5.3)

j =1 i=1

What is more, the flow variable tλ is denoted. The Hamiltonian equation F (ζ, λ) is ⎛ ∂F ⎞ − ∂q1j ⎛ ⎛ ⎞ ⎞ p1j p1j ⎜ ∂F ⎟ ⎜ ⎟ d ⎜ q2j ⎟ ⎜ ∂p2j ⎟ ⎜ q2j ⎟ ⎝ ⎠ = ⎜ ∂F ⎟ = M(λ, λj ) ⎝ ⎠, p p2j ⎜ ⎟ dtλ 2j ⎝ − ∂q2j ⎠ q1j q1j ∂F

(5.4)

∂p1j

where M(λ, λj ) = L∗

⎛

2 2 ζ L(λ) − L∗ + 2L2 , λ − λj

0 ⎜ ζ 2 L12 − L∗21 L2 = ⎜ ⎝ 0 2 ζ L32 − L∗23

0 0 2 0 ζ L14 − L∗41 0 0 2 0 ζ L12 − L∗21

⎞ 0 0⎟ ⎟. 0⎠ 0

is the adjoint matrix of L and satisfies the relations L∗11 = −L∗44 ,

L∗12 = L∗34 ,

L∗13 = −L∗24 ,

L∗21 = L∗43 ,

L∗22 = −L∗33 ,

L∗31 = −L∗42 .

Z. Qin / Physics Letters A 355 (2006) 452–459

457

Proposition 2. τj satisfies the Lax equation of F (λ) flow  dτj = M(λ, λj ), τj , dt

j = 1, . . . , N.

(5.5)

Proof. Notice M11 = −M44 ,

M12 = M34 ,

M13 = −M24 ,

M21 = M43 ,

and then Eq. (5.5) is directly calculated. This completes the proof.

M22 = −M33 ,

M31 = −M42

2

Theorem 1. Let M(λ, μ) =

2 2 ζ L − L∗ + 2L2 , λ−μ

then we have

 d L(μ) = M(λ, μ), L(μ) . dτλ

(5.6)

Proof. It only needs to prove that   

 2ζ 2 2 ∗ 2 ∗ d L1 (μ) + L(λ), L1 (λ) − L1 (μ) + L , L(λ) + L , L1 (μ) − L1 (λ) − 2 L2 , L1 (μ) = 0. dτλ λ−μ λ−μ λ−μ

(5.7)

Notice that 4 

Lij L∗lj

 =

j =1

det L(λ), i = l, 0, i = l.

(5.8)

On one hand, from (5.4) and (5.5) we get ⎛ 0 2 ⎜ d 2ζ (L22 − L11 ) − 2(L∗22 − L∗11 ) L1 (μ) = ⎜ ⎝ 0 dτλ 2 4L31 ζ + 4L∗13

0 0 0 0

⎞ 0 0 4L13 ζ 2 + 4L∗31 0⎟ ⎟. 0 0⎠ 2ζ 2 (L22 − L11 ) − 2(L∗22 − L∗11 ) 0

On the other hand ⎛

⎞ L12 0 L14 0 ⎜ L22 − L11 −L12  2ζ 2 −2L13 −L14 ⎟ ⎟, L(λ), L1 (λ) − L1 (μ) = −2ζ 2 ⎜ ⎝ L32 0 L12 0 ⎠ λ−μ −2L31 −L32 L22 − L11 −L12 ⎛ ⎞ L∗21 0 L∗41 0 ⎜ L∗22 − L∗11 −L∗21  2 ∗ −2L∗31 −L∗41 ⎟ ⎟, L , L1 (μ) − L1 (λ) = 2 ⎜ ∗ ⎝ L∗ ⎠ 0 L 0 λ−μ 23 21 ∗ ∗ ∗ ∗ ∗ −2L13 −L23 L22 − L11 −L21 ⎞ ⎛ 2 0 −ζ 2 L14 L∗41 0 −ζ L12 + L∗21 ⎜

 0 ζ 2 L12 − L∗21 0 ζ 2 L14 − L∗41 ⎟ ⎟ ⎜ 2 L2 , L1 (μ) = 2 ⎜ 2 ⎟. ∗ ∗ 2 ⎠ ⎝ −ζ L32 + L23 0 −ζ L12 L21 0 0

ζ 2 L32 − L∗23

0

ζ 2 L12 − L∗21

Substituting the above equations into (5.7) and by direct calculations, we can arrive at (5.6). This completes the proof. Theorem 2. The functions Fim are in involution in pair with respect to the Poisson bracket (5.3), that is {Fim , Fj n } = 0,

∀m, ∀n, i, j = 1, 2.

2

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Z. Qin / Physics Letters A 355 (2006) 452–459

Proof. According to Theorem 1, F2μ = det(L(μ)) is conserved integrals of tλ -flow defined by (5.4). Therefore, the integral F (η, μ) = det(ηI − L(μ)) is also conserved. 

 d F (μ) = 0, F (η, μ), F (ζ, λ) = dtλ

that is, ζ 2 η2 {F1μ , F1λ } + η2 {F1μ , F2λ } + ζ 2 {F2μ , F1λ } + {F2μ , F2λ } = 0. Since ζ, η are arbitrary, {F1μ , F1λ }, {F1μ , F2λ }, {F2μ , F1λ }, {F2μ , F2λ } are all zero. On the other hand,  {Fim , Fj n }λ−m μ−n . {Fiλ , Fj μ } = m,n0

It follows that Fim , i = 1, 2, m  1, are in involution in pair with respect to the Poisson bracket (5.3). This completes the proof.

2

Let us now continue to show the functional independence of the conserved integrals Fim , i = 1, 2, m = 1, 2, . . . , N . Theorem 3. The functions Fim (i = 1, 2, m = 1, . . . , N) are functionally independent in a dense open subset of R 4N . Proof. Without loss of generality, it only needs to consider one point P0 in R 4N . Let P0 be given by p1β = q1β = 1,

p2β = q2β = 0

(β = 1, 2, . . . , N).

Then at P0 by the direct and long calculation, we have ∂F1m = 0, ∂q1β

∂F1m = −2λm β, ∂q2β

∂F2m = 2N λm β, ∂q1β

∂F2m = 2λm+1 . β ∂q2β

(5.9)

By direct computation, we obtain N  ∂(F11 , . . . , F1N , F21 , . . . , F2N ) N J= λ2k = (4N ) ∂(q11 , . . . , q1N , q21 , . . . , q2N ) k=1



(λk − λl )2 .

1k
Hence, J is nonzero at P0 . Therefore, Fim (i = 1, 2, m = 1, . . . , N ) are functionally independent in a dense open subset of R 4N . This completes the proof. 2 In what follows let us sum up the above results. Theorem  4. The Hamiltonian system given by (4.5) is completely integrable in the Liouville sense in the symplectic space (R 4N , 2i=1 dpi ∧ dqi ). Acknowledgements The author is very grateful to Prof. Z.X. Zhou and Prof. R.G. Zhou for their enthusiastic guidance and help. This work is supported by Chinese National Research Project “Nonlinear Science”. References [1] C.W. Cao, X.G. Geng, in: C.H. Gu, Y.S. Li, G.Z. Tu, Y.B. Zeng (Eds.), Nonlinear Physics, Research Reports in Physics, Springer-Verlag, Berlin, 1990, pp. 68–78. [2] C.W. Cao, Sci. China Ser. A 33 (1990) 528. [3] C.W. Cao, X.G. Geng, J. Phys. A 21 (1990) 4117. [4] Z.M. Jiang, Phys. Lett. A 228 (1997) 275. [5] Z.M. Jiang, J. Nonlinear Math. Phys. 6 (5). [6] R.G. Zhou, Z.J. Qiao, Commun. Theor. Phys. 34 (2000) 329. [7] Z.J. Qiao, Phys. Lett. A 186 (1994) 97. [8] X.G. Geng, Physica A 180 (1992) 241. [9] X.G. Geng, Physica A 212 (1994) 132. [10] Y.T. Wu, X.G. Geng, J. Math. Phys. 40 (1999) 3409. [11] W.M. Ma, Z.X. Zhou, J. Math. Phys. 42 (2001) 4345. [12] Z.X. Zhou, W.X. Ma, R.G. Zhou, Nonlinearity 14 (2001) 701.

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