New extended Lie algebra and the generalized integrable Liouville hierarchy

Applied Mathematics and Computation 218 (2011) 643–650

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New extended Lie algebra and the generalized integrable Liouville hierarchy Weidong Zhao a, Huanhe Dong a,c, Hui Wang b,⇑ a

College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, China Department of Mathematics, Shanghai University, Shanghai 200444, China c Key Laboratory for Robot and Intelligent Technology of Shandong Province, Qingdao, 266510, China b

a r t i c l e

i n f o

a b s t r a c t f1 is presented, from Starting from a new Lie algebra G1, the corresponding loop algebra G which a Liouville integrable hierarchy is given by using of variational identity. It follows that an expanding Lie algebra G2 is obtained based on G1, furthermore, a related Lax intef2 . With the help of grable hierarchy is presented by making use of its related loop algebra G variational identity, it is not difficult to prove that the hierarchy has Hamiltonian structure and is Liouville integrable. It is also can be seen that the second hierarchy is the expanding model of the first one. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Lie algebra Loop algebra Variational identity Hamiltonian structures Liouville integrable

1. Introduction It was an important aspect of soliton theory to research for new integrable Hamiltonian systems, Lax integrable or Liouville integrable. Professor Tu proposed the famous trace identity [1], which was a powerful tool for generating Hamiltontian structure of some integrable soliton equation hierarchies. Professor Ma called it Tu scheme [2]. A host of Hamiltonian structures of soliton equations were presented with its help [3–9]. In order to use Tu scheme more rapidly, Guo and Yu introduce a kind of Lie algebra [10] An1 ¼ xlðn; CÞ ¼ fz ¼ ðxij Þnn jxij 2 Cg along with the commutator,

½Z; Y ¼ ZMY  YMZ;

8Z; Y; M 2 An1

ð1Þ

which satisfies that

½Z; Y ¼ ½Y; Z;

½aX þ bY; Z ¼ a½Z; X þ b½Y; X;

½½X; Y; Z þ ½½Y; Z; X þ ½½Z; X; Y ¼ 0:

An1

ð2Þ

However, based on and its corresponding Lie algebra, some integrable hierarchies have not Hamiltonian structures, that is to say the trace identity is not suitable for them. To overcome the limitation of the trace identity, the Generalized Tu Formula (GTF) and the variational identity appear as its development [11–20], they have their powerful scopes of applications respectively. Especially, Ma proposed the variational identity in [19], and applied it to discuss Hamiltonian structures of soliton equations associated with general Lie algebras, which enrich the theory of soliton. In Ref. [18], component-trace identities are introduced on a particular class of non-semisimple matrix Lie algebras, which can be applied to Hamiltonian structures of integrable couplings, in an effort to supplement the theory of multi-component Hamiltonian equations. In the following, we will see the application of the variational identity, the main ideas are as follows. ⇑ Corresponding author. E-mail address: [email protected] (H. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.001

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W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

First, set up an isospectral problem



ux ¼ UMu; ut ¼ VMu:

ð3Þ

Second, solving the stationary zero curvature equation

V x ¼ ½U; V;

ð4Þ

where

 U¼

 ;

U1 U2

 V¼

V1

 ð5Þ

V2

and

½U; V ¼





½U 1 ; V 1 

ð6Þ

½U 1 ; V 2  þ ½U 2 ; V 1 

with the commutator satisfy (1). From the zero curvature equation

U t  V x þ ½U; V ¼ 0

ð7Þ

a Lax integrable system appears

ut ¼ KðuÞ:

ð8Þ

In this paper, we first construct a new Lie algebra, and obtain its related Liouville integrable hierarchy with the help of variational identity. Then, we further extended the new Lie algebra to get a six-dimensional Lie algebra, and obtain an integrable coupling of hierarchy in Section 3. It is easy to verify that it is also Liouville integrable. 2. A new Lie algebra and a Liouville integrable hierarchy Consider a new Lie algebra G1 = span{e1, e2, e3}, where

0

1 1 0 1 B C e1 ¼ @ 0 2 0 A; 1 0 1

0

1 0 1 0 B C e2 ¼ @ 1 0 1 A; 0 1 0

0

1 0 1 0 B C e3 ¼ @ 1 0 1 A; 0 1 0

0

1 1 0 0 B C M 1 ¼ @ 0 12 0 A 0 0 0

ð9Þ

with the commutative relations

½e1 ; e2  ¼ 2e3 ;

½e1 ; e3  ¼ 2e2 ;

½e2 ; e3  ¼ e1

ð10Þ

and the corresponding loop algebra is given by

e 1 ¼ spanfe1 ðnÞ; e2 ðnÞ; e3 ðnÞg; G

ei ðnÞ ¼ ei k2n ;

i ¼ 1; 2; 3

(2m+2n)

along with the commutators [ei(m), ej(n)] = [ei, ej]k Take the isospectral problem



, m, n 2 Z, 1 6 i, j 6 6.

ux ¼ UM1 u; ut ¼ VM1 u; kt ¼ 0;

ð11Þ

where

 U¼  V¼ where

ak ¼

U1 U2 V1

 ¼ 

 ¼

V2 X

e1 ð1Þ þ qþr e2 ð0Þ þ qr e3 ð0Þ 2 2 2 2 kðu1 þu e2 ð0Þ þ u1 u e3 ð0ÞÞ 2 2

;

a1 e1 ð0Þ þ a2 e2 ð0Þ þ a3 e3 ð0Þ kðb1 e1 ð0Þ þ b2 e2 ð0Þ þ b3 e3 ð0ÞÞ

ak;m k2m ;

mP0

!

bk ¼

X

bk;m k2m ;

 ;

ð12Þ

k ¼ 1; 2; 3:

mP0

From (6), solving the stationary zero curvature equation

V x ¼ ½U; V;

ð13Þ

W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

645

we have

8 a1;mx ¼ qþr a3;m þ rq a2;m ; > > 2 2 > > > > a2;mx ¼ ðr  qÞa1;m þ 2a3;mþ1 ; > > > > > a3;mx ¼ ðq þ rÞa1;m þ 2a2;mþ1 ; > > > u1 þu2 qþr rq b ¼ ðr  qÞb þ 2b þ ðu  u Þa 1;m 3;mþ1 2 1 1;m ; > 2;mx > > > > > b3;mx ¼ ðr þ qÞb1;m þ 2b2;mþ1  ðu1 þ u2 Þa1;m ; > > > > a2;0 ¼ a3;0 ¼ 0; a1;0 ¼ a – 0; b2;0 ¼ b3;0 ¼ 0; b1;0 ¼ b – 0; > > > : 2 2 a; a3;1 ¼ qr a; b2;1 ¼ qþr b þ u1 þu a; b3;1 ¼ qr b þ u1 u a: a2;1 ¼ qþr 2 2 2 2 2 2

ð14Þ

Denoting

00

V þðnÞ

a1;m

a2;m þ a3;m

1 1

a1;m

C BB a2;m  a3;m A C C B @ a3;m  a2;m 2a1;m C B n B X C 2n2m a1;m a2;m þ a3;m a1;m Ck B 0 1 ¼ ; C B b2;m þ b3;m b1;m b1;m C m¼0 B C B B C @ k@ b3;m  b2;m 2b1;m b2;m  b3;m A A b1;m

b2;m þ b3;m

ðnÞ 2n V ðnÞ þ þ V  ¼ k V;

b1;m

then

  2a3;nþ1 e2 ð0Þ þ 2a2;nþ1 e3 ð0Þ ðnÞ : V ðnÞ þx þ ½U; V þ  ¼  kð2b3;nþ1 e2 ð0Þ þ 2b2;nþ1 e3 ð0ÞÞ

ð15Þ

Taking V ðnÞ ¼ V ðnÞ þ , thus the zero curvature equation ðnÞ U t  V ðnÞ  ¼ 0; x þ ½U; V

determined the system

0

1

q

0

2a2;nþ1 þ 2a3;nþ1

1

0

0

0

0

2

10

b3;nþ1  b2;nþ1

1

CB C B C B r C B 2a B C B 3;nþ1  2a2;nþ1 C B 0 0 2 0 CB b2;nþ1  b3;nþ1 C ut ¼ B C ¼ B CB C¼B C ¼ J 1 Pnþ1 : @ u1 A @ 2b2;nþ1 þ 2b3;nþ1 A @ 0 2 0 0 A@ a3;nþ1  a2;nþ1 A 2b3;nþ1  2b2;nþ1 2 0 0 0 u2 t a2;nþ1  a3;nþ1

ð16Þ

In terms of (14), we have

0 Pnþ1

 2@  2r @ 1 q

B B  q @ 1 q B 2 ¼B B 0 @ 0

 2r @ 1 u1  u22 @ 1 q

r 1 @ r 2 @ þ 2q q@ 1 r 2

 2q @ 1 u1  u21 @ 1 q

0

 @2  2r @ 1 q

0

 2q @ 1 q

1

r 1 @ u2 þ u22 @ 1 r 2 C q 1 @ u2 þ u21 @ 1 r C C 2 CP n r C r@ 1 r 2 A 1 q @ þ q@ r 2 2

¼ L1 P n :

where L1 is a recurrence operator. Therefore, the system (16) can be written as

1 1 0 rb  u2 a q B r C B qb  u a C 1 C B C B ut ¼ B C ¼ J 1 Ln1 B C: A @ u1 A @ r a 0

u2

ð17Þ

qa

t

Finally, in order to get the Hamiltonian structure of the generalized soliton-equation hierarchy, we introduce the bilinear form functional f(A, B) [17–20] to be

f ðA; BÞ ¼ trðA1 B1 Þ þ trðA1 B2 Þ þ trðA2 B1 Þ

ð18Þ

which is one of so-called tri-trace identities in [18], where

A¼



 A1 ; A2

and f(A, B) holds

B¼



 B1 ; B2

fN A; B 2 K

ð19Þ

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W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

(10) (20) (30) (40)

Symmetry: f(A, B) = f(B, A), Bilinearity: f(c1A1 + c2A2, B) = c1f(A1, B) + c2f(A2, B), e Communication: f ð½A; B; CÞ ¼ f ðA; ½B; CÞ; 8A; B; C 2 G. rBf(A, B) = A,rBf(A, Bx) = rBf(Ax, B) = Ax,

e N , the variational identity Assume U; V 2 K

   d @ @U kc f V; f ðV; U k Þ ¼ kc dui @k @ui

ð20Þ

was obtained as a special case of Ref. [18]. Thus, a direct calculate yields

  @U 1 ¼ 4b1 k2 þ 4a1 k þ ½u1 ða3  a2 Þ  u2 ða2 þ a3 Þ; f V; @k 2     @U 1 1 @U 1 1 ¼ ða3  a2 Þ þ kðb3  b2 Þ; f V; ¼  ða2 þ a3 Þ  kðb2 þ b3 Þ; f V; @q 2 2 @r 2 2     @U 1 @U 1 ¼ kða3  a2 Þ; f V; ¼  kða3 þ a2 Þ: f V; @u1 2 @u2 2

ð21Þ

Inserting the above results into (20) yields

1 1 ða3  a2 Þ þ 12 kðb3  b2 Þ 2 B 1 1 d 1 @ B  ða2 þ a3 Þ  2 kðb2 þ b3 Þ C C 4b1 k2 þ 4a1 k þ ½u1 ða3  a2 Þ  u2 ða2 þ a3 Þ ¼ kc kc B 2 C: 1 A du 2 @k @ kða3  a2 Þ 2 0





ð22Þ

 12 kða3 þ a2 Þ Comparing of the coefficients of k2n2 on the both side of (18) admits

1 b3;nþ1  b2;nþ1 C B d B b2;nþ1  b3;nþ1 C ðu1 ða3;nþ1  a2;nþ1 Þ  u2 ða3;nþ1 þ a2;nþ1 Þ þ 8b1;nþ2 Þ ¼ ðc  2n  1ÞB C: @ a3;nþ1  a2;nþ1 A du 0

a2;nþ1  a3;nþ1 Taking n = 0, gives c = 2. Thus, we get

0

b3;nþ1  b2;nþ1

1

C B b B 2;nþ1  b3;nþ1 C ¼B C; @ a3;nþ1  a2;nþ1 A

ð1Þ dHnþ1

du

a2;nþ1  a3;nþ1 ð1Þ Hnþ1

1 where ¼  2n1 ðu1 ða3;nþ1  a2;nþ1 Þ  u2 ða3;nþ1 þ a2;nþ1 Þ þ 8b1;nþ2 Þ . Therefore, the hierarchy (17) have Hamiltonian structure

ð1Þ

ut ¼ J 1

dHnþ1 : du

ð23Þ

It is easy to verity that J 1 L1 ¼ L1 J 1 . Therefore, the hierarchy (23) is Liouville integrable. Taking u1 = u2 = 0, Eq. (17) reduce to AKNS hierarchy

ut ¼ J

where J ¼



  br dHnþ1 ; ¼ JLn du bq 0 2

 2 ; L¼ 0

 @2  2r @ 1 q  2q @ 1 q

ð24Þ r r@ 1 r 2 @ þ 2q q@ 1 r 2

! .

3. A new extended Lie algebra and the generalized integrable hierarchy We expand Lie algebra G1 into the following Lie algebra

G2 ¼ spanfh1 ; h2 ; h3 ; h4 ; h5 ; h6 g;

647

W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

where

 h1 ¼  h5 ¼



e1

0

0 0

e1  e2

e2

0

 h2 ¼

;

 h6 ¼

;

e2 0 0 e3

 ; e2  e3 ; 0 0

 h3 ¼

e3

0

 ;

 h4 ¼

0 e3   M1 0 ; M2 ¼ 0 M1

0

e1

e1

0

 ; ð25Þ

f2 is given by Its corresponding loop algebra G

f2 ¼ spanfei ðnÞg6 ; G i¼1

ei ðnÞ ¼ ei k2n ;

i ¼ 1; 2; 3; 4; 5; 6;

along with the commutators

½h1 ðmÞ; h2 ðnÞ ¼ 2h3 ðm þ nÞ;

½h1 ðmÞ; h3 ðnÞ ¼ 2h2 ðm þ nÞ;

½h1 ðmÞ; h5 ðnÞ ¼ 2h6 ðm þ nÞ;

½h1 ðmÞ; h6 ðnÞ ¼ 2h5 ðm þ nÞ; ½h2 ðmÞ; h3 ðnÞ ¼ h1 ðm þ nÞ; ½h2 ðmÞ; h4 ðnÞ ¼ 2h6 ðm þ nÞ; ½h2 ðmÞ; h6 ðnÞ ¼ h4 ðm þ nÞ; ½h3 ðmÞ; h4 ðnÞ ¼ 2h5 ðm þ nÞ; ½h3 ðmÞ; h5 ðnÞ ¼ h1 ðm þ nÞ; ½h4 ðmÞ; h5 ðnÞ ¼ 2h3 ðm þ nÞ;

½h4 ðmÞ; h6 ðnÞ ¼ 2h2 ðm þ nÞ;

ð26Þ

½h5 ðmÞ; h6 ðnÞ ¼ h1 ðm þ nÞ;

½h1 ðmÞ; h4 ðnÞ ¼ ½h2 ðmÞ; h5 ðnÞ ¼ ½h3 ðmÞ; h6 ðnÞ ¼ 0: Consider an isospectral problem



ux ¼ UM2 u; ut ¼ VM2 u; kt ¼ 0;

ð27Þ

where

U¼



U1

 ¼

U2 

V¼

V1

2 2 h2 ð0Þ þ qr h3 ð0Þ þ u1 þu h5 ð0Þ þ u1 u h6 ð0Þ h1 ð1Þ þ qþr 2 2 2 2

¼

V2

4 4 6 6 kðu3 þu h2 ð0Þ þ u3 u h3 ð0Þ þ u5 þu h5 ð0Þ þ u5 u h6 ð0ÞÞ 2 2 2 2





! ;

 a1 h1 ð0Þ þ a2 h2 ð0Þ þ a3 h3 ð0Þ þ a4 h5 ð0Þ þ a5 h6 ð0Þ ; kðb1 h1 ð0Þ þ b2 h2 ð0Þ þ b3 h3 ð0Þ þ b4 h5 ð0Þ þ b5 h6 ð0ÞÞ

where

ak ¼

X

ak;m k2m ;

mP0

bk ¼

X

bk;m k2m ;

k ¼ 1; 2; 3; 4; 5:

mP0

From (6), the stationary zero curvature equation,

V x ¼ ½U; V

ð28Þ

gives rise to

8 2 > a ¼ rq a2;m þ qþrþu21 þu2 a3;m þ rqþu22 u1 a4;m þ u1 þu a5;m ; > 2 2 > 1;mx > > > a ¼ 2a3;mþ1 þ ðr  qÞa1;m ; > > 2;mx > > > a 3;mx ¼ 2a2;mþ1  ðq þ rÞa1;m ; > > > > > > > a4;mx ¼ 2a5;mþ1 þ ðu2  u1 Þa1;m ; > > > > > a5;mx ¼ 2a4;mþ1  ðu1 þ u2 Þa1;m ; > > qþrþu1 þu2 rq 5 2 3 6 3 6 > b3;m þ rqþu22 u1 b4;m þ u1 þu b5;m þ u4 u a2;m þ u5 þu a3;m þ u4 u a4;m þ u6 u a4;m þ u5 þu a5;m > 2 2 2 2 2 2 2 < b1;mx ¼ 2 b2;m þ b2;mx ¼ 2b3;mþ1 þ ðr  qÞb1;m þ ðu4  u3 Þa1;m ; > > > > b3;mx ¼ 2b2;mþ1  ðq þ rÞb1;m  ðu3 þ u4 Þa1;m ; > > > > > b4;mx ¼ 2b5;mþ1 þ ðu2  u1 Þb1;m þ ðu6  u5 Þa1;m ; > > > > > b5;mx ¼ 2b4;mþ1  ðu1 þ u2 Þb1;m  ðu5 þ u6 Þa1;m ; > > > > > a2;0 ¼ a3;0 ¼ a4;0 ¼ a5;0 ¼ 0; a1;0 ¼ a – 0; b2;0 ¼ b3;0 ¼ b4;0 ¼ b5;0 ¼ 0; > > > > 2 2 4 > b1;0 ¼ b – 0; a2;1 ¼ qþr a; a3;1 ¼ qr a; a4;1 ¼ u1 þu a; a5;1 ¼ u1 u a; b2;1 ¼ qþr b þ u3 þu a; > 2 2 2 2 2 2 > > : b ¼ qr b þ u3 u4 a; b ¼ u1 þu2 b þ u5 þu6 a; b ¼ u1 u2 b þ u5 u6 a: 3;1

2

2

4;1

2

2

5;1

2

2

ð29Þ Denoting 2n V ðnÞ þ ¼ V þk ; ðnÞ 2n V ðnÞ þ þ V  ¼ k V:

ð30Þ

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W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

A direct calculation reads

  2a3;nþ1 h2 ð0Þ þ 2a2;nþ1 h3 ð0Þ þ 2a5;nþ1 h5 ð0Þ þ 2a4;nþ1 h6 ð0Þ ðnÞ : V þx þ ½U; V ðnÞ  ¼  þ kð2b3;nþ1 h2 ð0Þ þ 2b2;nþ1 h3 ð0Þ þ 2b5;nþ1 h5 ð0Þ þ 2b4;nþ1 h6 ð0ÞÞ

ð31Þ

Taking V ðnÞ ¼ V ðnÞ þ , then the zero curvature equation ðnÞ U t  V ðnÞ ¼0 x þ ½U; V

admits

1 1 0 0 2a2;nþ1 þ 2a3;nþ1 0 0 0 0 q B r C B 2a3;nþ1  2a2;nþ1 C B 0 0 0 0 C B B C B C B B C B B u1 C B 2a5;nþ1 þ 2a4;nþ1 C B 0 0 0 0 C B B C B C B Bu C B 2a B 2C B 5;nþ1  2a4;nþ1 C B 0 0 0 0 ut ¼ B C ¼ B C¼B B u3 C B 2b3;nþ1 þ 2b2;nþ1 C B 0 2 0 0 C B B C B C B Bu C B 2b B 4C B 3;nþ1  2b2;nþ1 C B 2 0 0 0 C B B C B @ u5 A @ 2b5;nþ1 þ 2b4;nþ1 A @ 0 0 0 2 2b5;nþ1  2b4;nþ1 0 0 2 0 u6 t 0

0 2 0 2 0 0 0

0

0

0

0

2

0

0

0 0

0

0

0

0

0

0

0

0

10 1 b3;nþ1  b2;nþ1 0 C B 0 CB b3;nþ1  b2;nþ1 C C CB C 2 CB b5;nþ1  b4;nþ1 C CB C C B 0 C CB b5;nþ1  b4;nþ1 C CB C ¼ J 2 Pnþ1 : C B 0 CB a3;nþ1  a2;nþ1 C C C B 0 C CB a3;nþ1  a2;nþ1 C CB C 0 A@ a5;nþ1  a4;nþ1 A 0

ð32Þ

a5;nþ1  a4;nþ1

From (29), it is easy to obtain that



Pnþ1 ¼

 A B P n ¼ L2 P n ; 0 A

where

0 B B B A¼B B @

 @2  12 r@ 1 ðqÞ

0

 12 u2 @ 1 q

1 r@ 1 r 2 @ þ 12 q@ 1 ðqÞ 2 1 u @ 1 r 2 2

 2@  12 u2 @ 1 u1

 12 u1 @ 1 q

1 u @ 1 r 2 1

 12 u1 @ 1 u1

 12 q@ 1 q

 12 r@ 1 u3  12 u4 @ 1 q

B B  1 q@ 1 u3  1 u3 @ 1 q B 2 B ¼ B 2 1 B  1 u6 @ q  1 u2 @ 1 u3 2 @ 2  12 u5 @ 1 q  12 u1 @ 1 u3

 12 r@ 1 u1

1 r@ 1 u2 2 1 q@ 1 u2 2 1 u @ 1 u2 2 2

 12 q@ 1 u1

1 r@ 1 u4 2 1 q@ 1 u4 2 1 u @ 1 r 2 6 1 u @ 1 r 2 5

@ 2

þ 12 u1 @ 1 u2

1 C C C C; C A

þ 12 u4 @ 1 r

 12 r@ 1 u5  12 u4 @ 1 u1

þ 12 u3 @ 1 r

 12 q@ 1 u5  12 u3 @ 1 u1

þ 12 u2 @ 1 u4

 12 u2 @ 1 u5  12 u6 @ 1 u1

þ 12 u1 @ 1 u4

 12 u1 @ 1 u5  12 u5 @ 1 u1

1 r@ 1 u6 2 1 q@ 1 u6 2

þ 12 u4 @ 1 u2

1

C þ 12 u3 @ 1 u2 C C C  12 u2 @ 1 u5  12 u6 @ 1 u1 C A 1 u @ 1 u6 þ 12 u5 @ 1 u2 2 1

and L2 is a recurrence operator. Therefore, the system (32) can be reduced as

0

q

1

0

rb  u4 a

1

B r C B qb  u3 a C C B C B C B C B B u1 C B u2 b  u6 a C C B C B Bu C B u b  u a C 5 C B 2C B 1 ut ¼ B C ¼ J 2 Ln2 B C: C B u3 C B ra C B C B C Bu C B qa C B 4C B C B C B A @ u5 A @ u2 a u6

t

ð33Þ

u1 a

Now, we will establish the Hamiltonian structure of the hierarchy. From (18) and (19), a direct calculate reads that

  @U ¼ 8b1 k2 þ 8a1 k þ u3 ða3  a2 Þ þ u5 ða5  a4 Þ  u6 ða4 þ a5 Þ  u4 ða2 þ a3 Þ; f V; @k       @U @U @U ¼ ða3  a2 Þ þ kðb3  b2 Þ; f V; ¼ ða3  a2 Þ þ kðb3  b2 Þ; f V; ¼ ða5  a4 Þ þ kðb5  b4 Þ; f V; @q @r @u1     @U @U ¼ ða5  a4 Þ þ kðb5  b4 Þ; f V; ¼ kða3  a2 Þ; f V; @u2 @u3       @U @U @U ¼ kða3  a2 Þ; f V; ¼ kða5  a4 Þ; ¼ kða5  a4 Þ: V; f V; @u4 @u5 @u6 ð34Þ

W. Zhao et al. / Applied Mathematics and Computation 218 (2011) 643–650

649

Substituting the above results into the tri-trace identity yields

d ð8b1 k2 þ 8a1 k þ 2u3 ða3  a2 Þ þ u5 ða5  a4 Þ  u6 ða4 þ a5 Þ  u4 ða2 þ a3 ÞÞ du 1 0 ða3  a2 Þ þ kðb3  b2 Þ B ða3  a2 Þ þ kðb3  b2 Þ C C B B ða5  a4 Þ þ kðb5  b4 Þ C C B B @ ða5  a4 Þ þ kðb5  b4 Þ C C: ¼ kc kc B C kða3  a2 Þ @k B C B C B  a Þ kða 3 2 C B A @ kða5  a4 Þ kða5  a4 Þ

ð35Þ

Comparison of the coefficients of k2n2 on the both side of (30) gives

d ð8b1;nþ2 þ u3 ða3;nþ1  a2;nþ1 Þ þ u5 ða5;nþ1  a4;nþ1 Þ  u6 ða4;nþ1 þ a5;nþ1 Þ  u4 ða2;nþ1 þ a3;nþ1 ÞÞ du 1 0 b3;nþ1  b2;nþ1 B b3;nþ1  b2;nþ1 C C B B b5;nþ1  b4;nþ1 C C B B b5;nþ1  b4;nþ1 C C ¼ ðc  2n  1ÞB B a3;nþ1  a2;nþ1 C: C B B a3;nþ1  a2;nþ1 C C B @ a5;nþ1  a4;nþ1 A a5;nþ1  a4;nþ1

ð36Þ

Taking n = 0, gives c = 2. Thus, we have

0

dHnþ1 du

b3;nþ1  b2;nþ1

1

B b3;nþ1  b2;nþ1 C C B C B B b5;nþ1  b4;nþ1 C C B C B b B 5;nþ1  b4;nþ1 C ¼B C; B a3;nþ1  a2;nþ1 C C B C B a B 3;nþ1  a2;nþ1 C C B @ a5;nþ1  a4;nþ1 A a5;nþ1  a4;nþ1

ð2Þ Hnþ1

1 where ¼  2n1 ð8b1;nþ2 þ u3 ða3;nþ1  a2;nþ1 Þ þ u5 ða5;nþ1  a4;nþ1 Þ  u6 ða4;nþ1 þ a5;nþ1 Þ  u4 ða2;nþ1 þ a3;nþ1 Þ . Therefore, the hierarchy (32) have Hamiltonian structure

ð2Þ

ut ¼ J 2 Pnþ1 ¼ J 2

dHnþ1 : du

ð37Þ

After a complex calculation, we get J 2 L2 ¼ L2 J 2 . Therefore, the hierarchy (32) is Liouville integrable. When u1 = u2 = u5 = u6 = 0, (32) can be reduced to (16), from J1, L1 and J2, L2, it is easy to get that (37) is an integrable coupling of (23). 4. Conclusions In this paper, a new Lie algebra G1 and its extension algebra G2 and their related loop algebras are constructed. And based f2 and G f1 , two Lax integrable hierarchy are presented respectively. Then with the help of the variational identity, we get on G the Hamiltonian of the hierarchy which are presented before, and prove that they are all Liouville integrable. In fact, based on the two new Lie algebras, we can obtain other integrable hierarchies by designing different isospectral problem. Acknowledgement The Project supported by the Natural Science Foundation of China (Grant No. 60971022). References [1] G.Z. Tu, The trace identity a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys. 30 (1989) 330–338. [2] W.X. Ma, A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. J. Contemp. Math. 13 (1992) 79–89. [3] W.X. Ma, An approach for constructing nonisospectral hierarchies of evolution equations, J. Phys. A 25 (1992) 719–726.

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[4] H.H. Dong, X.B. Gong, A (2+1)-dimensional multi-component AKNS integrable hierarchy and its expanding model, Chaos. Soliton. Fract. 33 (2007) 945– 950. [5] H.H. Dong, N. Zhang, A multi-component matrix loop algebra and its application, Commun. Theor. Phys. 44 (2005) 997–1001. [6] T.C. Xia, F.C. You, Multi-component Dirac equations hierarchy and its multi-component integrable couplings system, Chin. Phys. 16 (2007) 605–610. [7] X.B. Hu, An approach to generate supper extensions of integrable system, J. Phys. A 30 (1997) 619–632. [8] E.G. Fan, Integrable systems of derivative nonlinear schrodinger type and their multi-Hamiltonian structure, J. Phys. A 34 (2001) 513–519. [9] Y.F. Zhang, F.K. Guo, Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy, Chaos. Soliton. Fract. 34 (2007) 490–495. [10] F.K. Guo, F.X. Yu, A class of Lie algebras, J. Shandong Univ. Sci. Technol. 22 (3) (2003) 87–88. [11] Y.F. Zhang, J. Liu, Induced Lie algebras of a six-dimensional matrix Lie algebra, Common. Theor. Phys. 50 (2008) 289–294. [12] F.K. Guo, Y.F. Zhang, A type of new loop algebra and a generalized Tu formula, Commun. Theor. Phys. 51 (2009) 39–46. [13] H.H. Dong, X.R. Wang, The quadratic-form identity for constructing Hamiltonian structures of the NLS-MKdV hierarchy and multi-component Levi hierarchy, Chaos. Soliton. Fract. 37 (1) (2008) 245–251. [14] F.K. Guo, Y.F. Zhang, Two unified formula, Phys. Lett. A 366 (2007) 403–410. [15] W.X. Ma, B. Fuchssteiner, Integrable theory of the perturbation equations, Chaos. Soliton. Fract. 7 (1996) 1227–1250. [16] W.X. Ma, X.X. Xu, Y.F. Zhang, Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. Lett. A 351 (2006) 125–130. [17] W.X. Ma, M. Chen, Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A 39 (2006) 10787–10801. [18] W.X. Ma, Y. Zhang, Component-trace identities for Hamiltonian structures, Appl. Anal. 89 (2010) 457–472. [19] W.X. Ma, Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal.: Theor. Methods Appl. 71 (2009) 1716– 1726. [20] W.X. Ma,Variational identities and Hamiltonian structures, in: Nonlinear. Mod. Math. Phys. 1–27, W.X. Ma, X.B. Hu, Q.P. Liu, (Eds.), AIP Conference Proceedings, vol. 1212, American Institute of Physics, Melville, NY, 2010.