A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure

Physics Letters A 375 (2011) 1504–1509

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Physics Letters A www.elsevier.com/locate/pla

A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure Fajun Yu ∗ School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China

a r t i c l e

i n f o

Article history: Received 4 January 2011 Received in revised form 21 January 2011 Accepted 18 February 2011 Available online 26 February 2011 Communicated by R. Wu Keywords: Nonlinear integrable coupling system Soliton hierarchy Non-semisimple Hamiltonian structure

a b s t r a c t Some integrable coupling systems of existing papers are linear integrable couplings. In the Letter, beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy. A direct application to the AKNS spectral problem leads to a novel nonlinear integrable couplings, then we consider the Hamiltonian structures of nonlinear integrable couplings of AKNS hierarchy with the component-trace identity. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The notion of integrable couplings was proposed while study of relations between Virasoro symmetry algebras and hereditary operators [1–3]. Many coupled equations appear in practical problems, such as biochemistry, physics, mechanics etc. For example, in order to systematically describe a kind of biochemistry models, Prigogine and Lefever proposed a coupled mathematical model in Ref. [4], which describes a biology–chemistry model; the wellknown shallow water wave mathematics model and coupled KdV model were given [5,6]. So that, many integrable coupled mathematical models can solve practical problems [7]. A few ways to construct integrable couplings are presented. The perturbation method for establishing integrable couplings was proposed in Refs. [2,3]. Guo, Zhang and Xia presented the enlarged Lie algebra method to obtain integrable couplings in [8–12]. Some interesting integrable couplings and associated Hamiltonian structures were obtained [8–15]. In order to get Hamiltonian structures of integrable couplings, Guo and Zhang proposed the quadratic-form identity [15]. The approaches for extending integrable equations through semi-direct sums of Lie algebra [16,17] were powerful on presenting continuous and discrete integrable Hamiltonian couplings. After this, Ma and Chen built the variational identity and generalized the

*

Address for correspondence: Department of Mathematics, Shenyang Normal University, Shenyang 110034, China. Tel.: +86 138 40329476; fax: +86 024 86593365. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.02.043

quadratic-form identity and obtained some integrable couplings and resulting Hamiltonian structures [18]. The reasons for study of integrable couplings lie in that there are much richer mathematical structures than scalar integrable equations. Moreover, the study of integrable couplings generalizes the symmetry problems and provides clues towards complete classification of integrable equations [1–3]. Recently, Ma and Gao [19] proposed the notion on coupling integrable couplings and further investigated the various coupling integrable couplings of the nonlinear Schrödinger equation and associated symmetry properties. The component-trace identities can generate Hamiltonian structures of integrable couplings including the perturbation equations, and Hamiltonian structures of integrable couplings of the KdV hierarchy was given in [20]. Some integrable coupling systems are linear integrable couplings in existing papers. For example, Ma, Xu and Zhang defined ¯ and prethe corresponding enlarged spectral matrix U¯ and W sented the first three systems of integrable couplings of AKNS hierarchy by using the semi-direct sums of Lie algebras in Ref. [16]. Sun take using the enlarged Lie algebra method to obtain integrable couplings of the AKNS equations [21]. However, these integrable coupling systems are linear integrable couplings. In fact, the nonlinear integrable coupling systems are more important. In [22], Ma and Zhu use the Lax pairs from special non-semisimple matrix Lie algebras and obtain a nonlinear integrable couplings of Volterra lattice hierarchy. Inspired by the previous work [22], we extend this method and present a scheme for constructing real nonlinear integrable couplings of the continuous soliton hierarchy. In this Letter, we present two Lax matrices U¯ and V¯ usually belong to a non-semisimple matrix Lie algebra g¯ , which can derive a nonlinear integrable couplings because the matrix V i ,x − [U i , V i ]

F. Yu / Physics Letters A 375 (2011) 1504–1509

often produces nonlinear terms. A direct application to the AKNS spectral problem leads to a novel nonlinear integrable couplings, then we obtain the Hamiltonian structures of nonlinear integrable couplings of AKNS hierarchy with the component-trace identity. 2. A nonlinear integrable couplings Integrable coupling system is presented by using Virasoro symmetry algebra [2]. In details, let

u t = K (u )

(1)

be a known integrable system, the following system



ut = K (u ), v t = S (u , v ),

is called an integrable coupling of the system (1), if v t = S (u , v ) is also integrable and S (u , v ) contains explicitly u or u-derivatives with respect to x. In such integrable couplings, the equation for the supplementary variable v is linear with respect to v. If the second equation of an integrable coupling

ut = K (u ),

v t = S (u , v ),

(3)

defines a nonlinear equation for v, then the whole system is called a nonlinear integrable couplings of ut = K (u ). To construct an integrable couplings of nonlinear soliton equation hierarchy, we enlarge the Lie algebra G and obtain the block ¯ type matrix of Lie algebras G. Let us next shed light on the general idea of constructing coupling systems by a particular class of block type matrix of Lie algebra. Consider the following matrix of Lie algebra:

G¯ = G ⊕ G c , with

⎛

(4)

··· .. . A 0 .. . 0 A .. .. .. . . . 0 ··· ··· ··· ⎛0 B B2 · · · 1 . ⎜ .. ⎜ . B1 B1 . . ⎜ . Gc = ⎜ ⎜ 0 0 B2 . . ⎜. . ⎝ . .. ... ... . 0 ··· ··· ··· A

⎜ .. ⎜. ⎜ G =⎜ ⎜0 ⎜. ⎝ ..

0

0

0

⎞

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ 0⎠ .. . .. .

⎛0 B B2 · · · Bν ⎞ 1 . . ⎟ . ⎜. ⎜ . B 1 B 1 . . .. ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ 0 0 B2 . . B2 ⎟ . ⎜. . ⎟ ⎝ . .. ... ... ⎠ . B1 0 · · · · · · · · · Bν According to the Lax pair

Φt = V¯ Φ,

(7)

and the compatibility of Eq. (7), then, the coupling system (3) becomes

⎧ ⎨ U 0,t = V 0,x + [U 0 , V 0 ], ⎩ U i ,t = V i ,x − [U i , V i ] −



[U k , V l ],

1  i  ν.

(8)

k+l=i , k,l0

This is a real nonlinear integrable couplings, and it is normally a nonlinear integrable coupling because the matrix V i ,x − [U i , V i ] often produces nonlinear terms. We consider another cases, the G¯ includes the matrix Lie algebra G and G a ,

G¯ = G ⊕ G a .

(9)

The notion of matrix means that G and G a satisfy

[G , G a ] ∈ G a ,

(10)

where [G , G a ] = {[ A , B ], A ∈ G , B ∈ G a }. The subscript a indicates a contribution to construct the integrable couplings. Consider a pair of new Lax matrices in block matrix of Lie algebra G¯ of G and G a :

U¯ = U + U c ,

V¯ = V + V a ,

U i , V i ∈ Ga

(i = 1, 2).

(11)

The new Lax pair form is presented as follows

⎛

U0 U¯ = ⎝ 0 0

A Bν ⎞

.. ⎟ . ⎟ ⎟ ⎟ B2 ⎟ , ⎟ ⎠

⎛

(5)

B1 Bν

where the matrices A, B i , 1  i  ν are arbitrary matrices of the same size as U 0 . So new enlarged spectral matrices U¯ and V¯ in G ⊕ G c can be chosen as

⎛U U1 U2 0 ⎜ .. U1 ⎜ . U0 + U1 ⎜ U¯ = ⎜ 0 U0 + U2 ⎜ 0 ⎜ . .. .. ⎝ . . . . 0 ··· ··· ⎛V V1 V2 0 ⎜ .. V1 ⎜ . V0 + V1 ⎜ V¯ = ⎜ 0 V0 + V2 ⎜ 0 ⎜ . .. .. ⎝ . . . . 0 ··· ···

The resulting Lie algebras are non-semisimple, since they have a non-trivial ideal Lie sub-algebra consisting of matrices of the form

Φx = U¯ Φ, (2)

1505

⎞ ··· Uν .. .. ⎟ . . ⎟ ⎟ .. ⎟, . U2 ⎟ ⎟ .. ⎠ . U1 · · · U0 + Uν ⎞ ··· Vν .. .. ⎟ . . ⎟ ⎟ .. ⎟. . V2 ⎟ ⎟ .. ⎠ . V1 · · · V0 + Vν

V0 V¯ = ⎝ 0 0

⎞

U1 U0 + U1 0

U2 ⎠, U1 U0 + U2

V1 V0+ V1 0

V2 ⎠, V1 V0+ V2

⎞

(12)

we take a pair of enlarged matrix spectral (12) into the stationary zero curvature equation

V¯ x − [U¯ , V¯ ] = 0,

(13)

the new stationary curvature equations are given

(6a)

⎧ ⎨ V 0x − [U 0 , V 0 ] = 0, V − [ U 1 , V 1 ] − [ U 0 , V 1 ] − [ U 1 , V 0 ] = 0, ⎩ 1x V 2x − [U 2 , V 2 ] − [U 1 , V 1 ] − [U 0 , V 2 ] − [U 2 , V 0 ] = 0.

(14)

Now we give a new form of zero curvature equations

(6b)

⎧ ⎨ U 0t − V 0x + [U 0 , V 0 ] = 0, (15) U − V 1x + [U 1 , V 1 ] + [U 0 , V 1 ] + [U 1 , V 0 ] = 0, ⎩ 1t U 2t − V 2x + [U 2 , V 2 ] + [U 1 , V 1 ] + [U 0 , V 2 ] + [U 2 , V 0 ] = 0. The first equation here exactly presents Eq. (3). And Eq. (15) provides two nonlinear coupling systems for Eq. (3). To summarize, block type matrix of G¯ with new Lie algebras provides a great

1506

F. Yu / Physics Letters A 375 (2011) 1504–1509

⎞

Then, we use the corresponding component-trace identity to furnish Hamiltonian structures for the integrable couplings described above. In the component-trace identity ·,· is a nondegenerate, symmetric and invariant bilinear form, over the nonsemisimple Lie algebra:

⎟ ⎠,

g¯ =

choice of candidates of integrable couplings for Eq. (15) associated ¯ with G. Furthermore, we consider higher dimension Lax pair, the new Lax pair 4 × 4 form is presented as follows

⎛

U0 0 ⎜ U¯ = ⎝ 0 0 ⎛ V0 ⎜ 0 V¯ = ⎝ 0 0

U1

U0 + U1 0 0 V1 V0 + V1 0 0

U2 U1 U + U2 0 V2 V1 V + V2 0

U3 U2 U1

U0 + U3 ⎞ V3 V2 ⎟ ⎠, V1 V0 + V3



(16a)

(16b)

(17)

U¯ =

U (u )

U a(v )

0

U (u ) + U a ( v )



(18)

can engender nonlinear integrable couplings. The set of all matrices above is closed under the matrix product, and thus it constitutes a matrix Lie under the matrix commutator. The resulting Lie algebras are non-semisimple, since they have a non-trivial ideal Lie sub-algebra consisting of matrices of the form



0 0

Ua Ua

 (19)

.

The component-trace identity over this kind of Lie algebras can furnish Hamiltonian structures for the associated integrable couplings. We will illustrate such an idea for generating nonlinear integrable Hamiltonian couplings by means of the AKNS hierarchy. Assume that an integrable coupling equation has a zerocurvature representation, where two Lax matrices U¯ and V¯ usually belong to a non-semisimple matrix Lie algebra g¯ . Let us introduce an enlarged spectral matrix

U¯ (u¯ ) =



U (u )

U a (v )

0

U (u ) + U a ( v )

 ,

(20)

where the new dependent variable u¯ consists of the original one u and supplementary one v. Now, upon choosing

V¯ (u¯ ) =



V (u )

V a (u¯ )

0

V (u ) + V a (u¯ )



V¯ (u¯ ) =





U (u ) 0

U a (v ) U (u ) + U a ( v )

V (u ) 0

V a (v ) V (u ) + V a ( v )

where



U=

,

(21)

−2 

r a c

b

−a

U a,t = V a,x − [U , V a ] − [U a , V ] − [U a , V a ].

 Va =

,



, 

1 u2

d f

,

u1 1

 ,



e

(23)

,

−d

where q and r are potentials, u 1 , u 2 are new dependent variables and u¯ = (q, r , u 1 , u 2 ) T . Set

a= d=

∞

ak λk ,

b=

∞

k =0

k =0

∞

∞

dk λk ,

e=

k =0

bk λk ,

c=

∞

ck λk ,

k =0

ek λk ,

f =

k =0

∞

f k λk .

(24)

k =0

The corresponding enlarged stationary zero curvature equation is obtained V¯ x = [U¯ , V¯ ], i.e.



V x = [U , V ],

(25)

V ax = [U , V a ] + [U a , V ] + [U a , V a ]. Obviously, we have



[U , V ] =

qcn − bn r −2λcn + 2ran

 [U , V a ] = [U a , V ] =

qf n − en r 2rdn − λ f n u 1 c n − bn u 2 u 2 an + an u 2



u 1 f n − en u 2 2u 2 dn

2λbn − 2qan rbn − cn q

λen − 2qdn ren − f n q

 ,

(26a)



(26b)

, 

−u 1 an − an u 1 , u 2 bn − c n u 1  −2u 1 dn . u 2 en − f n u 1

(26c) (26d)

Thus, substituting Eqs. (26) into Eq. (25), we obtain

yields

U t = V x − [U , V ],

 Ua =

,

λ

[U a , V a ] =

U¯ t = V¯ x + [U¯ , V¯ ]



q

2

 V =

λ



an enlarged zero curvature equation



A+B

    A, B ∈ g .

By using the theory on continuous integrable couplings, some integrable coupling systems of the known integrable hierarchies have been obtained, such as KdV hierarchy, KN hierarchy, TD hierarchy, etc., in Refs. [23–33]. In this Letter in order to obtain new nonlinear integrable coupling system, we illustrate a new approach by the non-semisimple matrix type Lie algebra to nonlinear soliton equation hierarchy. Let U¯ and V¯ , the new Lax pair form is presented as follows

U¯ (u¯ ) =

In this Letter, we would like to present a kind of Lie algebras which can generate nonlinear integrable couplings. More specifically, we would like to show that the following choice of spectral matrices:



B

3. A real nonlinear integrable couplings of the AKNS hierarchy

a new form of zero curvature equations is presented

⎧ U = V − [U , V ], 0t 0x 0 0 ⎪ ⎪ ⎪ ⎪ ⎨ U 1t = V 1x − [U 1 , V 1 ] − [U 0 , V 1 ] − [U 1 , V 0 ], U 2t = V 2x − [U 2 , V 2 ] − [U 1 , V 1 ] − [U 0 , V 2 ] − [U 2 , V 0 ], ⎪ ⎪ ⎪ ⎪ U 3t = V 3x − [U 3 , V 3 ] − [U 0 , V 3 ] − [U 1 , V 2 ] ⎩ − [U 2 , V 1 ] − [U 3 , V 0 ].

A 0

(22)

This is a nonlinear integrable coupling of Eq. (3) due to the assumption (22), and it is normally a nonlinear integrable coupling because the matrix V a,x − [U a , V a ] often produces nonlinear terms.

⎧ anx = qcn − rbn , ⎪ ⎪ ⎪ ⎪ bnx = 2bn+1 − 2qan , ⎪ ⎪ ⎪ ⎨ cnx = −2cn+1 + 2ran , d ⎪ nx = qf n − ren + u 1 cn − u 2 bn + u 1 f n − en u 2 , ⎪ ⎪ ⎪ ⎪ ⎪ enx = en+1 − 2qdn − 2u 1an − 2u 1 dn , ⎪ ⎩ f nx = 2rdn − f n+1 + 2u 2 an + 2u 2 dn .

(27)

F. Yu / Physics Letters A 375 (2011) 1504–1509

If set b0 = c 0 = 0, a0 = 12 , d0 = e 0 = f 0 = g 0 = 0, we see that all sets of functions an , bn , cn , dn , en , f n are uniquely determined. In particular, the first few sets are:

⎧ b 1 = q, c1 = r , a 1 = 0, ⎪ ⎪ ⎪ ⎪ b2 = qx , c 2 = −r x , a2 = −qr , ⎪ ⎪ ⎪ ⎪ 2 ⎪ b = q − 2q r , ⎪ 3 xx ⎪ ⎪ ⎪ ⎪ ⎪ c 3 = r xx − 2r 2 q, ⎨

d 1 = 0, e1 = u1 , f 1 = u2 , ⎪ ⎪ ⎪ d2 = −qu 2 − ru 1 − u 1 u 2 , ⎪ ⎪ ⎪ ⎪ ⎪ e 2 = u 1x , f 2 = −u 2x , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e 3 = u 1xx − 2u 1 (2q + u 1 )(r + u 2 ) − 2q2 u 2 , ⎪ ⎩ f 3 = −u 2xx − 2u 2 (2r + u 2 )(q + u 1 ) − 2r 2 u 1 .

(28)

U¯ tn

⎛

⎞

⎞⎛

⎞

⎛

⎞

b n +1 ⎜ c n +1 ⎟ = J⎝ ⎠, f n +1 e n +1

(29)

⎞

⎛

⎞

b n +1 bn ⎜ c n +1 ⎟ ⎜ cn ⎟ ⎝ ⎠ = L ⎝ ⎠, f n +1 fn e n +1 en

⎛

∂ − 2q∂ −1 r

⎜ −2r ∂ −1 r ⎜ ⎝ L 31

L=⎜

⎞ ··· ··· 0 .. ⎟ .⎟ 1 0 . . . . .. ⎟ , . . .⎟ ⎟ ⎠ 1 0 ⎛ ⎜ ⎜ ⎜ AN = ⎜ ⎜ ⎝

...,

−∂ + 2r ∂ −1 q

0

L 32

L 33

L 34 ⎠

L 42

L 43

L 44

0

0 ⎟ ⎟

⎟,

(30)

1

0

A = ( A 0 , A 1 , . . . , A N ),

B = (B0, B1, . . . , B N ) ∈ g,



Ck =

0  k  N,

Ai B j ,

(33)

(34)

i + j =k

L 31 = −2(r + u 2 )∂ −1 u 2 − 2u 2 ∂ −1 r ,

and the matrix commutator is given by

L 32 = −2(r + u 2 )∂ −1 u 1 + 2u 2 ∂ −1 q,

  [ A, B ] = A B − B A = . . . , [ A i , B j ], . . . .

(q + u 1 ),

(35)

i + j =k

L 34 = −2(r + u 2 )∂ −1 (r + u 2 ),

Let αi , 0  i  N, be given constants. We introduce a bilinear form on the Lie algebras g as follows:

L 41 = −2(q + u 1 )∂ −1 u 2 − 2u 1 ∂ −1 r , L 42 = −2(q + u 1 )∂ −1 u 1 + 2u 1 ∂ −1 q,

 A, B  =

(q + u 1 ),

N

αk tr(C k ) =

k =0

L 44 = ∂ − 2(q + u 1 )∂ −1 (r + u 2 ). By using of Eqs. (28) and (29), the first system in (29) is as following: when n = 2,

⎧ qt2 = −2q xx + 4q2 r , ⎪ ⎪ ⎪ ⎨ r = 2r − 4r 2 q, t2 xx ⎪ u = − u 1xx + 2u 1 (2q + u 1 )(r + u 2 ) + 2q2 u 2 , ⎪ 1t2 ⎪ ⎩ u 2t2 = −u 2xx − 2u 2 (2r + u 2 )(q + u 1 ) − 2r 2 u 1 .

(32)

1

A B = (C 0 , C 1 , . . . , C N ),

−1

0

⎞ ··· ··· 1 .. ⎟ .⎟ 0 0 . . . . .. ⎟ , . . .⎟ ⎟ ⎠ 0 0

0

the matrix product A B reads

with

−1

⎜ ⎜ ⎜ A1 = ⎜ ⎜ ⎝

⎞ ··· ··· 0 .. ⎟ .⎟ 1 1 . . . . .. ⎟ , . . .⎟ ⎟ ⎠ 0 1

0 1

where A i , 0  i  N, are square matrices of the same order. For convenience, we rewrite an element of the Lie algebra g as a vector of matrices: A = ( A 0 , A 1 , . . . , A N ). Then, for

⎞

0

L 33 = −∂ + 2(r + u 2 )∂

⎛

0

0 2q∂ −1 q

L 41

L 43 = 2(q + u 1 )∂

⎜ ⎜ ⎜ A0 = ⎜ ⎜ ⎝

1

0

where J is a Hamiltonian operator. According to Eq. (27), we have the following reduce relation

⎛

So the Lax pairs from special non-semisimple matrix Lie algebras is a simple and straightforward method to construct the nonlinear integrable coupling system of soliton hierarchy. The last two equations are real nonlinear equations in the system (31).

The Hamiltonian structures of integrable couplings of the KdV hierarchy is given by using the component-trace identities in [20]. In this section, we take using the component-trace identity [20] to obtain the Hamiltonian structure of integrable couplings of AKNS soliton hierarchy. Let us take a matrix Lie algebra g¯ consisting of the following matrices:

q b n +1 −1 0 0 0 ⎜ r ⎟ ⎜ 0 1 0 0 ⎟ ⎜ c n +1 ⎟ =⎝ ⎠ =⎝ ⎠⎝ ⎠ u1 0 0 0 −1 f n +1 0 0 1 0 u2 t e n +1

⎛

⎧ qt = −q xx + 2q2 r , ⎪ ⎪ ⎪ ⎨ r = 2r − 2r 2 q, t xx ⎪ u 1t = −2u 1xx − u 2 q x − 2qu 2x + u 1 qr , ⎪ ⎪ ⎩ u 2t2 = 2u 2xx + u 1 r x + 2ru 1x − u 2 qr .

4. Hamiltonian structure of the nonlinear integrable couplings (29)

Based on Tu method, it gives rise to

⎛

1507

k =0



αk tr

 Ai B j ,

A , B ∈ g , (36)

i + j =k

where A and B are given by (33) and C = A B = (C 0 , . . . , C N ) is the matrix product defined by (34). This is a linear combination of the traces of the components of A B. It is obvious that this bilinear form is non-degenerate on g iff α N = 0. Such a special case yields a non-degenerate bilinear form generated only by the last component trace:

(31)

We obtain a real nonlinear integrable couplings of the AKNS equations, which is different from the existing liner integrable couplings of the AKNS equations [21]

N

 A , B  = tr





Ai B j .

(37)

i+ j=N

This way, for a given spectral matrix

U¯ = U¯ (u , λ) = U 0 + U 1 + · · · + U N ∈ g ,

(38)

1508

F. Yu / Physics Letters A 375 (2011) 1504–1509

the variational identity

δ δu

 

We further introduce the enlarged Hamiltonian operator

(39)

where V¯ = V¯ (u , λ) = V 0 + V 1 + · · · + V N ∈ g¯ is a solution to

V¯ x = [U¯ , V¯ ], and

1 d 2 dλ

 tr

J







ln V¯ , V¯ ,



Vi

i+ j=N

J J1

J

J1

··· ···

(41)

∂U j ∂λ



dx = λ−γ

tr



k + j =1

  ∂U j ∂ γ . (42) Vi λ tr ∂λ ∂u i+ j=N

∂U j Vk ∂λ



∂U0 = tr V 1 ∂λ

tr

k + j =1

Vk

∂U j ∂ u¯







∂U2 ∂U1 ∂U0 + tr V 1 + tr V 2 tr V 0 dx ∂λ ∂λ ∂λ        ∂ γ ∂U2 ∂U1 ∂U0 + tr V 1 + tr V 2 . = λ−γ λ tr V 0 ∂λ ∂u ∂u ∂u (44)





= tr

d f

e g

 1 2

0

g

0



− 12 (49)

∂U0 ⎞ 1 ∂q )

⎜ ⎟ ⎜ tr( V 1 ∂∂Ur0 ) ⎟ ⎟ =⎜ ⎜ ∂U ⎟ ⎝ tr( V 0 ∂ u 11 ) ⎠ tr( V 0 ∂∂Uu 1 ) tr

2  d e   ⎞ 01

f g 00 ⎛ ⎞ f ⎜  d e  0 0  ⎟ ⎟ ⎜ tr ⎟ ⎜e⎟ ⎜ f g 10 = ⎜  ⎟ = ⎝ ⎠. c ⎜ tr a b  0 1  ⎟ ⎠ ⎝ c −a 00 b  a b  0 0 

tr

We will show that this identity (42) can be used to furnish Hamiltonian structures for the integrable couplings of any order. When N = 2, the identity (42) gives another tri-trace identity



2

⎛

(43)



1

2

⎛ tr( V



  



1

= d−

     ∂U0 ∂U1 ∂U0 + tr V 0 + tr V 1 tr V 0 dx ∂λ ∂λ ∂λ        ∂ γ ∂U0 ∂U1 ∂U0 + tr V 0 + tr V 1 . = λ−γ λ tr V 0 ∂λ ∂u ∂u ∂u

δ δu

(48)

JN



and

tity [20], since it only involves the traces of the components of the U and V ∂∂Uu . matrix products V ∂∂λ For N = 1, we consider a tri-trace identity, as follows:

  

⎟ ⎟ ⎟, ⎟ ⎠

where J i = |ε=0 J (uˆ N ), 1  i  N. Therefore, the Hamiltonian structures of the integrable coupling equations are just a consequence of the component-trace identity (32). The whole process provides a new approach to Hamiltonian structures of the integrable coupling equations by using component-trace identities. It is straightforward to compute the spectral problem (19), we have

We call this variational identity (42) the component-trace iden-

δ δu

.. . .. .



1 ∂ i! ∂ εi

yields the following result. Let g¯ be a matrix Lie algebra consisting of block matrices defined by (32). For a given spectral matrix U¯ = U¯ (u , λ) = U 0 + U 1 + · · · + U N ∈ g¯ , we have the variational identity: For general integer N, we can have a special variational identity

δ δu

⎜ ⎜ ¯J = ¯J i = ⎜ ⎜ ⎜ ⎝

(40)

γ is defined by

γ =−

J ⎞ J1 ⎟

⎛0

   ∂ U¯ ∂ γ ¯ ∂ U¯ , V¯ , dx = λ−γ λ V, ∂λ ∂λ ∂u

c −a

(50)

10

The component-trace identity (43) gives rise to

δ δ u¯

 

1 2

d−

1 2

⎛ ⎞ f



−γ

g dx = λ

∂ γ ⎜e⎟ λ ⎝ ⎠, c ∂λ

(51)

b checking a term of a special power of λ determines that application of the component-trace identity (42) yields

⎛

γ = 12 , an

⎞

This will be a tool for generating Hamiltonian structures of the nonlinear integrable coupling equations. Thus, the enlarged zero curvature equation

fi δ d i +1 − g i +1 ⎜ ei ⎟ , ⎝ ⎠= ci δ u¯ 2i − 1 bi

U¯ t − V¯ x + [U¯ , V¯ ] = 0,

Therefore, the hierarchy (29) of the coupling equations has the Hamiltonian structure

i = 0, 1 , 2 , . . . , N ,

(45)

where U¯ N and V¯ N are defined in (32) and (38). Then, a Hamiltonian structure of the integrable couplings (45) is given by

⎛U

0,t

⎞

⎛

⎜ ⎜ U 1,t ⎟ ⎜ ⎜ . ⎟ = ˆJ ⎜ ⎜ ⎝ . ⎠ . ⎝ U N ,t

δ H 0 (u ) δu δ H 1 (u ) δu

.. .

⎞

⎛

(46)

H i (u ) = tr

k + j =i

−1 0 0

0 ¯J = ⎜ ⎝

0 0

based on the generating function of Hamiltonian functions for the original equation

 ∂U j , Vk ∂λ

δ H¯ m , δ u¯

(52)

(53)

with the Hamiltonian operator

⎟ ⎟ ⎟, ⎟ ⎠

δ H N (u ) δu



u¯ tm = K¯ m = ¯J

i  0.

i = 0, 1 , 2 , . . . , N .

(47)

⎞

0 1 0 0 ⎟ ⎠ 0 0 −1 0 1 0

and the Hamiltonian functionals

¯m = H



2dm+1 − 2gm+1 2m − 1

dx,

m  0.

The above computation on the Hamiltonian structures of the nonlinear integrable couplings of the AKNS hierarchy is given for

F. Yu / Physics Letters A 375 (2011) 1504–1509

the fist time by using component-trace identity. We construct the Hamiltonian structure of a nonlinear integrable couplings of soliton hierarchy by using the non-semisimple matrix Lie algebras. Acknowledgements The author would like to express his sincere thanks to referees for his enthusiastic guidance and help. This work was supported by the Research Work of Liaoning Provincial Development of Education (L2010513). References [1] B. Fuchssteiner, in: A.A. Clarkson (Ed.), Applications of Analytic, Geometric Methods to Nonlinear Differential Equations, Kluwer, Dordrecht, 1993, pp. 125– 138. [2] W.X. Ma, B. Fuchssteiner, Chaos Solitons Fractals 7 (1996) 1227. [3] W.X. Ma, Methods Appl. Anal. 7 (2000) 21. [4] A. Pickering, J. Phys. A: Math. Gen. 26 (1993) 4395. [5] J.F. Zhang, Chinese Phys. Lett. 16 (1) (1999) 4. [6] E.G. Fan, H.Q. Zhang, Phys. Lett. A 245 (5) (1998) 389. [7] S.Y. Lou, Acta Phys. Sinica (Chinese) 49 (9) (2000) 1657. [8] Y.F. Zhang, T. Honwah, Chaos Solitons Fractals 23 (2005) 651.

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