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Invertible linear transformations and the Lie algebras
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Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 www.elsevier.com/locate/cnsns
Invertible linear transformations and the Lie algebras Yufeng Zhang
a,*
, Honwah Tam b, Fukui Guo
q
c
a
Mathematical School, Liaoning Normal University, Dalian 116029, PR China Department of Computer Sciences, Hong Kong Baptist University, Hong Kong, PR China Information School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China b
c
Received 22 July 2006; accepted 28 July 2006 Available online 25 September 2006
Abstract With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity. 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.20.Sv; 02.30.Rz Keywords: Invertible linear transformation; Lie algebra; Soliton hierarchy
As we know, the Lie algebras have important applications in mathematical physics. For example, the Lie algebra sl(2) describes the related information on spin electronics. By making use of the Lie algebras, the corresponding various loop algebras could be generated, which are devote to generating integrable soliton hierarchies of evolution equations under the framework of the zero curvature equations. In the paper, we construct a larger types of Lie algebras by invertible linear transformations to deduce integrable couplings of soliton hierarchies. Specially, we again construct a few types of new Lie algebras from the viewpoint of the classification of the Lie algebras. Then via linear transformations, some Lie algebras could be worked out. The second major point in the paper presents the corresponding various loop algebras of the above Lie algebras. As the applications of the loop algebras, two (2 + 1)-dimensional larger integrable systems are
q
This work was supported by The National Science Foundation of China (10471139) and Hong Kong Research Grant Council grant number HKBU RGC 2016/05p. * Corresponding author. E-mail address: [email protected] (Y. Zhang). 1007-5704/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.07.011
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
683
obtained, one of them can reduce to the well-known AKNS hierarchy. The third point is that the quasiHamiltonian structure and the Hamiltonian structure of two integrable systems are obtained, respectively by using the quadratic-form identity. The point is important because the known trace identity proposed by Tu Guizhang cannot produce the Hamiltonian structures of such larger integrable systems. But in the paper, the problem is completely overcome. The Lie algebras and F, F (see below) presented in the paper have extensive applications, that is to say, we could generate many integrable couplings and their Hamiltonian structures by use of these Lie algebras and F , F . 1. Introduction One method for generating integrable hierarchies of soliton equations and the corresponding Hamiltonian structures was proposed by Tu [1], Ma called the method Tu scheme [2]. Ma, Hu and Guo et al. further developed it and obtained some rich results [2–6]. The method was conducted by constructing appropriate Lie algebras and utilizing the zero curvature equations. As long as the Lie algebras are given, the corresponding loop algebras can be establishing by various ways [7]. Hence, there exist two approaches to obtain integrable hierarchies by using the Tu scheme. One is construction of new Lie algebras. Another is an improvement of the formalisms of the zero curvature equations. The later may be decided by establishing various types of Lax pairs [8,9]. In the paper, we only want to discuss the first approach. Some Lie algebras are derived from the groups. t U ¼ E2 ; det U ¼ 1g, where E2 For example, the matrix multiplication group SU ð2Þ ¼ fU ; U 2 GLð2; CÞ; U stands for a 2 · 2 unit matrix. The corresponding Lie algebra presents sl(2) = span{L1, L2, L3}, where ! ! ! 1 0 1 1 0 i 1 1 0 L1 ¼ ; L2 ¼ ; L3 ¼ ; 2 1 0 2 i 0 2 0 1 [L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2, i2 = 1. From the viewpoint of geometry, sl(2) represents the tangent-vector space of all unit elements at E2, which describes the spin electronics [10]. Hu [3,4] studied the Lie algebra sl(N, C) and its some subalgebras. Specially, he investigated the super-Lie algebra sl(m/n) and its applications, which give us a powerful guidance. Guo Fukui [5] constructed a special Lie algebra, from which the NLS-MKdV hierarchy was obtained by use of the zero curvature equation. In this paper, we give a way to generate Lie algebras by means of invertible linear transformation with a common property: If denote one of the Lie algebras by G, its two direct-sum subalgebras are noted as G1 and G2, then G ¼ G1 G2 ;
G1 ffi Ak ; ½G1 ; G2 G2 ;
ð1Þ
where Ak are the known lower-dimensional Lie algebras. It is use of (1) that integrable couplings of soliton hierarchies [11,12] are obtained. Hence, the Lie algebras presented in the paper are major applications to deduce integrable couplings. In addition, a few ways to construct loop algebras of the Lie algebras are presented which are used to generate integrable couplings. In terms of the classification of Lie algebras given by Ma [13], some explicit Lie algebras are constructed. Finally, some solutions of a partial-differential equation system are obtained, and their Hamiltonian form-expressions are also given. 2. Invertible linear transformations Let G be a n-dimensional Lie algebra, whose basis be T1, T2, . . . , Tn. If A is a n · n invertible matrix, there exists a linear transformation on G such that d : G ! G;
hi ¼
n X
aij T j ;
i ¼ 1; 2; . . . ; n;
j¼1
then call (2) an invertible linear transformation.
A ¼ ðaij Þnn ;
ð2Þ
684
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Specially, if A presents the 0 A1 B A2 A¼@ .. .A
form as follows: 1 C A;
Ai ¼ ð1Þ and
Aj ¼
1 1
1 ; 1
ð3Þ
1 6 i; j 6 m;
m
then h1, h2, . . . , hn are linear independent, which are maybe a basis of the Lie algebra G, whose commuting operations are different from those of the Lie algebra T1, T2, . . . , Tn. How we prove the guess problem properly is worthy of study. In what follows, we take a few examples to show the above version, which shows the such matrix A as above exists. Example 1. A subalgebra of the Lie algebra A3 is given by 6
slð4Þ1 ¼ spanfT i gi¼1 ; 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 B 0 1 0 0 C B0 0 0 0C B1 0 0 0C B C B C B C T1 ¼ B C; T 2 ¼ B C; T 3 ¼ B C; @0 0 1 0 A @0 0 0 1A @0 0 0 0A 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 B 0 0 0 1 C B0 0 0 0C B0 0 1 0C B C B C B C T4 ¼ B C; T 5 ¼ B C; T 6 ¼ B C; @0 0 0 0 A @0 0 0 0A @0 0 0 0A
ð4Þ
0 0 0 0 0 0 0 0 0 0 0 0 ½T 1 ; T 2 ¼ 2T 2 ; ½T 1 ; T 3 ¼ 2T 3 ; ½T 2 ; T 3 ¼ T 1 ; ½T 1 ; T 4 ¼ 0; ½T 1 ; T 5 ¼ 2T 5 ; ½T 1 ; T 6 ¼ 2T 6 ; ½T 2 ; T 4 ¼ 2T 5 ; ½T 2 ; T 5 ¼ 0; ½T 2 ; T 6 ¼ T 4 ; ½T 3 ; T 4 ¼ 2T 6 ; ½T 3 ; T 5 ¼ T 4 ; ð5Þ ½T 3 ; T 6 ¼ ½T 4 ; T 5 ¼ ½T 4 ; T 6 ¼ ½T 5 ; T 6 ¼ 0: Let
0
1 B 0 1 B0 f1 B 0 1B B .. C F ¼ @ . A ¼ A1 T ¼ B 2B B0 B f6 @0 0
0 1 1 0 0 0
0 1 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 1
1 0 0 C C0 T 1 1 C 0 C . C CB @ .. A; 0 C C C 1 A T6 1
ð6Þ
then 0
1
1B B0 f1 ¼ B 2@0 0 0 0 B 1B0 f4 ¼ B 2@0 0 ½f1 ; f2 ¼ f3 ; ½f2 ; f5 ¼ f4 ;
1
0
0
0
1
0
0 C C C; 0 A
0 1 0 0 1 1 0 1 0 0 0 1 C C C; 0 0 0 A 0 0
0
0
0
1B B 1 f2 ¼ B 2@ 0 0 0 0 0 B 1B0 0 f5 ¼ B 2@0 0 0
0
1
1
0
0
0
0
0C C C; 1A
0
1
0
0
1B B1 f3 ¼ B 2@0
0
0
0C C C; 1A
0 0 0 1 0 1 0 0 1 0 B 1 0C 1B0 C C; f 6 ¼ B 2@0 0 0A
0
0
0
0
1
0
0
0 0 0
0 0 0 1 0 1 0 1 1 0 C C C; 0 0A 0
0
½f1 ; f3 ¼ f2 ; ½f2 ; f3 ¼ f1 ; ½f1 ; f4 ¼ 0; ½f1 ; f5 ¼ f6 ; ½f1 ; f6 ¼ f5 ; ½f2 ; f4 ¼ f5 ; ½f2 ; f6 ¼ 0; ½f3 ; f4 ¼ f6 ; ½f3 ; f5 ¼ 0; ½f3 ; f6 ¼ f4 ; ½f4 ; f5 ¼ ½f4 ; f6 ¼ ½f5 ; f6 ¼ 0:
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
685
6
Denote slð4Þ2 ¼ spanffi gi¼1 ; we see that (6) is an isomorphic map from the Lie algebra sl(4)1 to the sl (4)2 in the Lie algebra A3. ) Obviously, the Lie algebra sl (4)2 indeed possesses various commuting relations with the Lie algebra sl(4)1. Example 2. Consider a subalgebra of the Lie algebra A2 as follows [5]: slð3Þ1 ¼ spanfwi g5i¼1 ;
ð7Þ
where 0
1
B w1 ¼ B @0 0 0 0 B w4 ¼ B @0 0
0
0
1
0
0 1
0
1
0
C B C B C 1 0 C A; w 2 ¼ @ 0 0 0 A; 0 0 0 0 0 1 0 1 0 1 0 0 0 C B C B C 0 0C A; w5 ¼ @ 0 0 1 A; 0 0 0 0 0
½w1 ; w2 ¼ 2w2 ; ½w2 ; w4 ¼ 0;
½w1 ; w3 ¼ 2w3 ; ½w2 ; w5 ¼ w4 ;
0
B w3 ¼ B @1 0
½w2 ; w3 ¼ w1 ;
½w3 ; w4 ¼ w5 ;
0
0
1
0
C 0C A;
0
0
½w1 ; w4 ¼ w4 ;
½w1 ; w5 ¼ w5 ;
½w3 ; w5 ¼ ½w4 ; w5 ¼ 0:
If 0
g1
1
0
w1
1
01 2
0
B C B C B B g2 C B w2 C B 0 1 2 B C B C B B C B C B B g 3 C ¼ A2 B w 3 C ¼ B 0 1 B C B C B 2 B C B C B B g4 C B w4 C B 0 0 @ A @ A @ g5 w5 0 0
0
0
1 2
0
12
0
0
1
0
0
0
10
w1
1
CB C B C 0C CB w2 C CB C B C 0C CB w3 C: CB C B C 0C A@ w4 A 1
w5
Obviously, jA2j 5 0. Hence, (8) is an invertible linear transformation 0 1 0 1 0 1 0 0 0 1 0 0 1 B B C C 1B 1 1 B C B 0 1 0 C g1 ¼ B A; g2 ¼ 2 @ 1 0 0 A; g3 ¼ 2 @ 1 0 2@ 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 B C B C B C B g 4 ¼ @ 0 0 0 A; g 5 ¼ @ 0 0 1 C A; 0 0 0 0 0 0 ½g1 ; g2 ¼ g3 ; 1 ½g2 ; g4 ¼ g4 ; 2
½g1 ; g3 ¼ g2 ;
½g2 ; g3 ¼ g1 ;
1 ½g3 ; g4 ¼ g5 ; 2
ð8Þ
in A2. It is easy to see that 1 0 C 0C A; 0
1 ½g1 ; g4 ¼ g5 ; 2
1 ½g2 ; g5 ¼ g5 ; 2
1 ½g1 ; g5 ¼ g4 ; 2 1 ½g3 ; g5 ¼ g4 ; ½g4 ; g5 ¼ 0: 2
5
Denote slð3Þ2 ¼ spanfgi gi¼1 ; then sl (3)1 is different from sl(3)2. Example 3. Consider a subalgebra of the Lie algebra A5 slð6Þ1 ¼ spanf Te i g9i¼1 ;
ð9Þ
686
where
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 B 0 1 0 0 0 0 C B0 0 0 0 0 0C B1 0 0 0 0 0C C C C B B B C C C B B B B0 0 1 0 0 0 C B0 0 0 1 0 0C B0 0 0 0 0 0C C; Te 2 ¼ B C; Te 3 ¼ B C Te 1 ¼ B B 0 0 0 1 0 0 C B0 0 0 0 0 0C B 0 0 1 0 0 0 C; C C C B B B C C C B B B @0 0 0 0 1 0 A @0 0 0 0 0 1A @0 0 0 0 0 0A 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 B 0 0 0 1 0 0 C B0 0 0 0 0 0C B0 0 1 0 0 0C C C C B B B C C C B B B B0 0 0 0 1 0 C B0 0 0 0 0 1C B0 0 0 0 0 0C C C C B B B e e e T4 ¼ B C; T 5 ¼ B 0 0 0 0 0 0 C; T 6 ¼ B 0 0 0 0 1 0 C; C C B 0 0 0 0 0 1 C B B C C C B B B @0 0 0 0 0 0 A @0 0 0 0 0 0A @0 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 B 0 0 0 0 0 1 C B0 0 0 0 0 0C B0 0 0 0 1 0C C C C B B B C C C B B B B0 0 0 0 0 0 C B0 0 0 0 0 0C B0 0 0 0 0 0C C; Te 8 ¼ B C; Te 9 ¼ B C Te 7 ¼ B B0 0 0 0 0 0 C B0 0 0 0 0 0C B 0 0 0 0 0 0 C; C C C B B B C C C B B B @0 0 0 0 0 0 A @0 0 0 0 0 0A @0 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ½ Te 1 ; Te 2 ¼ 2 Te 2 ; ½ Te 1 ; Te 3 ¼ 2 Te 3 ; ½ Te 2 ; Te 3 ¼ Te 1 ; ½ Te 1 ; Te 4 ¼ 0; ½ Te 1 ; Te 5 ¼ 2 Te 5 ; ½ Te 1 ; Te 6 ¼ 2 Te 6 ; 0
½ Te 1 ; Te 7 ¼ 0; ½ Te 1 ; Te 8 ¼ 2 Te 8 ; ½ Te 1 ; Te 9 ¼ 2 Te 9 ; ½ Te 2 ; Te 5 ¼ 0; ½ Te 2 ; Te 4 ¼ 2 Te 5 ; ½ Te 2 ; Te 6 ¼ Te 4 ; ½ Te 2 ; Te 7 ¼ 2 Te 8 ; ½ Te 2 ; Te 8 ¼ 0; ½ Te 2 ; Te 9 ¼ Te 7 ; ½ Te 3 ; Te 4 ¼ 2 Te 6 ; ½ Te 3 ; Te 5 ¼ Te 4 ; ½ Te 3 ; Te 6 ¼ 0; ½ Te 3 ; Te 7 ¼ 2 Te 9 ; ½ Te 3 ; Te 8 ¼ Te 7 ; ½ Te 3 ; Te 9 ¼ 0; ½ Te 4 ; Te 5 ¼ 2 Te 8 ; ½ Te 4 ; Te 6 ¼ 2 Te 9 ; ½ Te 4 ; Te 7 ¼ ½ Te 4 ; Te 8 ¼ ½ Te 4 ; Te 9 ¼ 0; ½ Te 5 ; Te 6 ¼ Te 7 ; ½ Te 5 ; Te 7 ¼ ½ Te 5 ; Te 8 ¼ ½ Te 5 ; Te 9 ¼ ½ Te 6 ; Te 7 ¼ ½ Te 6 ; Te 8 ¼ ½ Te 6 ; Te 9 ¼ ½ Te 7 ; Te 8 ¼ ½ Te 7 ; Te 9 ¼ ½ Te 8 ; Te 9 ¼ 0: Let
1 h1 B . C B .. C B C B C B C H ¼ B h6 C ¼ A3 Te ; B h9 C B C B C @ h7 A h8 0 1 0 0 B 0 1 1 B B B0 1 1 B B0 0 0 1B B A3 ¼ B 0 0 0 2B B0 0 0 B B0 0 0 B B @0 0 0 0 0 0 0
ð10Þ
0 0 0 1 0 0 0 0 0
0 0 0 0 1 1 0 0 0
0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 1 0 0
1 0 0 0 0 C C C 0 0 C C 0 0 C C C 0 0 C; C 0 0 C C 0 0 C C C 1 1 A 1 1
Te ¼ ð Te 1 ; . . . ; Te 6 ; Te 9 ; Te 7 ; Te 8 ÞT ;
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
then we have that 0 1 0 B 0 1 B B 1B B0 0 h1 ¼ B 2B0 0 B B @0 0
1
0
0
0
0
0
0
0
0
1
0
0
B 1 0 C C B C B C 0 C 1B B 0 C; h2 ¼ B C B 0 2 0 C B C B @ 0 0 A
0 1 0 0
0
1
0 1 1 0 0 0 1 0 0 0 0 B 0 0 0 1 0 0 C B0 C B B C B B C B 1B0 0 0 0 1 0 C 1B B0 h4 ¼ B C ; h5 ¼ B 2 B 0 0 0 0 0 1 C 2B0 C B B C B B @0 0 0 0 0 0 A @0 0
0
0
0
0
0 0 0 0
0 B0 B B 1B B0 h7 ¼ B 2B0 B B @0 0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0 1 0
0
0
0
0
0
B1 0C C B C B C 0C 1B B0 C; h3 ¼ B C B0 2 0C B C B @0 1A
0
0
0 1 0 1 0 0 0 C B 0C B0 C B C 1C 1B B0 C ; h6 ¼ B 2B0 0C C B C B @0 0A
0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0
0 0 0 0 0
687
0 1 0 0 0 0
1
0 0 0 0 0C C C 0 0 1 0 0C C C; 0 1 0 0 0C C C 0 0 0 0 1A
0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 C C C 0 0 0 1 C C C; 0 0 1 0 C C C 0 0 0 0 A
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 C C B 0 0 0 1 0 C B B0 0 0 0 1 0C C C B0 0 0 0 B C 0 0 0 0 0C 1B 1B C B B0 0 0 0 0 0C C ; h8 ¼ B C ; h9 ¼ B 0 0 0 0 2B0 0 0 0 0 0C 2B 0 0 0 0 0C B0 0 0 0 C C B C C @ B @0 0 0 0 0 0A 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
1
0
1
C 0 1 C C C 0 0 C; C 0 0 C A 0 0
½h1 ; h2 ¼ h3 ; ½h1 ; h3 ¼ h2 ; ½h2 ; h3 ¼ h1 ; ½h1 ; h4 ¼ 0; ½h1 ; h5 ¼ h6 ; ½h1 ; h6 ¼ h5 ; ½h1 ; h7 ¼ h8 ; ½h1 ; h8 ¼ h7 ; ½h1 ; h9 ¼ 0; ½h2 ; h4 ¼ h5 ; ½h2 ; h5 ¼ h4 ; ½h2 ; h6 ¼ ½h2 ; h7 ¼ 0; ½h2 ; h8 ¼ h9 ; ½h2 ; h9 ¼ h8 ; ½h3 ; h4 ¼ h6 ; ½h3 ; h5 ¼ 0; ½h3 ; h6 ¼ h4 ; ½h3 ; h7 ¼ h9 ; ½h3 ; h8 ¼ 0; ½h3 ; h9 ¼ h7 ; ½h4 ; h5 ¼ h7 ; ½h4 ; h6 ¼ h8 ; ½h4 ; h7 ¼ ½h4 ; h8 ¼ ½h4 ; h9 ¼ 0; ½h5 ; h6 ¼ h9 ; ½h5 ; h7 ¼ ½h5 ; h8 ¼ ½h5 ; h9 ¼ ½h6 ; h7 ¼ ½h6 ; h8 ¼ ½h6 ; h9 ¼ ½h7 ; h8 ¼ ½h7 ; h9 ¼ ½h8 ; h9 ¼ 0: Note slð6Þ2 ¼ spanfhi g9i¼1 , then sl (6)1 and sl(6)2 are various. The Lie algebras in the above examples have the common property (1). As for example 1, let G = sl(4)1, G1 = span{T1, T2, T3}, G2 = span{T4, T5, T6}, then they satisfy (1). With the help of G1 (actually its resulting loop algebra), we may establish the zero curvature equation U t V xðnÞ þ ½U ; V ðnÞ ¼ 0 derived from the compatibility of the isospectral Lax pair ( ux ¼ U u; ut ¼ V ðnÞ u:
ð11Þ
ð12Þ
By means of the subalgebra G2, similarly we obtain the zero curvature equation ðnÞ
ðnÞ
U 1t V 1x þ ½U 1 ; V 1 ¼ 0; the corresponding Lax pair reads ( ux ¼ U 1 u; ðnÞ
ut ¼ V 1 u:
ð13Þ
ð14Þ
688
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Thus, the equations as below constitute the integrable couplings ( U t V xðnÞ þ ½U ; V ðnÞ ¼ 0; ðnÞ
ð15Þ
ðnÞ
U 1t V 1x þ ½U 1 ; V 1 ¼ 0:
The main goal for presenting this paper lies in giving a way to construct Lie algebras such that new integrable couplings and their Hamiltonian structures could be worked out. Since looking for the Hamiltonian structures of the integrable couplings first turns to establish an isomorphism between the matrix Lie algebra G and the column-vector Lie algebra Rn by using the quadraticform identity [14]. Hence,in the following, we want to take sl(6)i (i = 1, 2) for examples to illustrate the idea. P P 8a ¼ 9i¼1 ai Te i ; b ¼ 9i¼1 bi Te i 2 slð6Þ1 ; in terms of commuting relations among Te i ði ¼ 1; 2; . . . ; 9Þ, define ½a; bT1 ¼ ða1 ; . . . ; a9 ÞR1 ðbÞ; where
0
0
B b B 3 B B b2 B B B 0 B R1 ðbÞ ¼ B B 0 B B 0 B B 0 B B @ 0 0
ð16Þ
2b2
2b3
0
2b5
2b6
0
2b8
2b9
2b1 0
0 2b1
b6 b5
2b4 0
0 2b4
b9 b8
2b7 0
0 2b7
0
0
0
2b2
2b3
0
2b5
0 0
0 0
b3 b2
2b1 0
0 2b1
b6 b5
2b4 0
0 0
0 0
0 0
0 0
0 0
0 b3
2b2 2b1
0
0
0
0
0
b2
0
1
C C C C C C 2b6 C C 0 C C: C 2b4 C C 2b3 C C C 0 A 2b1
9
It is easy to verify that R is a Lie algebra equipped with the commutator [a, b]1. Define a functional [14]: fa; bg ¼ aT Fb;
ð17Þ
~ 1 ; which is a corresponding loop algebra of the Lie algebra sl(6)1, F satisfies [14]: where a; b 2 slð6Þ T
R1 ðbÞF ¼ ðR1 ðbÞF Þ ;
F ¼ F T:
ð18Þ
In order to employ the quadratic-form identity to deduce the Hamiltonian structures of integrable couplings, we need to solve the matrix Eq. (18) for F. A direct calculation, we obtain 1 0 2 0 0 2 0 0 2 0 0 B0 0 1 0 0 1 0 0 1C C B C B B0 1 0 0 1 0 0 1 0C C B C B B2 0 0 2 0 0 0 0 0C C B C F ¼B B 0 0 1 0 0 1 0 0 0 C: C B B0 1 0 0 1 0 0 0 0C C B B2 0 0 0 0 0 0 0 0C C B C B @0 0 1 0 0 0 0 0 0A 0 1 0 0 0 0 0 0 0 P9 P9 T As for the Lie algebra sl(6)2, take a ¼ i¼1 ai hi ; b ¼ i¼1 bi hi ; after calculations, we have ½a; b2 ¼ ða2 b3 a3 b2 ; a1 b3 a3 b1 ; a1 b2 a2 b1 ; a2 b5 a5 b2 þ a3 b6 a6 b3 ; a4 b2 a2 b4 þ a6 b1 a1 b6 ; a3 b4 a4 b3 þ a5 b1 a1 b 5 ; a 1 b 8 a 8 b 1 þ a 4 b 5 a 5 b 4 þ a 9 b 3 a 3 b 9 ; a 1 b 7 a 7 b 1 þ a 6 b 4 a 4 b 6 þ a 9 b 2 a 2 b 9 ; a 2 b 8 a 8 b 2 þ a 5 b 6 a 6 b 5 þ a7 b3 a3 b7 Þ ¼ ða1 ; . . . ; a9 ÞR2 ðbÞ,
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
0
0
B b B 3 B B b2 B B B 0 B R2 ðbÞ ¼ B B 0 B B 0 B B 0 B B @ 0 0
b3
b2
0
b6
b5
b8
b7
0
b1
b5
b4
0
0
b9
b1 0
0 0
b6 0
0 b2
b4 b3
b9 b5
0 b6
0 0
0 0
b2 b3
0 b1
b1 0
b4 0
0 b4
0
0
0
0
0
0
b1
0 0
0 0
0 0
0 0
0 0
b1 b3
0 b2
0
689
1
b8 C C C b7 C C C 0 C C b6 C C: C b6 C C b3 C C C b2 A 0
The constant symmetric matrix F satisfies T R2 ðbÞF ¼ ðR2 ðbÞF Þ
which gives rise to 0 1 0 B 0 1 B B B 0 0 B B 0 B 1 B F ¼ B 0 0 B B 1 B 0 B B 0 1 B B 0 @ 0 1 0
ð19Þ
0 0
1 0
0 0
0 1
0 1
0 0
1 0
0 1
1 0
0 0
0 0
1 0
1
0
1
0
0
0
0 0
0 0
0 0
1 0
0 0
0 0
1 0
0 0
0 0
0 0
0 0
0 0
1 1 0 C C C 0 C C C 0 C C 0 C C C 0 C C 0 C C C 0 A 0
3. Various loop algebras of the Lie algebras An important application of the loop algebras is devoting to producing new integrable hierarchies of soliton equations. In general, various loop algebras usually can give various integrable soliton hierarchies under the framework of the zero curvature equations. Guo Fukui once proposed the useful methods for obtaining loop algebras [7]. In the section, we present some ways of generating loop algebras from the Lie algebras presented in the paper. Take sl(6)i (i = 1, 2) for examples. As for the Lie algebra sl(6)1, its simplest loop algebra is given by ~ 11 ¼ spanf Te i ðnÞg9i¼1 ; slð6Þ
ð20Þ
Te i ðnÞ ¼ Te i kn ; n ¼ 0; 1; 2; . . . ; the commutator is defined as ½ Te i ðmÞ; Te j ðnÞ ¼ ½ Te i ; Te j kmþn ;
i 6¼ j; m; n 2 Z:
ð21Þ
Another loop algebra with powers of k being k2n, k2n+1 is given by ~slð6Þ12 ¼ spanf Te i ðnÞg9i¼1 ;
ð22Þ
where Te 1 ðnÞ ¼ Te 1 k2n ; Te 2 ðnÞ ¼ Te 2 k2nþ1 ; Te 6 ðnÞ ¼ Te 6 k2nþ1 ; Te 7 ðnÞ ¼ Te 7 k2n ;
Te 3 ðnÞ ¼ Te 3 k2nþ1 ; Te 8 ðnÞ ¼ Te 8 k2nþ1 ;
Te 4 ðnÞ ¼ Te 4 k2n ; Te 5 ðnÞ ¼ Te 5 k2nþ1 ; Te 9 ðnÞ ¼ Te 9 k2nþ1 ; ½ Te 1 ðmÞ; Te 2 ðnÞ ¼ 2 Te 2 ðm þ nÞ;
½ Te 1 ðmÞ; Te 3 ðnÞ ¼ 2 Te 3 ðm þ nÞ; ½ Te 2 ðmÞ; Te 3 ðnÞ ¼ Te 1 ðm þ n þ 1Þ; ½ Te 1 ðmÞ; Te 4 ðnÞ ¼ 0; ½ Te 1 ðmÞ; Te 5 ðnÞ ¼ 2 Te 5 ðm þ nÞ; ½ Te 1 ðmÞ; Te 6 ðnÞ ¼ 2 Te 6 ðm þ nÞ; ½ Te 1 ðmÞ; Te 7 ðnÞ ¼ 0;
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Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
½ Te 1 ðmÞ; Te 8 ðnÞ ¼ 2 Te 8 ðm þ nÞ; ½ Te 1 ðmÞ; Te 9 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 2 ðmÞ; Te 5 ðnÞ ¼ 0; ½ Te 2 ðmÞ; Te 4 ðnÞ ¼ 2 Te 5 ðm þ nÞ; ½ Te 2 ; Te 6 ðnÞ ¼ Te 4 ðm þ n þ 1Þ; ½ Te 2 ðmÞ; Te 7 ðnÞ ¼ 2 Te 8 ðm þ nÞ; ½ Te 2 ðmÞ; Te 8 ðnÞ ¼ 0; ½ Te 2 ðmÞ; Te 9 ðnÞ ¼ Te 7 ðm þ n þ 1Þ; ½ Te 3 ðmÞ; Te 4 ðnÞ ¼ 2 Te 6 ðm þ nÞ; ½ Te 3 ðmÞ; Te 5 ðnÞ ¼ Te 4 ðm þ n þ 1Þ; ½ Te 3 ðmÞ; Te 8 ðnÞ ¼ Te 7 ðm þ n þ 1Þ;
½ Te 3 ðmÞ; Te 6 ðnÞ ¼ 0; ½ Te 3 ðmÞ; Te 9 ðnÞ ¼ 0;
½ Te 3 ðmÞ; Te 7 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 4 ðmÞ; Te 5 ðnÞ ¼ 2 Te 8 ðm þ nÞ;
½ Te 4 ðmÞ; Te 6 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 4 ðmÞ; Te 7 ðnÞ ¼ ½ Te 4 ðmÞ; ½ Te 5 ðmÞ; Te 6 ðnÞ ¼ Te 7 ðm þ n þ 1Þ;
Te 8 ðnÞ ¼ ½ Te 4 ðmÞ; Te 9 ðnÞ ¼ 0;
½ Te 5 ðmÞ; Te 7 ðnÞ ¼ ½ Te 5 ðmÞ; Te 8 ðnÞ ¼ ½ Te 5 ðmÞ; Te 9 ðnÞ ¼ ½ Te 6 ðmÞ; Te 7 ðnÞ ¼ ½ Te 6 ðmÞ; Te 8 ðnÞ ¼ ½ Te 6 ðmÞ; Te 9 ðnÞ ¼ ½ Te 7 ðmÞ; Te 8 ðnÞ ¼ ½ Te 7 ðmÞ; Te 9 ðnÞ ¼ ½ Te 8 ðmÞ; Te 9 ðnÞ ¼ 0; m; n 2 Z; n ¼ 0; 1 2; . . . The general loop algebra which corresponds to the loop algebra (22) reads that ~slð6Þ13 ¼ spanf Te i ðj; mÞg9i¼1 ; Te i ðj; mÞ ¼ Te i k2nþj ; j ¼ 0; 1: The resulting commuting operations are given by ( i þ j 6 1; 2 Te 2 ði þ j; m þ nÞ; e e ½ T 1 ði; mÞ; T 2 ðj; nÞ ¼ 2 Te 2 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 3 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 3 ðj; nÞ ¼ e 2 T 3 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 1 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 3 ðj; nÞ ¼ ½ Te 1 ði; mÞ; Te 4 ðj; nÞ ¼ 0; e T 1 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 5 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 5 ðj; nÞ ¼ 2 Te 5 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 6 ði þ j; m þ nÞ; e e ½ T 1 ði; mÞ; T 6 ðj; nÞ ¼ ½ Te 1 ði; mÞ; Te 7 ðj; nÞ ¼ 0; e 2 T 6 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 8 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 8 ðj; nÞ ¼ 2 Te 8 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2 ( i þ j 6 1; 2 Te 9 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 9 ðj; nÞ ¼ e 2 T 9 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 5 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 5 ðj; nÞ ¼ 0; ½ Te 2 ði; mÞ; Te 4 ðj; nÞ ¼ 2 Te 5 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 4 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 6 ðj; nÞ ¼ e T 4 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 8 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 2 ði; mÞ; Te 8 ðj; nÞ ¼ 0; 2 Te 8 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 7 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 9 ðj; nÞ ¼ e T 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2;
ð23Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
( ½ Te 3 ði; mÞ; Te 4 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 5 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 7 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 8 ðj; nÞ ¼ ( ½ Te 4 ði; mÞ; Te 5 ðj; nÞ ¼ ( ½ Te 4 ði; mÞ; Te 6 ðj; nÞ ¼
2 Te 6 ði þ j; m þ nÞ; 2 Te 6 ði þ j 2; m þ n þ 1Þ;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; Te 4 ði þ j; m þ nÞ; e T 4 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; 2 Te 9 ði þ j; m þ nÞ; 2 Te 9 ði þ j 2; m þ n þ 1Þ;
½ Te 3 ði; mÞ; Te 6 ðj; nÞ ¼ 0;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; Te 7 ði þ j; m þ nÞ; e T 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; 2 Te 8 ði þ j; m þ nÞ; 2 Te 8 ði þ j 2; m þ n þ 1Þ;
691
½ Te 3 ði; mÞ; Te 9 ðj; nÞ ¼ 0;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; 2 Te 9 ði þ j; m þ nÞ; e 2 T 9 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2;
½ Te 4 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 4 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 4 ði; mÞ; Te 9 ðj; nÞ ¼ 0; ( i þ j 6 1; Te 7 ði þ j; m þ nÞ; ½ Te 5 ði; mÞ; Te 6 ðj; nÞ ¼ Te 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ½ Te 5 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 5 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 5 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 7 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 7 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 8 ði; mÞ; Te 9 ðj; nÞ ¼ 0: Another often used loop algebra of the Lie algebra (9) is that ~ 14 ¼ spanf Te i ðj; nÞg9i¼1 ; slð6Þ
j ¼ 0; 1; 2;
Te i ðj; nÞ ¼ Te i k3nþj ; i ¼ 1; 2; . . . ; 9: The commutator is defined as ( i þ j < 3; ½ Te k ; Te l ði þ j; m þ nÞ; ½ Te k ði; mÞ; Te l ðj; nÞ ¼ ½ Te k ; Te l ði þ j 3; m þ n þ 1Þ; i þ j P 3; For example,
(
½ Te 1 ði; mÞ; Te 2 ðj; nÞ ¼
ð24Þ
1 6 k; l 6 9; m; n 2 Z:
i þ j < 3; 2 Te 2 ði þ j; m þ nÞ; e 2 T 2 ði þ j 3; m þ n þ 1Þ; i þ j P 3:
In general, the loop algebra corresponding to the Lie algebra (9) is given by ~ 15 ¼ spanfT i ðj; nÞg9i¼1 ; slð6Þ where Te i ðj; nÞ ¼ Te i k
ð25Þ
Nnþj
; j ¼ 0; 1; 2; . . . ; N 1; N is an arbitrary natural number, the commutator is defined as ( i þ j < N; ½ Te k ; Te l ði þ j; m þ nÞ; ½ Te k ði; mÞ; Te l ðj; nÞ ¼ 1 6 k; l 6 9; m; n 2 Z: ½ Te k ; Te l ði þ j N ; m þ n þ 1Þ; i þ j P N ;
As for the Lie algebra sl(6)2, its simplest loop algebra is given by ~ 21 ¼ spanfhi ðnÞg9i¼1 ; hi ðnÞ ¼ hi kn ; slð6Þ
ð26Þ
the commutator is defined as [hi(m), hj(n)] = [hi, hj]km+n, 1 6 k, l 6 9. ~ 22 ; slð6Þ ~ 23 ; slð6Þ ~ 24 and slð6Þ ~ 25 which correspond to slð6Þ ~ 1i ði ¼ 2; 3; 4; 5Þ Similarly, the loop algebras slð6Þ respectively, are obtained. Here we omit them.
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Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
It is easy to see that the spectral matrices generated by the Lie algebras sl(3)i (i = 1, 2) are presented as U 22 U 1 U¼ ð27Þ ; U 1 ¼ ða; bÞT ; 0 0 the spectral matrices given by the Lie algebra sl(4)i, sl(6)i (i = 1, 2) are presented, respectively U 22 U 1 a b U¼ ; ; U1 ¼ c d 0 U 22 0 1 U U1 U2 B C U ¼ @ 0 U U 1 A; U i ði ¼ 1; 2Þ are 2 2 matrices: 0 The general case 0 U B B B0 U ¼B B .. @ . 0
0
0
0
0
ð31Þ
0
Furthermore, the more case of (29) was exhibited [13] 0 1 U U a1 U a2 B C U ¼ @ 0 U U a3 A: 0
ð30Þ
U
and the general case of (27) was given by Ma [13] 0 1 U U a1 U a2 B C U ¼ @ 0 U U a3 A; 0
ð29Þ
U
of (28) and (29) was constructed by Ma [13] 1 U a1 . . . U av .. C .. C . C U . C; .. .. C . U a1 A . ...
ð28Þ
ð32Þ
U a4
In terms of the spectral matrices (30)–(32), a kind of classification of the Lie algebras was proposed in [13]. However, the explicit matrix forms of (31) and (32) were not given. In what follows, we want to explicitly construct a few Lie algebras to satisfy (31) and (32), which fill up Ma’s work. Take 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 C C C B B B B 0 1 0 0 0 C B0 0 0 0 0C B1 0 0 0 0C C C C B B B C C C B B e1 ¼ B B 0 0 1 0 0 C; e 2 ¼ B 0 0 0 1 0 C; e 3 ¼ B 0 0 0 0 0 C; C C C B B B @ 0 0 0 1 0 A @0 0 0 0 0A @0 0 1 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 C C C B B B B 0 0 0 1 0 C B0 0 0 0 0C B0 0 1 0 0C C C C B B B C C C B B e4 ¼ B B 0 0 0 0 0 C; e 5 ¼ B 0 0 0 0 0 C; e 6 ¼ B 0 0 0 0 0 C; C C C B B B @0 0 0 0 0A @0 0 0 0 0A @0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
0
0 0 0 0 1
B B0 B e7 ¼ B B0 B @0 0
1
0
0 0 0 0 0
C B 0 0 0 0C B0 C B B0 ; e 0 0 0 0C ¼ 8 C B C B 0 0 0 0A @0 0 0 0 0 0
1
0
0 0 0 0 0
C B 0 0 0 1C B0 C B B0 ; e 0 0 0 0C ¼ 9 C B C B 0 0 0 0A @0 0 0 0 0 0
1
0
693
0 0 0 0 0
C B 0 0 0 0C B0 C B B0 ; e 0 0 0 1C ¼ 10 C B C B 0 0 0 0A @0 0 0 0 0 0
1
C 0 0 0 0C C 0 0 0 0C C; C 0 0 0 1A 0 0 0 0
it is easy to find that ½e1 ; e2 ¼ 2e2 ; ½e2 ; e4 ¼ 2e5 ;
½e1 ; e3 ¼ 2e3 ; ½e2 ; e5 ¼ 0;
½e2 ; e3 ¼ e1 ; ½e2 ; e6 ¼ e4 ;
½e1 ; e4 ¼ 0;
½e1 ; e5 ¼ 2e5 ;
½e3 ; e4 ¼ 2e6 ;
½e3 ; e5 ¼ e4 ;
½e1 ; e6 ¼ 2e6 ;
½e3 ; e6 ¼ ½e4 ; e5 ¼ ½e4 ; e6 ¼ ½e5 ; e6 ¼ 0; ½e1 ; e7 ¼ e7 ; ½e1 ; e8 ¼ e8 ; ½e1 ; e9 ¼ e9 ; ½e1 ; e10 ¼ e10 ; ½e2 ; e7 ¼ 0; ½e2 ; e8 ¼ e7 ; ½e2 ; e9 ¼ 0; ½e2 ; e10 ¼ e9 ; ½e3 ; e7 ¼ e8 ; ½e3 ; e8 ¼ 0; ½e3 ; e9 ¼ e10 ; ½e3 ; e10 ¼ ½e4 ; e7 ¼ ½e4 ; e8 ¼ 0; ½e4 ; e9 ¼ e7 ; ½e4 ; e10 ¼ e8 ; ½e5 ; e7 ¼ ½e5 ; e8 ¼ ½e5 ; e9 ¼ 0; ½e5 ; e10 ¼ e7 ; ½e6 ; e7 ¼ ½e6 ; e8 ¼ 0;
½e6 ; e9 ¼ e8 ; ½e6 ; e10 ¼ 0;
½ei ; ej ¼ 0
ð7 6 i; j 6 10Þ:
Let U = ae1 + be2 + ce3 + de4 + ee5 + fe6 + ge7 + he8 + ke9 + we10, then 1 0 a b d e g C B B c a f d h C C B U ¼B kC C: B0 0 a b C B @ 0 0 c a w A 0 0 0 0 0 10
Assume a linear map d : spanfei gi¼1 0 1 0 0 0 B 0 1 1 0 B B B0 1 1 0 B 1 0 B0 0 0 1 w1 B B Bw C B 2 C 1B0 0 0 0 B . C¼ B B . C 2B0 0 0 0 B @ . A B B0 0 0 0 w10 B B0 0 0 0 B B @0 0 0 0 0 0 0 0
ð33Þ
10
! spanfwi gi¼1 ,
1
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 1
0 1
0 0
0 0
0 0
1 0
0
0
0C C C 0C C0 1 e1 0C C CB e C 0 CB 2 C CB . C; B C 0C C@ .. A C 0C C e10 0C C C 0A
1 0 0
0 0
2 0
0 2
0 0
0 0
0 0
0 0
0 0
2 0
ð34Þ
2
then w1 ¼ 12 e1 ; w2 ¼ 12 ðe2 e3 Þ; w3 ¼ 12 ðe2 þ e3 Þ; w4 ¼ e4 ; w5 ¼ 12 ðe5 þ e6 Þ; w6 ¼ 12 ðe5 e6 Þ; w7 ¼ e7 ; w8 ¼ e8 ; w9 ¼ e9 ; 10 10 w10 ¼ e10 . It is easy to see that spanfwi gi¼1 is also a Lie algebra different form spanfei gi¼1 . If let 1 0 a1 a2 þ a3 a4 a5 þ a6 2a7 C B a1 a5 a6 a4 2a8 C B a2 þ a3 10 X C 1B ð35Þ U¼ ai e i ¼ B 0 0 a1 a2 þ a3 2a9 C C: B 2B C i¼1 0 0 a2 þ a3 a1 2a10 A @ 0 0 0 0 0 Then (33) and (35) are the spectral cases of (31), which have extensive applications on generating integrable couplings. In the following, we construct explicit Lie algebras whose spectral matrices are of the form (32). First, consider a simplest case.
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Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Let 0
1
B p1 ¼ @ 0 0
0 0
B p4 ¼ @ 0 0
0
0
1
C 1 0 A; 0 0 1 0 1 C 0 0 A; 0
0
0
1 0
B p2 ¼ @ 0 0
0
0 B p5 ¼ @ 0
0
0
1
0
C 0 0 A; 0 0 1 0 0 C 0 1 A; 0
1
0
0
0
B p3 ¼ @ 1
0
C 0 A;
0
0
0
0 1
0 B p6 ¼ @ 0
0
0
0
C 0 A;
0
0
1
0
then we have ½p1 ; p2 ¼ 2p2 ; ½p2 ; p3 ¼ p1 ;
½p1 ; p3 ¼ 2p3 ; ½p2 ; p4 ¼ 0;
½p2 ; p5 ¼ p4 ;
½p3 ; p5 ¼ ½p3 ; p6 ¼ ½p4 ; p5 ¼ 0; P6 Set U ¼ i¼1 ai pi ; then 0 1 a1 a2 a4 B C U ¼ @ a3 a1 a5 A: 0
0
½p1 ; p4 ¼ p4 ;
½p1 ; p5 ¼ p5 ;
½p2 ; p6 ¼ 0;
½p4 ; p6 ¼ p4 ;
½p1 ; p6 ¼ 0;
½p3 ; p4 ¼ p5 ;
½p5 ; p6 ¼ p5 :
ð36Þ
a6
Make a linear transformation 0
6
d : spanfpi gi¼1
q1
1
0
1 0
q6
1
0 0
1
1
0 0
0
1
0 0
0
0
2 0
0
0
0 2
CB C 0 CB p 2 C CB C B C 0C CB p 3 C CB C; B C 0C CB p 4 C CB C 0 A@ p 5 A
0 0
0
0 0
2
B B C B0 B q2 C B B C Bq C 1B0 B B 3C 6 ! spanfqi gi¼1 ; B C ¼ B B q4 C 2 B 0 B B C B B C @0 @ q5 A
0
10
0
p1
p6
6
then spanfqi gi¼1 is a Lie algebra, whose commuting operations are as follows: 1 ½q1 ; q5 ¼ q5 ; 2 1 1 1 ½q1 ; q6 ¼ 0; ½q2 ; q4 ¼ q5 ; ½q2 ; q5 ¼ q4 ; ½q2 ; q6 ¼ 0; ½q3 ; q4 ¼ q5 ; 2 2 2 1 1 1 ½q3 ; q5 ¼ q4 ; ½q3 ; q6 ¼ ½q4 ; q5 ¼ 0; ½q4 ; q6 ¼ q4 ; ½q5 ; q6 ¼ q5 : 2 2 2 P6 If let U ¼ i¼1 bi qi ; then it is easy to see that 0 1 b1 b2 þ b3 b4 1B C b1 b5 A : U ¼ @ b2 b3 2 0 0 b6 ½q1 ; q2 ¼ q3 ;
½q1 ; q3 ¼ q2 ;
½q2 ; q3 ¼ q1 ;
1 ½q1 ; q4 ¼ q4 ; 2
ð37Þ
Obviously, (36) and (37) are the same-type matrices, and are the special cases of (32). Next, we consider the more complicated cases of (32). There are two special cases basic on a subalgebra of the Lie algebra A5. First, we present the subalgebra. Let 5 ¼ spanfvi g6 ; A i¼1
ð38Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
695
where 0
1
B B0 B B0 B v1 ¼ B B0 B B @0 0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0 B B0 B B0 B v4 ¼ B B0 B B @0
1 0
0
0
0
0
0
1
0
0
0
0
0 0
1
0
0 0
0
0
0 0
0
0
0 0
0
0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0
0 0
0
0
0 B B1 B B0 B v5 ¼ B B0 B B @0
0
0
0 0
0
0
B B0 B B0 B v2 ¼ B B0 B B @0
0
0 0
½v2 ; v3 ¼ v6 ;
0
0 0
C 0C C 0C C C; 0C C C 0A
C 0C C 0C C C; 0C C C 0A
½v1 ; v2 ¼ 0;
1
0
½v1 ; v3 ¼ 0;
0 0
0
0
0 0
0
1
0 0
0
0
0 0
0
0
0 0
0
½v3 ; v6 ¼ 2v3 ;
½v4 ; v6 ¼ ½v5 ; v6 ¼ 0:
If set U ¼
0
0
½v2 ; v6 ¼ 2v2 ;
½v4 ; v5 ¼ v1 ;
1
0
½v1 ; v4 ¼ 2v4 ;
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 B B0 B B0 B v6 ¼ B B0 B B @0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
1
1 0 1 0 C 0 C C 0 C C C; 0 C C C 0 A
0
0
0 0
0
1
B B0 B B0 B v3 ¼ B B0 B B @0 0
0 0
1
0
C 0C C 0C C C; 0C C C 1A
C 0C C 0C C C; 0C C C 0A
½v1 ; v6 ¼ ½v2 ; v4 ¼ ½v2 ; v5 ¼ 0; P6
0
C 0 0C C 0 0C C C; 0 0C C C 0 0A
½v1 ; v5 ¼ 2v5 ;
½v3 ; v4 ¼ ½v3 ; v5 ¼ 0;
then it is easy to see that 1 a4 0 0 0 0 C a1 0 0 0 0 C C 0 a1 a4 0 0 C C C; 0 a5 a1 0 0 C C C 0 0 0 a6 a2 A
i¼1 ai vi ;
0
a1 B B a5 B B0 B U ¼B B0 B B @0 0
0
0
0
a3
a6
which is a special case of (32). Now we begin to enlarge the Lie algebra (38) into two bigger Lie algebras with aspect to dimensional numbers Case 1: Set 0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
B B0 B B0 B v7 ¼ B B0 B B @0 0
B B0 B B0 B v9 ¼ B B0 B B @0 0
1
1
0
0
0
C 0C C 0C C C; 0C C C 0A
B B0 B B0 B v8 ¼ B B0 B B @0
0
0
0
0 0
0
0
0
0 0
0
1
0 0
0
0
0 0
0
0
0 0
0
0
C 0C C 0C C C; 0C C C 0A
0 0 0 B B0 B B0 B v10 ¼ B B0 B B @0
0 0
0
0
0
1
0
1
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
C 1C C 0C C C; 0C C C 0A
0
0
0 0
0
1 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C 0 0C C 0 0C C C; 0 0C C C 0 0A
0 0
0
0
0 0
696
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 10
then we find that spanfvi gi¼1 ¼: A51 is a Lie algebra, where ½v1 ; v7 ¼ v7 ;
½v2 ; v7 ¼ ½v4 ; v7 ¼ 0;
½v5 ; v7 ¼ v8 ;
½v6 ; v7 ¼ v7 ;
½v1 ; v8 ¼ v8 ;
½v2 ; v8 ¼ 0;
½v3 ; v8 ¼ v9 ;
½v4 ; v8 ¼ v7 ;
½v7 ; v8 ¼ 0;
½v1 ; v9 ¼ v9 ;
½v2 ; v9 ¼ v8 ;
½v5 ; v9 ¼ 0;
½v6 ; v9 ¼ v9 ;
½v7 ; v9 ¼ ½v8 ; v9 ¼ 0;
½v2 ; v10 ¼ v7 ;
½v3 ; v10 ¼ ½v4 ; v10 ¼ 0;
½v6 ; v10 ¼ v10 ; Set U ¼
P10
then
0
a4
0
0
a10
a1
0
0
a9
0
a1
a4
0
0
a5
a1
0
0
0
0
a6
0
0
0
a3
0 0
B B a5 B B0 B U ¼B B0 B B @0 0
½v5 ; v8 ¼ 0;
½v6 ; v8 ¼ v8 ;
½v3 ; v9 ¼ 0;
½v4 ; v9 ¼ v10 ;
½v1 ; v10 ¼ v10 ;
½v5 ; v10 ¼ v9 ;
½v7 ; v10 ¼ ½v8 ; v10 ¼ ½v9 ; v10 ¼ 0:
i¼1 ai vi ;
a1
½v3 ; v7 ¼ v10 ;
a7
1
C a8 C C 0 C C C: 0 C C C a2 A a6
Case 2: Let 0
0
B B0 B B0 B v11 ¼ B B0 B B @0 0
0 0
B B0 B B0 B v13 ¼ B B0 B B @0 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 1 0 0 C 0 0C C 0 1C C C; 0 0C C C 0 0A
0
0
0
0 0
0
C 0 0C C 1 0C C C; 0 0C C C 0 0A
0
B B0 B B0 B v12 ¼ B B0 B B @0 0
0 0
B B0 B B0 B v14 ¼ B B0 B B @0 0
0
1
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0 0
0 0
0 0 0 0
0
0
0 0
0
0
0 0
0
0
0 1
0
0
0 0
0 1 0 C 0C C 0C C C; 0C C C 0A
0
0
0 0
0
C 0C C 0C C C; 1C C C 0A
then span{v1, v2, v3, v4, v5, v6, v11, v12, v13, v14} ¼: A52 is also a Lie algebra, where ½v1 ; v11 ¼ v11 ; ½v6 ; v11 ¼ v11 ;
½v2 ; v11 ¼ v13 ;
½v3 ; v11 ¼ ½v4 ; v11 ¼ 0;
½v1 ; v12 ¼ v12 ;
½v2 ; v12 ¼ 0;
½v4 ; v12 ¼ v13 ;
½v5 ; v12 ¼ 0;
½v6 ; v12 ¼ v12 ;
½v1 ; v13 ¼ v13 ;
½v2 ; v13 ¼ 0;
½v3 ; v13 ¼ v11 ;
½v5 ; v13 ¼ v12 ;
½v6 ; v13 ¼ v13 ;
½v4 ; v14 ¼ v11 ;
½v5 ; v14 ¼ 0;
½v11 ; v13 ¼ ½v11 ; v14 ¼ ½v12 ; v14 ¼ ½v13 ; v14 ¼ 0:
½v3 ; v12 ¼ v14 ;
½v11 ; v12 ¼ 0;
½v1 ; v14 ¼ v14 ; ½v6 ; v14 ¼ v14 ;
½v5 ; v11 ¼ v14 ;
½v4 ; v13 ¼ 0; ½v2 ; v14 ¼ v12 ;
½v3 ; v14 ¼ 0;
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
The corresponding spectral 0 0 a1 a4 B B a5 a1 0 B B 0 a1 B0 U ¼B B0 0 a5 B B B0 0 0 @ 0
0
0
matrix presents 0
0
0
0
a4
a11
a1
a14
0
a6
0
a3
0
697
1
C 0 C C C a13 C C: a12 C C C a2 C A a6
Remark 1. There are many Lie algebras whose spectral matrices are of the forms (31) and (32), we do not want 5 ; A51 to go on constructing them in the paper. It is easy to find that the semi-direct sum of the Lie algebras A and A52 is also a Lie algebra, the resulting spectral matrix belongs to the form (32). Similar to (34), some new 5 ; A51 and A52 are worked out by use of invertible linear transformations. Lie algebras generated by A 4. Some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy Li [15] presented the Lax pair from the self-dual Yang-Mills equation ( ux ¼ Mu; ut ¼ P ðkÞuy þ N u;
ð39Þ
whose compatibility is the following (2 + 1)-dimensional partial-differential equation hierarchy M t N x þ ½M; N P ðkÞM y ¼ 0;
ð40Þ
where P(k) = a0 + a1k + , k is a spectral parameter. If let P(k) = 0, then (40) reduces to the standard zero curvature equation M t N x þ ½M; N ¼ 0:
ð41Þ
In the section, we want to take the loop algebras (20) and (26) for examples to deduce some solutions of Eq. (40) which present the (2 + 1)-dimensional nonlinear evolution equations. Example 4. Consider the spectral problem ux ¼ Mu; M ¼ i Te 1 ð1Þ iqr Te 1 ð0Þ þ q Te 2 ð0Þ þ r Te 3 ð0Þ þ u1 Te 5 ð0Þ þ u2 Te 6 ð0Þ þ u3 Te 8 ð0Þ þ u4 Te 9 ð0Þ: P P9 Let N ¼ mP0 ð i¼1 aim Te i ðmÞÞ; then the stationary linear equation N x ¼ ½M; N generates a set of solutions for aim (i = 1, 2, . . . , 9): 8 a1mx ¼ qa3m ra2m ; > > > > > > 2ia2;mþ1 ¼ a2mx þ 2iqra2m þ 2qa1m ; > > > > > > 2ia3;mþ1 ¼ a3mx þ 2iqra3m þ 2ra1m ; > > > > > > a4mx ¼ qa6m ra5m þ u1 a3m u2 a2m ; > < 2ia5;mþ1 ¼ a5mx þ 2iqra5m þ 2qa4m þ 2u1 a1m ; > > > > 2ia6;mþ1 ¼ a6mx þ 2iqra6m þ 2ra4m þ 2u2 a1m ; > > > > > > a7mx ¼ qa9m ra8m þ u1 a6m u2 a5m þ u3 a3m u4 a2m ; > > > > > > 2ia8;mþ1 ¼ a8mx þ 2iqra8m þ 2qa7m þ 2u1 a4m þ 2u3 a1m ; > > > : 2ia9;mþ1 ¼ a9mx þ 2iqra9m þ 2ra7m þ 2u2 a4m þ 2u4 a1m :
ð42Þ ð43Þ
ð44Þ
698
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 ðnÞ
Case 1: Take N þ ¼
Pn
m¼0 ð
P9
e
i¼1 aim T i ðn
ðnÞ mÞÞ ¼ kn N N ; then a direct calculation reads that
ðnÞ ðnÞ N þx þ ½M; N þ ¼ 2ia2;nþ1 Te 2 ð0Þ þ 2ia3;nþ1 Te 3 ð0Þ 2ia5;nþ1 Te 5 ð0Þ þ 2ia6;nþ1 Te 6 ð0Þ 2ia8;nþ1 Te 8 ð0Þ þ 2ia9;nþ1 Te 9 ð0Þ: ðnÞ
Denote N ðnÞ ¼ N þ ; then the (2 + 1)-dimensional partial-differential equation system M t N xðnÞ þ ½M; N ðnÞ M y ¼ 0 admits the following solution for M 8 1 0 1 0 1 0 a2;nþ1 q q > > > > C C C B B B > > B r C B a3;nþ1 C B r C > > C B C B C B > > C B C B C B > < ut ¼ B u1 C ¼ 2iB a5;nþ1 C þ B u1 C ; C B C Bu C B a B 2C B 6;nþ1 C B u2 C > C C B C B B > > @ u3 A @ a8;nþ1 A @ u3 A > > > > > a9;nþ1 u4 t u4 y > > > : ðqrÞt ¼ ðqrÞy :
ð45Þ
ð46Þ
It is easy to see that (qr)t = 2ira2,n+12iqa3,n+1 + (qr)y. Hence, a1,n+1x = 0. Take a1n = 0, then Eq. (44) is still solvable. A (2 + 1)-dimensional integrable solitary hierarchy is obtained 1 0 1 0 a2;nþ1 q B r C B a3;nþ1 C C B C B C B C B B u1 C B a5;nþ1 C C ¼ 2iB C; ~ uz ¼: B ð47Þ C Bu C B a B 2C B 6;nþ1 C C B C B @ u3 A @ a8;nþ1 A u4
ty
a9;nþ1
which is a (2 + 1)-dimensional enlarging AKNS hierarchy. ðnÞ Case 2: Take N ðnÞ ¼ N þ 2a1;nþ1 Te 1 ð0Þ, then Eq. (45) admits the solution 8 1 0 1 0 2ia2;nþ1 4qa1;nþ1 q > > > > C Br C B 2ia > 3;nþ1 þ 4ra1;nþ1 C > B C B > > C C B B > > B u1 C B 2ia5;nþ1 4u1 a1;nþ1 C > <~ C C; B B uz ¼: B C ¼ B C B u2 C B 2ia6;nþ1 þ 4u2 a1;nþ1 C > C B C B > > @ u3 A @ 2ia8;nþ1 4u3 a1;nþ1 A > > > > > 2ia9;nþ1 þ 4u4 a1;nþ1 u4 ty > > > : ðqrÞty ¼ 2ia1;nþ1x :
ð48Þ
In the following, we deduce the Hamiltonian expression of (47). e 9 ¼ spanfcðnÞ ¼ ckn ; c 2 R9 g along with the commutator [X(m), Y(n)]1 = Consider the loop algebra R m+n 9 [X, Y]1k , X, Y 2 R . Let ¼ ðik iqr; q; r; 0; u1 ; u2 ; 0; u3 ; u4 ÞT ; M n X T T ðnÞ w þ wy ; N ðnÞ ¼ wt ¼ N ða1m ; a2m ; . . . ; a9m Þ knm þ ð2a1;nþ1 ; 0; 0; 0; 0; 0; 0; 0; 0Þ :
wx ¼ Mw;
m¼0
Then the zero curvature equation t N ðnÞ þ ½M; N ðnÞ M y ¼ 0 M x
ð49Þ ð50Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
699
also gives rise to the system (47). Hence, it can be expressed as 10 1 0 0 1 a6;nþ1 þ a9;nþ1 q 0 0 0 0 0 1 C B 0 0 0 B r C B 0 1 0 C CB a5;nþ1 þ a8;nþ1 C B B C CB C B B C B 0 0 0 B u1 C 1 0 1 CB a3;nþ1 þ a6;nþ1 C CB C ¼ 2iB C ¼: JP nþ1 ; ~ uz ¼ B C B 0 0 1 0 Bu C B 1 0 C CB a2;nþ1 þ a5;nþ1 C B B 2C CB C B B C A @ 0 1 0 1 0 @ u3 A a3;nþ1 1 A@ a2;nþ1 1 0 1 0 1 0 u4 ty where J is a Hamiltonian operator. A direct calculation gives 1 0 2irða4;nþ1 þ a7;nþ1 Þ þ a3;nþ1 þ a6;nþ1 þ a9;nþ1 C B 2iqða 4;nþ1 þ a7;nþ1 Þ þ a2;nþ1 þ a5;nþ1 þ a8;nþ1 C B C B C B a3;nþ1 þ a6;nþ1 C ¼: Rnþ1 B C B a2;nþ1 þ a5;nþ1 C B C B A @ a3;nþ1 a2;nþ1
0
1
2iro q þ 1 B 2iqo1 q B B B 0 ¼B B 0 B B @ 0 0
0
2iro1 r 2iqo1 r þ 1
2iro1 u1 2iqo1 u1
2iro1 u2 2iqo1 u2
2iro1 u3 þ 1 2iqo1 u3
0
1
0
0
0 0
0 0
1 0
0 1
0
0
0
0
10
a6;nþ1 þ a9;nþ1
1
1 2iro1 u4 2iqo1 u4 þ 1 C C C C 0 C C 0 C C A 0 1
a6;nþ1 þ a9;nþ1 C Ba B 5;nþ1 þ a8;nþ1 C C B B a3;nþ1 þ a6;nþ1 C C ¼: QP nþ1 : B B C B a2;nþ1 þ a5;nþ1 C C B A @ a3;nþ1 a2;nþ1 Thus, we have 0 0 1 q 0 B 0 B r C B B C B B C B 0 B u1 C C ¼ 2iB ~ uz ¼ B B 0 Bu C B B 2C B B C @ 0 @ u3 A 1 u4 ty
0
0
0
0
0
0
0
1
0 0
0 1
1 0
0 1
1 0
0 1
1 0
0 1
1
1
C B 0 C CB a5;nþ1 þ a8;nþ1 C CB C 1 CB a3;nþ1 þ a6;nþ1 C CB C ¼: JP nþ1 ¼ 2iJQ1 Rnþ1 : C B 0 C CB a2;nþ1 þ a5;nþ1 C CB C A a3;nþ1 1 A@ a2;nþ1 0
From the formula (17) and F, a direct computation gives ; oM ¼ 2irða4 þ a7 Þ þ a3 þ a6 þ a9 ; N oq o ; M ¼ 2iqða4 þ a7 Þ þ a2 þ a5 þ a8 ; N or
ð51Þ
700
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
oM oM N; N; ¼ a3 þ a6 ; ¼ a2 þ a5 ; ou1 ou2 oM oM oM N; N; N; ¼ 2iða4 þ a7 Þ: ¼ a3 ; ¼ a2 ; ou3 ou4 ok Substituting the above results into the quadratic-form identity yields 1 0 2irða4 þ a7 Þ þ a3 þ a6 þ a9 C B B 2iqða4 þ a7 Þ þ a2 þ a5 þ a8 C C B C B a3 þ a6 d C c o c B 2iða4 þ a7 Þ ¼ k kB C: C d~ u ok B a þ a 2 5 C B C B a3 A @
ð52Þ
a2 Comparing the coefficients of k
n1
in (52) gives rise to
d 2iða4;nþ1 þ a7;nþ1 Þ ¼ ðn þ cÞRn d~ u 2iða
þa7;nþ1 Þ
n . It is easy to see that c = 0. Thus, Rn ¼ dH ; H n ¼ 4;nþ1n d~ u structure of the higher-dimensional integrable hierarchy (47)
~ uz ¼ 2iJQ1 Rnþ1 ¼ 2iJQ1
. Hence, we obtain the quasi-Hamiltonian
dH nþ1 : d~ u
ð53Þ
Remark 2. Because Q is not a sky-symmetric matrix, JQ1 is not Hamiltonian. Hence, we call (53) the quasiHamiltonian structure. Of course, if we omit the term qr Te 1 ð0Þ in (42), it is easy to obtain the Hamiltonian structure of the corresponding enlarging higher-dimensional AKNS hierarchy. Example 5. Consider the isospectral problem ux ¼ Mu; M ¼ h2 ð1Þ þ uh3 ð0Þ þ vh1 ð0Þ þ u1 h4 ð0Þ þ u2 h5 ð0Þ þ u3 h8 ð0Þ þ u4 h9 ð0Þ: ð54Þ P Set N ¼ mP0 ðb1m h1 ðmÞ þ b2m h2 ðmÞ þ b3m h3 ðmÞ þ b4m h4 ðmÞ þ b5m h5 ðmÞ þ b6m h6 ðmÞ þ b7m h7 ðmÞþ b8m h8 ðmÞ þ b9m h9 ðmÞÞ, a solution of the linear equation for N N x ¼ ½M; N is given by 8 b3;mþ1 ¼ b1mx ub2m ; > > > > > > b2mx ¼ ub1m þ vb3m ; > > > > > b1;mþ1 ¼ b3mx vb2m ; > > > > > > b5;mþ1 ¼ b4mx þ ub6m u2 b2m ; > > > > > b4;mþ1 ¼ b5mx þ vb6m u1 b2m ; > > > > < b6mx ¼ ub4m vb5m u1 b3m þ u2 b1m ; > b7mx ¼ ub9m þ vb8m þ u1 b5m u2 b4m u3 b1m þ u4 b3m ; > > > > > > b9;mþ1 ¼ b8mx vb7m þ u1 b6m u4 b2m ; > > > >b > 8;mþ1 ¼ b9mx ub7m þ u2 b6m u3 b2m ; > > > > > b2;0 ¼ a ¼ const:; b1;0 ¼ b3;0 ¼ 0; b1;1 ¼ av; b3;1 ¼ au; > > > > > > b2;1 ¼ 0; b4;0 ¼ b5;0 ¼ b6;0 ¼ b7;0 ¼ b8;0 ¼ b9;0 ¼ 0; > > > : b4;1 ¼ au1 ; b5;1 ¼ au2 ; b6;1 ¼ b7;1 ¼ 0; b8;1 ¼ au3 ; b9;1 ¼ au4 ; . . .
ð55Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 ðnÞ
Let N þ ¼
Pn
m¼0 ð
P9
ðnÞ
i¼1 bim hi ðn
701
mÞÞ ¼ kn N N ðnÞ ; then
ðnÞ
N þx þ ½M; N þ ¼ b1;nþ1 h3 ð0Þ þ b3;nþ1 h1 ð0Þ b4;nþ1 h5 ð0Þ þ b5;nþ1 h4 ð0Þ þ b8;nþ1 h9 ð0Þ b9;nþ1 h8 ð0Þ: A solution of the (2 + 1)-dimensional partial-differential equation system for M ðnÞ
ðnÞ
M t N þx þ ½M; N þ M y ¼ 0 is obtained 0
u
1
0
0
1
0
C B B b B 3;nþ1 C B 0 C B B B b5;nþ1 C B 0 C B B ¼B C¼B B b4;nþ1 C B 0 C B B C B B @ b9;nþ1 A @ 0
B C BvC B C B u1 C B C ~ uz ¼ B C B u2 C B C B C @ u3 A u4
b1;nþ1
ty
b8;nþ1
0 1
10
b3;nþ1 þ b5;nþ1 b8;nþ1
1
0
0
0
0
0
0
0
0
1
0
1
0
1 1
0
CB C 0 CB b1;nþ1 þ b4;nþ1 b9;nþ1 C CB C C B b1;nþ1 b4;nþ1 1 0 C CB C CB C ¼: JW nþ1 ; C B b b 0 1 C 3;nþ1 5;nþ1 CB C CB C b3;nþ1 0 0 A@ A
1
0
1
0
0
1
0
b1;nþ1 ð56Þ
where J is a Hamiltonian operator. In order to get Hamiltonian structure of (56), a Lax pair is constructed by e 9 ¼ spanfxðnÞ ¼ xkn ; x 2 R9 g with the commuting operation [x(m), y(n)]2 = using the loop algebra R m+n 9 [x, y]2k , x, y 2 R : 8 e u; M e ¼ ðv; k; u; u1 ; u2 ; 0; 0; u3 ; u4 ÞT ; > < ux ¼ M n P T e ðnÞ u þ uy ; N e ðnÞ ¼ > ðb1m ; b2m ; . . . ; b9m Þ knm ; : ut ¼ N m¼0
etN e ðnÞ þ ½M; N e ðnÞ M e y ¼ 0 also admits the solution (56). Taking use of F , a direct whose compatibility M x computation gives ( ) ( ) e e oM oM e e N; N; ¼ b3 þ b 5 b 8 ; ¼ b1 þ b4 b9 ; ou ov ( ) ( ) e e o M o M e; e; N N ¼ b1 b4 ; ¼ b3 b5 ; ou1 ou2 ( ) ( ) ( ) e e e o M o M o M e; e; e; N N N ¼ b2 b 6 b7 ; ¼ b3 ; ¼ b1 ; ou3 ou4 ok e ¼ ðb1 ; b2 ; . . . ; b9 Þ ; bi ¼ where N form identity gives T
P
mP0 bim k
0
m
; i ¼ 1; 2; . . . ; 9. Inserting the above results into the quadratic-
b3 þ b5 b8
1
C B B b1 þ b4 b9 C C B B b1 b4 C d C c o c B ðb2 b6 b7 Þ ¼ k kB C: d~ u ok B b3 b5 C C B C B b3 A @
ð57Þ
b1 Comparing the coefficients of kn1 in (57) leads to ( ; W nþ1 ¼ dHd~nþ1 u 1 H nþ1 ¼ nþc ðb2;nþ2 b6;nþ2 b7;nþ2 Þ;
d ðb2;nþ1 d~ u
b6;nþ1 b7;nþ1 Þ ¼ ðn þ cÞW n , ð58Þ
702
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
From the recurrence relations (55), we have c = 0. Hence, the Hamiltonian structure of the (2 + 1)-dimensional soliton equation hierarchy is produced ~ uz ¼ JW nþ1 ¼ J
dH nþ1 : d~ u
ð59Þ
References [1] Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30:330–8. [2] Ma Wen-Xiu. A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chinese J Contemp Math 1992;13(1):79–89. [3] Hu Xingbiao. A powerful approach to generate new integrable systems. J Phys A 1994;27:2497–514. [4] Hu Xingbiao. Approach to generate super-extensions of integrable systems. J Phys A 1997;30:619–32. [5] Guo Fukui. NLS-MKdV hierarchy of equations that are integrable and in the Hamiltonian forms. Acta Math Sinica 1997;40(6):801–4; Guo Fukui, Zhang Yufeng. A type of expanding integrable model of the AKNS hierarchy. Acta Phys Sin 2002;51(5):951–4. [6] Fan Engui. Integrable systems of derived nonlinear Schro¨dinger type and their multi-Hamiltonian structure. J Phys A 2001;34:513–9. e 1 and integrable Hamiltonian hierarchy of equations. Acta Math Phys Sin [7] Fukui Guo. Subalgebras of the loop algebra A 1999;19(5):507–12. [8] Ablowitz MJ, Chakravarty S, Halburd RG. Integrable systems and reductions of the self-dual Yang-Mills equations. J Math Phys 2003;44(8):3147–73. [9] Zhang Yufeng, Tam Honwah. Reduced equations of the self-dual Yang-Mills equations and applications. Chaos Solitons Fractals, in press. [10] Yu Yangzheng, Feng Chengtian. Geometrical methods in physics. Beijing: China Higher Education Press; 1998. [11] Ma Wen-Xiu, Fuchssteiner B. Integrable theory of the perturbation equations. Chaos Solitons Fractals 1996;7:1227–50. [12] Ma Wen-Xiu. Integrable couplings of soliton equations by perturbation I. A general theory and application to the KdV hierarchy. Methods Appl Anal 2000;7:21–55. [13] Ma Wen-Xiu, Xu Xixiang, Zhang Yufeng. Semi-direct sum Lie algebras and continuous integrable couplings. Phys Lett A 2006;351:125–30. [14] Guo Fukui, Zhang Yufeng. The quadratic-form identity for constructing the Hamiltonian structure systems. J Phys A 2005;38:8537–48. [15] Li Yishen. Soliton and integrable system. Shanghai: Shanghai Scientific and Technological Education Publishing House; 1999.
Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 www.elsevier.com/locate/cnsns
Invertible linear transformations and the Lie algebras Yufeng Zhang
a,*
, Honwah Tam b, Fukui Guo
q
c
a
Mathematical School, Liaoning Normal University, Dalian 116029, PR China Department of Computer Sciences, Hong Kong Baptist University, Hong Kong, PR China Information School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China b
c
Received 22 July 2006; accepted 28 July 2006 Available online 25 September 2006
Abstract With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity. 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.20.Sv; 02.30.Rz Keywords: Invertible linear transformation; Lie algebra; Soliton hierarchy
As we know, the Lie algebras have important applications in mathematical physics. For example, the Lie algebra sl(2) describes the related information on spin electronics. By making use of the Lie algebras, the corresponding various loop algebras could be generated, which are devote to generating integrable soliton hierarchies of evolution equations under the framework of the zero curvature equations. In the paper, we construct a larger types of Lie algebras by invertible linear transformations to deduce integrable couplings of soliton hierarchies. Specially, we again construct a few types of new Lie algebras from the viewpoint of the classification of the Lie algebras. Then via linear transformations, some Lie algebras could be worked out. The second major point in the paper presents the corresponding various loop algebras of the above Lie algebras. As the applications of the loop algebras, two (2 + 1)-dimensional larger integrable systems are
q
This work was supported by The National Science Foundation of China (10471139) and Hong Kong Research Grant Council grant number HKBU RGC 2016/05p. * Corresponding author. E-mail address: [email protected] (Y. Zhang). 1007-5704/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.07.011
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
683
obtained, one of them can reduce to the well-known AKNS hierarchy. The third point is that the quasiHamiltonian structure and the Hamiltonian structure of two integrable systems are obtained, respectively by using the quadratic-form identity. The point is important because the known trace identity proposed by Tu Guizhang cannot produce the Hamiltonian structures of such larger integrable systems. But in the paper, the problem is completely overcome. The Lie algebras and F, F (see below) presented in the paper have extensive applications, that is to say, we could generate many integrable couplings and their Hamiltonian structures by use of these Lie algebras and F , F . 1. Introduction One method for generating integrable hierarchies of soliton equations and the corresponding Hamiltonian structures was proposed by Tu [1], Ma called the method Tu scheme [2]. Ma, Hu and Guo et al. further developed it and obtained some rich results [2–6]. The method was conducted by constructing appropriate Lie algebras and utilizing the zero curvature equations. As long as the Lie algebras are given, the corresponding loop algebras can be establishing by various ways [7]. Hence, there exist two approaches to obtain integrable hierarchies by using the Tu scheme. One is construction of new Lie algebras. Another is an improvement of the formalisms of the zero curvature equations. The later may be decided by establishing various types of Lax pairs [8,9]. In the paper, we only want to discuss the first approach. Some Lie algebras are derived from the groups. t U ¼ E2 ; det U ¼ 1g, where E2 For example, the matrix multiplication group SU ð2Þ ¼ fU ; U 2 GLð2; CÞ; U stands for a 2 · 2 unit matrix. The corresponding Lie algebra presents sl(2) = span{L1, L2, L3}, where ! ! ! 1 0 1 1 0 i 1 1 0 L1 ¼ ; L2 ¼ ; L3 ¼ ; 2 1 0 2 i 0 2 0 1 [L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2, i2 = 1. From the viewpoint of geometry, sl(2) represents the tangent-vector space of all unit elements at E2, which describes the spin electronics [10]. Hu [3,4] studied the Lie algebra sl(N, C) and its some subalgebras. Specially, he investigated the super-Lie algebra sl(m/n) and its applications, which give us a powerful guidance. Guo Fukui [5] constructed a special Lie algebra, from which the NLS-MKdV hierarchy was obtained by use of the zero curvature equation. In this paper, we give a way to generate Lie algebras by means of invertible linear transformation with a common property: If denote one of the Lie algebras by G, its two direct-sum subalgebras are noted as G1 and G2, then G ¼ G1 G2 ;
G1 ffi Ak ; ½G1 ; G2 G2 ;
ð1Þ
where Ak are the known lower-dimensional Lie algebras. It is use of (1) that integrable couplings of soliton hierarchies [11,12] are obtained. Hence, the Lie algebras presented in the paper are major applications to deduce integrable couplings. In addition, a few ways to construct loop algebras of the Lie algebras are presented which are used to generate integrable couplings. In terms of the classification of Lie algebras given by Ma [13], some explicit Lie algebras are constructed. Finally, some solutions of a partial-differential equation system are obtained, and their Hamiltonian form-expressions are also given. 2. Invertible linear transformations Let G be a n-dimensional Lie algebra, whose basis be T1, T2, . . . , Tn. If A is a n · n invertible matrix, there exists a linear transformation on G such that d : G ! G;
hi ¼
n X
aij T j ;
i ¼ 1; 2; . . . ; n;
j¼1
then call (2) an invertible linear transformation.
A ¼ ðaij Þnn ;
ð2Þ
684
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Specially, if A presents the 0 A1 B A2 A¼@ .. .A
form as follows: 1 C A;
Ai ¼ ð1Þ and
Aj ¼
1 1
1 ; 1
ð3Þ
1 6 i; j 6 m;
m
then h1, h2, . . . , hn are linear independent, which are maybe a basis of the Lie algebra G, whose commuting operations are different from those of the Lie algebra T1, T2, . . . , Tn. How we prove the guess problem properly is worthy of study. In what follows, we take a few examples to show the above version, which shows the such matrix A as above exists. Example 1. A subalgebra of the Lie algebra A3 is given by 6
slð4Þ1 ¼ spanfT i gi¼1 ; 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 B 0 1 0 0 C B0 0 0 0C B1 0 0 0C B C B C B C T1 ¼ B C; T 2 ¼ B C; T 3 ¼ B C; @0 0 1 0 A @0 0 0 1A @0 0 0 0A 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 B 0 0 0 1 C B0 0 0 0C B0 0 1 0C B C B C B C T4 ¼ B C; T 5 ¼ B C; T 6 ¼ B C; @0 0 0 0 A @0 0 0 0A @0 0 0 0A
ð4Þ
0 0 0 0 0 0 0 0 0 0 0 0 ½T 1 ; T 2 ¼ 2T 2 ; ½T 1 ; T 3 ¼ 2T 3 ; ½T 2 ; T 3 ¼ T 1 ; ½T 1 ; T 4 ¼ 0; ½T 1 ; T 5 ¼ 2T 5 ; ½T 1 ; T 6 ¼ 2T 6 ; ½T 2 ; T 4 ¼ 2T 5 ; ½T 2 ; T 5 ¼ 0; ½T 2 ; T 6 ¼ T 4 ; ½T 3 ; T 4 ¼ 2T 6 ; ½T 3 ; T 5 ¼ T 4 ; ð5Þ ½T 3 ; T 6 ¼ ½T 4 ; T 5 ¼ ½T 4 ; T 6 ¼ ½T 5 ; T 6 ¼ 0: Let
0
1 B 0 1 B0 f1 B 0 1B B .. C F ¼ @ . A ¼ A1 T ¼ B 2B B0 B f6 @0 0
0 1 1 0 0 0
0 1 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 1
1 0 0 C C0 T 1 1 C 0 C . C CB @ .. A; 0 C C C 1 A T6 1
ð6Þ
then 0
1
1B B0 f1 ¼ B 2@0 0 0 0 B 1B0 f4 ¼ B 2@0 0 ½f1 ; f2 ¼ f3 ; ½f2 ; f5 ¼ f4 ;
1
0
0
0
1
0
0 C C C; 0 A
0 1 0 0 1 1 0 1 0 0 0 1 C C C; 0 0 0 A 0 0
0
0
0
1B B 1 f2 ¼ B 2@ 0 0 0 0 0 B 1B0 0 f5 ¼ B 2@0 0 0
0
1
1
0
0
0
0
0C C C; 1A
0
1
0
0
1B B1 f3 ¼ B 2@0
0
0
0C C C; 1A
0 0 0 1 0 1 0 0 1 0 B 1 0C 1B0 C C; f 6 ¼ B 2@0 0 0A
0
0
0
0
1
0
0
0 0 0
0 0 0 1 0 1 0 1 1 0 C C C; 0 0A 0
0
½f1 ; f3 ¼ f2 ; ½f2 ; f3 ¼ f1 ; ½f1 ; f4 ¼ 0; ½f1 ; f5 ¼ f6 ; ½f1 ; f6 ¼ f5 ; ½f2 ; f4 ¼ f5 ; ½f2 ; f6 ¼ 0; ½f3 ; f4 ¼ f6 ; ½f3 ; f5 ¼ 0; ½f3 ; f6 ¼ f4 ; ½f4 ; f5 ¼ ½f4 ; f6 ¼ ½f5 ; f6 ¼ 0:
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
685
6
Denote slð4Þ2 ¼ spanffi gi¼1 ; we see that (6) is an isomorphic map from the Lie algebra sl(4)1 to the sl (4)2 in the Lie algebra A3. ) Obviously, the Lie algebra sl (4)2 indeed possesses various commuting relations with the Lie algebra sl(4)1. Example 2. Consider a subalgebra of the Lie algebra A2 as follows [5]: slð3Þ1 ¼ spanfwi g5i¼1 ;
ð7Þ
where 0
1
B w1 ¼ B @0 0 0 0 B w4 ¼ B @0 0
0
0
1
0
0 1
0
1
0
C B C B C 1 0 C A; w 2 ¼ @ 0 0 0 A; 0 0 0 0 0 1 0 1 0 1 0 0 0 C B C B C 0 0C A; w5 ¼ @ 0 0 1 A; 0 0 0 0 0
½w1 ; w2 ¼ 2w2 ; ½w2 ; w4 ¼ 0;
½w1 ; w3 ¼ 2w3 ; ½w2 ; w5 ¼ w4 ;
0
B w3 ¼ B @1 0
½w2 ; w3 ¼ w1 ;
½w3 ; w4 ¼ w5 ;
0
0
1
0
C 0C A;
0
0
½w1 ; w4 ¼ w4 ;
½w1 ; w5 ¼ w5 ;
½w3 ; w5 ¼ ½w4 ; w5 ¼ 0:
If 0
g1
1
0
w1
1
01 2
0
B C B C B B g2 C B w2 C B 0 1 2 B C B C B B C B C B B g 3 C ¼ A2 B w 3 C ¼ B 0 1 B C B C B 2 B C B C B B g4 C B w4 C B 0 0 @ A @ A @ g5 w5 0 0
0
0
1 2
0
12
0
0
1
0
0
0
10
w1
1
CB C B C 0C CB w2 C CB C B C 0C CB w3 C: CB C B C 0C A@ w4 A 1
w5
Obviously, jA2j 5 0. Hence, (8) is an invertible linear transformation 0 1 0 1 0 1 0 0 0 1 0 0 1 B B C C 1B 1 1 B C B 0 1 0 C g1 ¼ B A; g2 ¼ 2 @ 1 0 0 A; g3 ¼ 2 @ 1 0 2@ 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 B C B C B C B g 4 ¼ @ 0 0 0 A; g 5 ¼ @ 0 0 1 C A; 0 0 0 0 0 0 ½g1 ; g2 ¼ g3 ; 1 ½g2 ; g4 ¼ g4 ; 2
½g1 ; g3 ¼ g2 ;
½g2 ; g3 ¼ g1 ;
1 ½g3 ; g4 ¼ g5 ; 2
ð8Þ
in A2. It is easy to see that 1 0 C 0C A; 0
1 ½g1 ; g4 ¼ g5 ; 2
1 ½g2 ; g5 ¼ g5 ; 2
1 ½g1 ; g5 ¼ g4 ; 2 1 ½g3 ; g5 ¼ g4 ; ½g4 ; g5 ¼ 0: 2
5
Denote slð3Þ2 ¼ spanfgi gi¼1 ; then sl (3)1 is different from sl(3)2. Example 3. Consider a subalgebra of the Lie algebra A5 slð6Þ1 ¼ spanf Te i g9i¼1 ;
ð9Þ
686
where
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 B 0 1 0 0 0 0 C B0 0 0 0 0 0C B1 0 0 0 0 0C C C C B B B C C C B B B B0 0 1 0 0 0 C B0 0 0 1 0 0C B0 0 0 0 0 0C C; Te 2 ¼ B C; Te 3 ¼ B C Te 1 ¼ B B 0 0 0 1 0 0 C B0 0 0 0 0 0C B 0 0 1 0 0 0 C; C C C B B B C C C B B B @0 0 0 0 1 0 A @0 0 0 0 0 1A @0 0 0 0 0 0A 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 B 0 0 0 1 0 0 C B0 0 0 0 0 0C B0 0 1 0 0 0C C C C B B B C C C B B B B0 0 0 0 1 0 C B0 0 0 0 0 1C B0 0 0 0 0 0C C C C B B B e e e T4 ¼ B C; T 5 ¼ B 0 0 0 0 0 0 C; T 6 ¼ B 0 0 0 0 1 0 C; C C B 0 0 0 0 0 1 C B B C C C B B B @0 0 0 0 0 0 A @0 0 0 0 0 0A @0 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 B 0 0 0 0 0 1 C B0 0 0 0 0 0C B0 0 0 0 1 0C C C C B B B C C C B B B B0 0 0 0 0 0 C B0 0 0 0 0 0C B0 0 0 0 0 0C C; Te 8 ¼ B C; Te 9 ¼ B C Te 7 ¼ B B0 0 0 0 0 0 C B0 0 0 0 0 0C B 0 0 0 0 0 0 C; C C C B B B C C C B B B @0 0 0 0 0 0 A @0 0 0 0 0 0A @0 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ½ Te 1 ; Te 2 ¼ 2 Te 2 ; ½ Te 1 ; Te 3 ¼ 2 Te 3 ; ½ Te 2 ; Te 3 ¼ Te 1 ; ½ Te 1 ; Te 4 ¼ 0; ½ Te 1 ; Te 5 ¼ 2 Te 5 ; ½ Te 1 ; Te 6 ¼ 2 Te 6 ; 0
½ Te 1 ; Te 7 ¼ 0; ½ Te 1 ; Te 8 ¼ 2 Te 8 ; ½ Te 1 ; Te 9 ¼ 2 Te 9 ; ½ Te 2 ; Te 5 ¼ 0; ½ Te 2 ; Te 4 ¼ 2 Te 5 ; ½ Te 2 ; Te 6 ¼ Te 4 ; ½ Te 2 ; Te 7 ¼ 2 Te 8 ; ½ Te 2 ; Te 8 ¼ 0; ½ Te 2 ; Te 9 ¼ Te 7 ; ½ Te 3 ; Te 4 ¼ 2 Te 6 ; ½ Te 3 ; Te 5 ¼ Te 4 ; ½ Te 3 ; Te 6 ¼ 0; ½ Te 3 ; Te 7 ¼ 2 Te 9 ; ½ Te 3 ; Te 8 ¼ Te 7 ; ½ Te 3 ; Te 9 ¼ 0; ½ Te 4 ; Te 5 ¼ 2 Te 8 ; ½ Te 4 ; Te 6 ¼ 2 Te 9 ; ½ Te 4 ; Te 7 ¼ ½ Te 4 ; Te 8 ¼ ½ Te 4 ; Te 9 ¼ 0; ½ Te 5 ; Te 6 ¼ Te 7 ; ½ Te 5 ; Te 7 ¼ ½ Te 5 ; Te 8 ¼ ½ Te 5 ; Te 9 ¼ ½ Te 6 ; Te 7 ¼ ½ Te 6 ; Te 8 ¼ ½ Te 6 ; Te 9 ¼ ½ Te 7 ; Te 8 ¼ ½ Te 7 ; Te 9 ¼ ½ Te 8 ; Te 9 ¼ 0: Let
1 h1 B . C B .. C B C B C B C H ¼ B h6 C ¼ A3 Te ; B h9 C B C B C @ h7 A h8 0 1 0 0 B 0 1 1 B B B0 1 1 B B0 0 0 1B B A3 ¼ B 0 0 0 2B B0 0 0 B B0 0 0 B B @0 0 0 0 0 0 0
ð10Þ
0 0 0 1 0 0 0 0 0
0 0 0 0 1 1 0 0 0
0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 1 0 0
1 0 0 0 0 C C C 0 0 C C 0 0 C C C 0 0 C; C 0 0 C C 0 0 C C C 1 1 A 1 1
Te ¼ ð Te 1 ; . . . ; Te 6 ; Te 9 ; Te 7 ; Te 8 ÞT ;
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
then we have that 0 1 0 B 0 1 B B 1B B0 0 h1 ¼ B 2B0 0 B B @0 0
1
0
0
0
0
0
0
0
0
1
0
0
B 1 0 C C B C B C 0 C 1B B 0 C; h2 ¼ B C B 0 2 0 C B C B @ 0 0 A
0 1 0 0
0
1
0 1 1 0 0 0 1 0 0 0 0 B 0 0 0 1 0 0 C B0 C B B C B B C B 1B0 0 0 0 1 0 C 1B B0 h4 ¼ B C ; h5 ¼ B 2 B 0 0 0 0 0 1 C 2B0 C B B C B B @0 0 0 0 0 0 A @0 0
0
0
0
0
0 0 0 0
0 B0 B B 1B B0 h7 ¼ B 2B0 B B @0 0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0 1 0
0
0
0
0
0
B1 0C C B C B C 0C 1B B0 C; h3 ¼ B C B0 2 0C B C B @0 1A
0
0
0 1 0 1 0 0 0 C B 0C B0 C B C 1C 1B B0 C ; h6 ¼ B 2B0 0C C B C B @0 0A
0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0
0 0 0 0 0
687
0 1 0 0 0 0
1
0 0 0 0 0C C C 0 0 1 0 0C C C; 0 1 0 0 0C C C 0 0 0 0 1A
0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 C C C 0 0 0 1 C C C; 0 0 1 0 C C C 0 0 0 0 A
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 C C B 0 0 0 1 0 C B B0 0 0 0 1 0C C C B0 0 0 0 B C 0 0 0 0 0C 1B 1B C B B0 0 0 0 0 0C C ; h8 ¼ B C ; h9 ¼ B 0 0 0 0 2B0 0 0 0 0 0C 2B 0 0 0 0 0C B0 0 0 0 C C B C C @ B @0 0 0 0 0 0A 0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
1
0
1
C 0 1 C C C 0 0 C; C 0 0 C A 0 0
½h1 ; h2 ¼ h3 ; ½h1 ; h3 ¼ h2 ; ½h2 ; h3 ¼ h1 ; ½h1 ; h4 ¼ 0; ½h1 ; h5 ¼ h6 ; ½h1 ; h6 ¼ h5 ; ½h1 ; h7 ¼ h8 ; ½h1 ; h8 ¼ h7 ; ½h1 ; h9 ¼ 0; ½h2 ; h4 ¼ h5 ; ½h2 ; h5 ¼ h4 ; ½h2 ; h6 ¼ ½h2 ; h7 ¼ 0; ½h2 ; h8 ¼ h9 ; ½h2 ; h9 ¼ h8 ; ½h3 ; h4 ¼ h6 ; ½h3 ; h5 ¼ 0; ½h3 ; h6 ¼ h4 ; ½h3 ; h7 ¼ h9 ; ½h3 ; h8 ¼ 0; ½h3 ; h9 ¼ h7 ; ½h4 ; h5 ¼ h7 ; ½h4 ; h6 ¼ h8 ; ½h4 ; h7 ¼ ½h4 ; h8 ¼ ½h4 ; h9 ¼ 0; ½h5 ; h6 ¼ h9 ; ½h5 ; h7 ¼ ½h5 ; h8 ¼ ½h5 ; h9 ¼ ½h6 ; h7 ¼ ½h6 ; h8 ¼ ½h6 ; h9 ¼ ½h7 ; h8 ¼ ½h7 ; h9 ¼ ½h8 ; h9 ¼ 0: Note slð6Þ2 ¼ spanfhi g9i¼1 , then sl (6)1 and sl(6)2 are various. The Lie algebras in the above examples have the common property (1). As for example 1, let G = sl(4)1, G1 = span{T1, T2, T3}, G2 = span{T4, T5, T6}, then they satisfy (1). With the help of G1 (actually its resulting loop algebra), we may establish the zero curvature equation U t V xðnÞ þ ½U ; V ðnÞ ¼ 0 derived from the compatibility of the isospectral Lax pair ( ux ¼ U u; ut ¼ V ðnÞ u:
ð11Þ
ð12Þ
By means of the subalgebra G2, similarly we obtain the zero curvature equation ðnÞ
ðnÞ
U 1t V 1x þ ½U 1 ; V 1 ¼ 0; the corresponding Lax pair reads ( ux ¼ U 1 u; ðnÞ
ut ¼ V 1 u:
ð13Þ
ð14Þ
688
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Thus, the equations as below constitute the integrable couplings ( U t V xðnÞ þ ½U ; V ðnÞ ¼ 0; ðnÞ
ð15Þ
ðnÞ
U 1t V 1x þ ½U 1 ; V 1 ¼ 0:
The main goal for presenting this paper lies in giving a way to construct Lie algebras such that new integrable couplings and their Hamiltonian structures could be worked out. Since looking for the Hamiltonian structures of the integrable couplings first turns to establish an isomorphism between the matrix Lie algebra G and the column-vector Lie algebra Rn by using the quadraticform identity [14]. Hence,in the following, we want to take sl(6)i (i = 1, 2) for examples to illustrate the idea. P P 8a ¼ 9i¼1 ai Te i ; b ¼ 9i¼1 bi Te i 2 slð6Þ1 ; in terms of commuting relations among Te i ði ¼ 1; 2; . . . ; 9Þ, define ½a; bT1 ¼ ða1 ; . . . ; a9 ÞR1 ðbÞ; where
0
0
B b B 3 B B b2 B B B 0 B R1 ðbÞ ¼ B B 0 B B 0 B B 0 B B @ 0 0
ð16Þ
2b2
2b3
0
2b5
2b6
0
2b8
2b9
2b1 0
0 2b1
b6 b5
2b4 0
0 2b4
b9 b8
2b7 0
0 2b7
0
0
0
2b2
2b3
0
2b5
0 0
0 0
b3 b2
2b1 0
0 2b1
b6 b5
2b4 0
0 0
0 0
0 0
0 0
0 0
0 b3
2b2 2b1
0
0
0
0
0
b2
0
1
C C C C C C 2b6 C C 0 C C: C 2b4 C C 2b3 C C C 0 A 2b1
9
It is easy to verify that R is a Lie algebra equipped with the commutator [a, b]1. Define a functional [14]: fa; bg ¼ aT Fb;
ð17Þ
~ 1 ; which is a corresponding loop algebra of the Lie algebra sl(6)1, F satisfies [14]: where a; b 2 slð6Þ T
R1 ðbÞF ¼ ðR1 ðbÞF Þ ;
F ¼ F T:
ð18Þ
In order to employ the quadratic-form identity to deduce the Hamiltonian structures of integrable couplings, we need to solve the matrix Eq. (18) for F. A direct calculation, we obtain 1 0 2 0 0 2 0 0 2 0 0 B0 0 1 0 0 1 0 0 1C C B C B B0 1 0 0 1 0 0 1 0C C B C B B2 0 0 2 0 0 0 0 0C C B C F ¼B B 0 0 1 0 0 1 0 0 0 C: C B B0 1 0 0 1 0 0 0 0C C B B2 0 0 0 0 0 0 0 0C C B C B @0 0 1 0 0 0 0 0 0A 0 1 0 0 0 0 0 0 0 P9 P9 T As for the Lie algebra sl(6)2, take a ¼ i¼1 ai hi ; b ¼ i¼1 bi hi ; after calculations, we have ½a; b2 ¼ ða2 b3 a3 b2 ; a1 b3 a3 b1 ; a1 b2 a2 b1 ; a2 b5 a5 b2 þ a3 b6 a6 b3 ; a4 b2 a2 b4 þ a6 b1 a1 b6 ; a3 b4 a4 b3 þ a5 b1 a1 b 5 ; a 1 b 8 a 8 b 1 þ a 4 b 5 a 5 b 4 þ a 9 b 3 a 3 b 9 ; a 1 b 7 a 7 b 1 þ a 6 b 4 a 4 b 6 þ a 9 b 2 a 2 b 9 ; a 2 b 8 a 8 b 2 þ a 5 b 6 a 6 b 5 þ a7 b3 a3 b7 Þ ¼ ða1 ; . . . ; a9 ÞR2 ðbÞ,
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
0
0
B b B 3 B B b2 B B B 0 B R2 ðbÞ ¼ B B 0 B B 0 B B 0 B B @ 0 0
b3
b2
0
b6
b5
b8
b7
0
b1
b5
b4
0
0
b9
b1 0
0 0
b6 0
0 b2
b4 b3
b9 b5
0 b6
0 0
0 0
b2 b3
0 b1
b1 0
b4 0
0 b4
0
0
0
0
0
0
b1
0 0
0 0
0 0
0 0
0 0
b1 b3
0 b2
0
689
1
b8 C C C b7 C C C 0 C C b6 C C: C b6 C C b3 C C C b2 A 0
The constant symmetric matrix F satisfies T R2 ðbÞF ¼ ðR2 ðbÞF Þ
which gives rise to 0 1 0 B 0 1 B B B 0 0 B B 0 B 1 B F ¼ B 0 0 B B 1 B 0 B B 0 1 B B 0 @ 0 1 0
ð19Þ
0 0
1 0
0 0
0 1
0 1
0 0
1 0
0 1
1 0
0 0
0 0
1 0
1
0
1
0
0
0
0 0
0 0
0 0
1 0
0 0
0 0
1 0
0 0
0 0
0 0
0 0
0 0
1 1 0 C C C 0 C C C 0 C C 0 C C C 0 C C 0 C C C 0 A 0
3. Various loop algebras of the Lie algebras An important application of the loop algebras is devoting to producing new integrable hierarchies of soliton equations. In general, various loop algebras usually can give various integrable soliton hierarchies under the framework of the zero curvature equations. Guo Fukui once proposed the useful methods for obtaining loop algebras [7]. In the section, we present some ways of generating loop algebras from the Lie algebras presented in the paper. Take sl(6)i (i = 1, 2) for examples. As for the Lie algebra sl(6)1, its simplest loop algebra is given by ~ 11 ¼ spanf Te i ðnÞg9i¼1 ; slð6Þ
ð20Þ
Te i ðnÞ ¼ Te i kn ; n ¼ 0; 1; 2; . . . ; the commutator is defined as ½ Te i ðmÞ; Te j ðnÞ ¼ ½ Te i ; Te j kmþn ;
i 6¼ j; m; n 2 Z:
ð21Þ
Another loop algebra with powers of k being k2n, k2n+1 is given by ~slð6Þ12 ¼ spanf Te i ðnÞg9i¼1 ;
ð22Þ
where Te 1 ðnÞ ¼ Te 1 k2n ; Te 2 ðnÞ ¼ Te 2 k2nþ1 ; Te 6 ðnÞ ¼ Te 6 k2nþ1 ; Te 7 ðnÞ ¼ Te 7 k2n ;
Te 3 ðnÞ ¼ Te 3 k2nþ1 ; Te 8 ðnÞ ¼ Te 8 k2nþ1 ;
Te 4 ðnÞ ¼ Te 4 k2n ; Te 5 ðnÞ ¼ Te 5 k2nþ1 ; Te 9 ðnÞ ¼ Te 9 k2nþ1 ; ½ Te 1 ðmÞ; Te 2 ðnÞ ¼ 2 Te 2 ðm þ nÞ;
½ Te 1 ðmÞ; Te 3 ðnÞ ¼ 2 Te 3 ðm þ nÞ; ½ Te 2 ðmÞ; Te 3 ðnÞ ¼ Te 1 ðm þ n þ 1Þ; ½ Te 1 ðmÞ; Te 4 ðnÞ ¼ 0; ½ Te 1 ðmÞ; Te 5 ðnÞ ¼ 2 Te 5 ðm þ nÞ; ½ Te 1 ðmÞ; Te 6 ðnÞ ¼ 2 Te 6 ðm þ nÞ; ½ Te 1 ðmÞ; Te 7 ðnÞ ¼ 0;
690
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
½ Te 1 ðmÞ; Te 8 ðnÞ ¼ 2 Te 8 ðm þ nÞ; ½ Te 1 ðmÞ; Te 9 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 2 ðmÞ; Te 5 ðnÞ ¼ 0; ½ Te 2 ðmÞ; Te 4 ðnÞ ¼ 2 Te 5 ðm þ nÞ; ½ Te 2 ; Te 6 ðnÞ ¼ Te 4 ðm þ n þ 1Þ; ½ Te 2 ðmÞ; Te 7 ðnÞ ¼ 2 Te 8 ðm þ nÞ; ½ Te 2 ðmÞ; Te 8 ðnÞ ¼ 0; ½ Te 2 ðmÞ; Te 9 ðnÞ ¼ Te 7 ðm þ n þ 1Þ; ½ Te 3 ðmÞ; Te 4 ðnÞ ¼ 2 Te 6 ðm þ nÞ; ½ Te 3 ðmÞ; Te 5 ðnÞ ¼ Te 4 ðm þ n þ 1Þ; ½ Te 3 ðmÞ; Te 8 ðnÞ ¼ Te 7 ðm þ n þ 1Þ;
½ Te 3 ðmÞ; Te 6 ðnÞ ¼ 0; ½ Te 3 ðmÞ; Te 9 ðnÞ ¼ 0;
½ Te 3 ðmÞ; Te 7 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 4 ðmÞ; Te 5 ðnÞ ¼ 2 Te 8 ðm þ nÞ;
½ Te 4 ðmÞ; Te 6 ðnÞ ¼ 2 Te 9 ðm þ nÞ; ½ Te 4 ðmÞ; Te 7 ðnÞ ¼ ½ Te 4 ðmÞ; ½ Te 5 ðmÞ; Te 6 ðnÞ ¼ Te 7 ðm þ n þ 1Þ;
Te 8 ðnÞ ¼ ½ Te 4 ðmÞ; Te 9 ðnÞ ¼ 0;
½ Te 5 ðmÞ; Te 7 ðnÞ ¼ ½ Te 5 ðmÞ; Te 8 ðnÞ ¼ ½ Te 5 ðmÞ; Te 9 ðnÞ ¼ ½ Te 6 ðmÞ; Te 7 ðnÞ ¼ ½ Te 6 ðmÞ; Te 8 ðnÞ ¼ ½ Te 6 ðmÞ; Te 9 ðnÞ ¼ ½ Te 7 ðmÞ; Te 8 ðnÞ ¼ ½ Te 7 ðmÞ; Te 9 ðnÞ ¼ ½ Te 8 ðmÞ; Te 9 ðnÞ ¼ 0; m; n 2 Z; n ¼ 0; 1 2; . . . The general loop algebra which corresponds to the loop algebra (22) reads that ~slð6Þ13 ¼ spanf Te i ðj; mÞg9i¼1 ; Te i ðj; mÞ ¼ Te i k2nþj ; j ¼ 0; 1: The resulting commuting operations are given by ( i þ j 6 1; 2 Te 2 ði þ j; m þ nÞ; e e ½ T 1 ði; mÞ; T 2 ðj; nÞ ¼ 2 Te 2 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 3 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 3 ðj; nÞ ¼ e 2 T 3 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 1 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 3 ðj; nÞ ¼ ½ Te 1 ði; mÞ; Te 4 ðj; nÞ ¼ 0; e T 1 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 5 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 5 ðj; nÞ ¼ 2 Te 5 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 6 ði þ j; m þ nÞ; e e ½ T 1 ði; mÞ; T 6 ðj; nÞ ¼ ½ Te 1 ði; mÞ; Te 7 ðj; nÞ ¼ 0; e 2 T 6 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 8 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 8 ðj; nÞ ¼ 2 Te 8 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2 ( i þ j 6 1; 2 Te 9 ði þ j; m þ nÞ; ½ Te 1 ði; mÞ; Te 9 ðj; nÞ ¼ e 2 T 9 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 5 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 5 ðj; nÞ ¼ 0; ½ Te 2 ði; mÞ; Te 4 ðj; nÞ ¼ 2 Te 5 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 4 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 6 ðj; nÞ ¼ e T 4 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; 2 Te 8 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 2 ði; mÞ; Te 8 ðj; nÞ ¼ 0; 2 Te 8 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ( i þ j 6 1; Te 7 ði þ j; m þ nÞ; ½ Te 2 ði; mÞ; Te 9 ðj; nÞ ¼ e T 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2;
ð23Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
( ½ Te 3 ði; mÞ; Te 4 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 5 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 7 ðj; nÞ ¼ ( ½ Te 3 ði; mÞ; Te 8 ðj; nÞ ¼ ( ½ Te 4 ði; mÞ; Te 5 ðj; nÞ ¼ ( ½ Te 4 ði; mÞ; Te 6 ðj; nÞ ¼
2 Te 6 ði þ j; m þ nÞ; 2 Te 6 ði þ j 2; m þ n þ 1Þ;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; Te 4 ði þ j; m þ nÞ; e T 4 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; 2 Te 9 ði þ j; m þ nÞ; 2 Te 9 ði þ j 2; m þ n þ 1Þ;
½ Te 3 ði; mÞ; Te 6 ðj; nÞ ¼ 0;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; Te 7 ði þ j; m þ nÞ; e T 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; 2 Te 8 ði þ j; m þ nÞ; 2 Te 8 ði þ j 2; m þ n þ 1Þ;
691
½ Te 3 ði; mÞ; Te 9 ðj; nÞ ¼ 0;
i þ j 6 1; i þ j ¼ 2;
i þ j 6 1; 2 Te 9 ði þ j; m þ nÞ; e 2 T 9 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2;
½ Te 4 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 4 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 4 ði; mÞ; Te 9 ðj; nÞ ¼ 0; ( i þ j 6 1; Te 7 ði þ j; m þ nÞ; ½ Te 5 ði; mÞ; Te 6 ðj; nÞ ¼ Te 7 ði þ j 2; m þ n þ 1Þ; i þ j ¼ 2; ½ Te 5 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 5 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 5 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 7 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 6 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 7 ði; mÞ; Te 8 ðj; nÞ ¼ ½ Te 7 ði; mÞ; Te 9 ðj; nÞ ¼ ½ Te 8 ði; mÞ; Te 9 ðj; nÞ ¼ 0: Another often used loop algebra of the Lie algebra (9) is that ~ 14 ¼ spanf Te i ðj; nÞg9i¼1 ; slð6Þ
j ¼ 0; 1; 2;
Te i ðj; nÞ ¼ Te i k3nþj ; i ¼ 1; 2; . . . ; 9: The commutator is defined as ( i þ j < 3; ½ Te k ; Te l ði þ j; m þ nÞ; ½ Te k ði; mÞ; Te l ðj; nÞ ¼ ½ Te k ; Te l ði þ j 3; m þ n þ 1Þ; i þ j P 3; For example,
(
½ Te 1 ði; mÞ; Te 2 ðj; nÞ ¼
ð24Þ
1 6 k; l 6 9; m; n 2 Z:
i þ j < 3; 2 Te 2 ði þ j; m þ nÞ; e 2 T 2 ði þ j 3; m þ n þ 1Þ; i þ j P 3:
In general, the loop algebra corresponding to the Lie algebra (9) is given by ~ 15 ¼ spanfT i ðj; nÞg9i¼1 ; slð6Þ where Te i ðj; nÞ ¼ Te i k
ð25Þ
Nnþj
; j ¼ 0; 1; 2; . . . ; N 1; N is an arbitrary natural number, the commutator is defined as ( i þ j < N; ½ Te k ; Te l ði þ j; m þ nÞ; ½ Te k ði; mÞ; Te l ðj; nÞ ¼ 1 6 k; l 6 9; m; n 2 Z: ½ Te k ; Te l ði þ j N ; m þ n þ 1Þ; i þ j P N ;
As for the Lie algebra sl(6)2, its simplest loop algebra is given by ~ 21 ¼ spanfhi ðnÞg9i¼1 ; hi ðnÞ ¼ hi kn ; slð6Þ
ð26Þ
the commutator is defined as [hi(m), hj(n)] = [hi, hj]km+n, 1 6 k, l 6 9. ~ 22 ; slð6Þ ~ 23 ; slð6Þ ~ 24 and slð6Þ ~ 25 which correspond to slð6Þ ~ 1i ði ¼ 2; 3; 4; 5Þ Similarly, the loop algebras slð6Þ respectively, are obtained. Here we omit them.
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Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
It is easy to see that the spectral matrices generated by the Lie algebras sl(3)i (i = 1, 2) are presented as U 22 U 1 U¼ ð27Þ ; U 1 ¼ ða; bÞT ; 0 0 the spectral matrices given by the Lie algebra sl(4)i, sl(6)i (i = 1, 2) are presented, respectively U 22 U 1 a b U¼ ; ; U1 ¼ c d 0 U 22 0 1 U U1 U2 B C U ¼ @ 0 U U 1 A; U i ði ¼ 1; 2Þ are 2 2 matrices: 0 The general case 0 U B B B0 U ¼B B .. @ . 0
0
0
0
0
ð31Þ
0
Furthermore, the more case of (29) was exhibited [13] 0 1 U U a1 U a2 B C U ¼ @ 0 U U a3 A: 0
ð30Þ
U
and the general case of (27) was given by Ma [13] 0 1 U U a1 U a2 B C U ¼ @ 0 U U a3 A; 0
ð29Þ
U
of (28) and (29) was constructed by Ma [13] 1 U a1 . . . U av .. C .. C . C U . C; .. .. C . U a1 A . ...
ð28Þ
ð32Þ
U a4
In terms of the spectral matrices (30)–(32), a kind of classification of the Lie algebras was proposed in [13]. However, the explicit matrix forms of (31) and (32) were not given. In what follows, we want to explicitly construct a few Lie algebras to satisfy (31) and (32), which fill up Ma’s work. Take 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 C C C B B B B 0 1 0 0 0 C B0 0 0 0 0C B1 0 0 0 0C C C C B B B C C C B B e1 ¼ B B 0 0 1 0 0 C; e 2 ¼ B 0 0 0 1 0 C; e 3 ¼ B 0 0 0 0 0 C; C C C B B B @ 0 0 0 1 0 A @0 0 0 0 0A @0 0 1 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 C C C B B B B 0 0 0 1 0 C B0 0 0 0 0C B0 0 1 0 0C C C C B B B C C C B B e4 ¼ B B 0 0 0 0 0 C; e 5 ¼ B 0 0 0 0 0 C; e 6 ¼ B 0 0 0 0 0 C; C C C B B B @0 0 0 0 0A @0 0 0 0 0A @0 0 0 0 0A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
0
0 0 0 0 1
B B0 B e7 ¼ B B0 B @0 0
1
0
0 0 0 0 0
C B 0 0 0 0C B0 C B B0 ; e 0 0 0 0C ¼ 8 C B C B 0 0 0 0A @0 0 0 0 0 0
1
0
0 0 0 0 0
C B 0 0 0 1C B0 C B B0 ; e 0 0 0 0C ¼ 9 C B C B 0 0 0 0A @0 0 0 0 0 0
1
0
693
0 0 0 0 0
C B 0 0 0 0C B0 C B B0 ; e 0 0 0 1C ¼ 10 C B C B 0 0 0 0A @0 0 0 0 0 0
1
C 0 0 0 0C C 0 0 0 0C C; C 0 0 0 1A 0 0 0 0
it is easy to find that ½e1 ; e2 ¼ 2e2 ; ½e2 ; e4 ¼ 2e5 ;
½e1 ; e3 ¼ 2e3 ; ½e2 ; e5 ¼ 0;
½e2 ; e3 ¼ e1 ; ½e2 ; e6 ¼ e4 ;
½e1 ; e4 ¼ 0;
½e1 ; e5 ¼ 2e5 ;
½e3 ; e4 ¼ 2e6 ;
½e3 ; e5 ¼ e4 ;
½e1 ; e6 ¼ 2e6 ;
½e3 ; e6 ¼ ½e4 ; e5 ¼ ½e4 ; e6 ¼ ½e5 ; e6 ¼ 0; ½e1 ; e7 ¼ e7 ; ½e1 ; e8 ¼ e8 ; ½e1 ; e9 ¼ e9 ; ½e1 ; e10 ¼ e10 ; ½e2 ; e7 ¼ 0; ½e2 ; e8 ¼ e7 ; ½e2 ; e9 ¼ 0; ½e2 ; e10 ¼ e9 ; ½e3 ; e7 ¼ e8 ; ½e3 ; e8 ¼ 0; ½e3 ; e9 ¼ e10 ; ½e3 ; e10 ¼ ½e4 ; e7 ¼ ½e4 ; e8 ¼ 0; ½e4 ; e9 ¼ e7 ; ½e4 ; e10 ¼ e8 ; ½e5 ; e7 ¼ ½e5 ; e8 ¼ ½e5 ; e9 ¼ 0; ½e5 ; e10 ¼ e7 ; ½e6 ; e7 ¼ ½e6 ; e8 ¼ 0;
½e6 ; e9 ¼ e8 ; ½e6 ; e10 ¼ 0;
½ei ; ej ¼ 0
ð7 6 i; j 6 10Þ:
Let U = ae1 + be2 + ce3 + de4 + ee5 + fe6 + ge7 + he8 + ke9 + we10, then 1 0 a b d e g C B B c a f d h C C B U ¼B kC C: B0 0 a b C B @ 0 0 c a w A 0 0 0 0 0 10
Assume a linear map d : spanfei gi¼1 0 1 0 0 0 B 0 1 1 0 B B B0 1 1 0 B 1 0 B0 0 0 1 w1 B B Bw C B 2 C 1B0 0 0 0 B . C¼ B B . C 2B0 0 0 0 B @ . A B B0 0 0 0 w10 B B0 0 0 0 B B @0 0 0 0 0 0 0 0
ð33Þ
10
! spanfwi gi¼1 ,
1
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 1
0 1
0 0
0 0
0 0
1 0
0
0
0C C C 0C C0 1 e1 0C C CB e C 0 CB 2 C CB . C; B C 0C C@ .. A C 0C C e10 0C C C 0A
1 0 0
0 0
2 0
0 2
0 0
0 0
0 0
0 0
0 0
2 0
ð34Þ
2
then w1 ¼ 12 e1 ; w2 ¼ 12 ðe2 e3 Þ; w3 ¼ 12 ðe2 þ e3 Þ; w4 ¼ e4 ; w5 ¼ 12 ðe5 þ e6 Þ; w6 ¼ 12 ðe5 e6 Þ; w7 ¼ e7 ; w8 ¼ e8 ; w9 ¼ e9 ; 10 10 w10 ¼ e10 . It is easy to see that spanfwi gi¼1 is also a Lie algebra different form spanfei gi¼1 . If let 1 0 a1 a2 þ a3 a4 a5 þ a6 2a7 C B a1 a5 a6 a4 2a8 C B a2 þ a3 10 X C 1B ð35Þ U¼ ai e i ¼ B 0 0 a1 a2 þ a3 2a9 C C: B 2B C i¼1 0 0 a2 þ a3 a1 2a10 A @ 0 0 0 0 0 Then (33) and (35) are the spectral cases of (31), which have extensive applications on generating integrable couplings. In the following, we construct explicit Lie algebras whose spectral matrices are of the form (32). First, consider a simplest case.
694
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
Let 0
1
B p1 ¼ @ 0 0
0 0
B p4 ¼ @ 0 0
0
0
1
C 1 0 A; 0 0 1 0 1 C 0 0 A; 0
0
0
1 0
B p2 ¼ @ 0 0
0
0 B p5 ¼ @ 0
0
0
1
0
C 0 0 A; 0 0 1 0 0 C 0 1 A; 0
1
0
0
0
B p3 ¼ @ 1
0
C 0 A;
0
0
0
0 1
0 B p6 ¼ @ 0
0
0
0
C 0 A;
0
0
1
0
then we have ½p1 ; p2 ¼ 2p2 ; ½p2 ; p3 ¼ p1 ;
½p1 ; p3 ¼ 2p3 ; ½p2 ; p4 ¼ 0;
½p2 ; p5 ¼ p4 ;
½p3 ; p5 ¼ ½p3 ; p6 ¼ ½p4 ; p5 ¼ 0; P6 Set U ¼ i¼1 ai pi ; then 0 1 a1 a2 a4 B C U ¼ @ a3 a1 a5 A: 0
0
½p1 ; p4 ¼ p4 ;
½p1 ; p5 ¼ p5 ;
½p2 ; p6 ¼ 0;
½p4 ; p6 ¼ p4 ;
½p1 ; p6 ¼ 0;
½p3 ; p4 ¼ p5 ;
½p5 ; p6 ¼ p5 :
ð36Þ
a6
Make a linear transformation 0
6
d : spanfpi gi¼1
q1
1
0
1 0
q6
1
0 0
1
1
0 0
0
1
0 0
0
0
2 0
0
0
0 2
CB C 0 CB p 2 C CB C B C 0C CB p 3 C CB C; B C 0C CB p 4 C CB C 0 A@ p 5 A
0 0
0
0 0
2
B B C B0 B q2 C B B C Bq C 1B0 B B 3C 6 ! spanfqi gi¼1 ; B C ¼ B B q4 C 2 B 0 B B C B B C @0 @ q5 A
0
10
0
p1
p6
6
then spanfqi gi¼1 is a Lie algebra, whose commuting operations are as follows: 1 ½q1 ; q5 ¼ q5 ; 2 1 1 1 ½q1 ; q6 ¼ 0; ½q2 ; q4 ¼ q5 ; ½q2 ; q5 ¼ q4 ; ½q2 ; q6 ¼ 0; ½q3 ; q4 ¼ q5 ; 2 2 2 1 1 1 ½q3 ; q5 ¼ q4 ; ½q3 ; q6 ¼ ½q4 ; q5 ¼ 0; ½q4 ; q6 ¼ q4 ; ½q5 ; q6 ¼ q5 : 2 2 2 P6 If let U ¼ i¼1 bi qi ; then it is easy to see that 0 1 b1 b2 þ b3 b4 1B C b1 b5 A : U ¼ @ b2 b3 2 0 0 b6 ½q1 ; q2 ¼ q3 ;
½q1 ; q3 ¼ q2 ;
½q2 ; q3 ¼ q1 ;
1 ½q1 ; q4 ¼ q4 ; 2
ð37Þ
Obviously, (36) and (37) are the same-type matrices, and are the special cases of (32). Next, we consider the more complicated cases of (32). There are two special cases basic on a subalgebra of the Lie algebra A5. First, we present the subalgebra. Let 5 ¼ spanfvi g6 ; A i¼1
ð38Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
695
where 0
1
B B0 B B0 B v1 ¼ B B0 B B @0 0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0 B B0 B B0 B v4 ¼ B B0 B B @0
1 0
0
0
0
0
0
1
0
0
0
0
0 0
1
0
0 0
0
0
0 0
0
0
0 0
0
0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0
0 0
0
0
0 B B1 B B0 B v5 ¼ B B0 B B @0
0
0
0 0
0
0
B B0 B B0 B v2 ¼ B B0 B B @0
0
0 0
½v2 ; v3 ¼ v6 ;
0
0 0
C 0C C 0C C C; 0C C C 0A
C 0C C 0C C C; 0C C C 0A
½v1 ; v2 ¼ 0;
1
0
½v1 ; v3 ¼ 0;
0 0
0
0
0 0
0
1
0 0
0
0
0 0
0
0
0 0
0
½v3 ; v6 ¼ 2v3 ;
½v4 ; v6 ¼ ½v5 ; v6 ¼ 0:
If set U ¼
0
0
½v2 ; v6 ¼ 2v2 ;
½v4 ; v5 ¼ v1 ;
1
0
½v1 ; v4 ¼ 2v4 ;
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 B B0 B B0 B v6 ¼ B B0 B B @0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
1
1 0 1 0 C 0 C C 0 C C C; 0 C C C 0 A
0
0
0 0
0
1
B B0 B B0 B v3 ¼ B B0 B B @0 0
0 0
1
0
C 0C C 0C C C; 0C C C 1A
C 0C C 0C C C; 0C C C 0A
½v1 ; v6 ¼ ½v2 ; v4 ¼ ½v2 ; v5 ¼ 0; P6
0
C 0 0C C 0 0C C C; 0 0C C C 0 0A
½v1 ; v5 ¼ 2v5 ;
½v3 ; v4 ¼ ½v3 ; v5 ¼ 0;
then it is easy to see that 1 a4 0 0 0 0 C a1 0 0 0 0 C C 0 a1 a4 0 0 C C C; 0 a5 a1 0 0 C C C 0 0 0 a6 a2 A
i¼1 ai vi ;
0
a1 B B a5 B B0 B U ¼B B0 B B @0 0
0
0
0
a3
a6
which is a special case of (32). Now we begin to enlarge the Lie algebra (38) into two bigger Lie algebras with aspect to dimensional numbers Case 1: Set 0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
B B0 B B0 B v7 ¼ B B0 B B @0 0
B B0 B B0 B v9 ¼ B B0 B B @0 0
1
1
0
0
0
C 0C C 0C C C; 0C C C 0A
B B0 B B0 B v8 ¼ B B0 B B @0
0
0
0
0 0
0
0
0
0 0
0
1
0 0
0
0
0 0
0
0
0 0
0
0
C 0C C 0C C C; 0C C C 0A
0 0 0 B B0 B B0 B v10 ¼ B B0 B B @0
0 0
0
0
0
1
0
1
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
C 1C C 0C C C; 0C C C 0A
0
0
0 0
0
1 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C 0 0C C 0 0C C C; 0 0C C C 0 0A
0 0
0
0
0 0
696
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 10
then we find that spanfvi gi¼1 ¼: A51 is a Lie algebra, where ½v1 ; v7 ¼ v7 ;
½v2 ; v7 ¼ ½v4 ; v7 ¼ 0;
½v5 ; v7 ¼ v8 ;
½v6 ; v7 ¼ v7 ;
½v1 ; v8 ¼ v8 ;
½v2 ; v8 ¼ 0;
½v3 ; v8 ¼ v9 ;
½v4 ; v8 ¼ v7 ;
½v7 ; v8 ¼ 0;
½v1 ; v9 ¼ v9 ;
½v2 ; v9 ¼ v8 ;
½v5 ; v9 ¼ 0;
½v6 ; v9 ¼ v9 ;
½v7 ; v9 ¼ ½v8 ; v9 ¼ 0;
½v2 ; v10 ¼ v7 ;
½v3 ; v10 ¼ ½v4 ; v10 ¼ 0;
½v6 ; v10 ¼ v10 ; Set U ¼
P10
then
0
a4
0
0
a10
a1
0
0
a9
0
a1
a4
0
0
a5
a1
0
0
0
0
a6
0
0
0
a3
0 0
B B a5 B B0 B U ¼B B0 B B @0 0
½v5 ; v8 ¼ 0;
½v6 ; v8 ¼ v8 ;
½v3 ; v9 ¼ 0;
½v4 ; v9 ¼ v10 ;
½v1 ; v10 ¼ v10 ;
½v5 ; v10 ¼ v9 ;
½v7 ; v10 ¼ ½v8 ; v10 ¼ ½v9 ; v10 ¼ 0:
i¼1 ai vi ;
a1
½v3 ; v7 ¼ v10 ;
a7
1
C a8 C C 0 C C C: 0 C C C a2 A a6
Case 2: Let 0
0
B B0 B B0 B v11 ¼ B B0 B B @0 0
0 0
B B0 B B0 B v13 ¼ B B0 B B @0 0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 1 0 0 C 0 0C C 0 1C C C; 0 0C C C 0 0A
0
0
0
0 0
0
C 0 0C C 1 0C C C; 0 0C C C 0 0A
0
B B0 B B0 B v12 ¼ B B0 B B @0 0
0 0
B B0 B B0 B v14 ¼ B B0 B B @0 0
0
1
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0 0
0 0
0 0 0 0
0
0
0 0
0
0
0 0
0
0
0 1
0
0
0 0
0 1 0 C 0C C 0C C C; 0C C C 0A
0
0
0 0
0
C 0C C 0C C C; 1C C C 0A
then span{v1, v2, v3, v4, v5, v6, v11, v12, v13, v14} ¼: A52 is also a Lie algebra, where ½v1 ; v11 ¼ v11 ; ½v6 ; v11 ¼ v11 ;
½v2 ; v11 ¼ v13 ;
½v3 ; v11 ¼ ½v4 ; v11 ¼ 0;
½v1 ; v12 ¼ v12 ;
½v2 ; v12 ¼ 0;
½v4 ; v12 ¼ v13 ;
½v5 ; v12 ¼ 0;
½v6 ; v12 ¼ v12 ;
½v1 ; v13 ¼ v13 ;
½v2 ; v13 ¼ 0;
½v3 ; v13 ¼ v11 ;
½v5 ; v13 ¼ v12 ;
½v6 ; v13 ¼ v13 ;
½v4 ; v14 ¼ v11 ;
½v5 ; v14 ¼ 0;
½v11 ; v13 ¼ ½v11 ; v14 ¼ ½v12 ; v14 ¼ ½v13 ; v14 ¼ 0:
½v3 ; v12 ¼ v14 ;
½v11 ; v12 ¼ 0;
½v1 ; v14 ¼ v14 ; ½v6 ; v14 ¼ v14 ;
½v5 ; v11 ¼ v14 ;
½v4 ; v13 ¼ 0; ½v2 ; v14 ¼ v12 ;
½v3 ; v14 ¼ 0;
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
The corresponding spectral 0 0 a1 a4 B B a5 a1 0 B B 0 a1 B0 U ¼B B0 0 a5 B B B0 0 0 @ 0
0
0
matrix presents 0
0
0
0
a4
a11
a1
a14
0
a6
0
a3
0
697
1
C 0 C C C a13 C C: a12 C C C a2 C A a6
Remark 1. There are many Lie algebras whose spectral matrices are of the forms (31) and (32), we do not want 5 ; A51 to go on constructing them in the paper. It is easy to find that the semi-direct sum of the Lie algebras A and A52 is also a Lie algebra, the resulting spectral matrix belongs to the form (32). Similar to (34), some new 5 ; A51 and A52 are worked out by use of invertible linear transformations. Lie algebras generated by A 4. Some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy Li [15] presented the Lax pair from the self-dual Yang-Mills equation ( ux ¼ Mu; ut ¼ P ðkÞuy þ N u;
ð39Þ
whose compatibility is the following (2 + 1)-dimensional partial-differential equation hierarchy M t N x þ ½M; N P ðkÞM y ¼ 0;
ð40Þ
where P(k) = a0 + a1k + , k is a spectral parameter. If let P(k) = 0, then (40) reduces to the standard zero curvature equation M t N x þ ½M; N ¼ 0:
ð41Þ
In the section, we want to take the loop algebras (20) and (26) for examples to deduce some solutions of Eq. (40) which present the (2 + 1)-dimensional nonlinear evolution equations. Example 4. Consider the spectral problem ux ¼ Mu; M ¼ i Te 1 ð1Þ iqr Te 1 ð0Þ þ q Te 2 ð0Þ þ r Te 3 ð0Þ þ u1 Te 5 ð0Þ þ u2 Te 6 ð0Þ þ u3 Te 8 ð0Þ þ u4 Te 9 ð0Þ: P P9 Let N ¼ mP0 ð i¼1 aim Te i ðmÞÞ; then the stationary linear equation N x ¼ ½M; N generates a set of solutions for aim (i = 1, 2, . . . , 9): 8 a1mx ¼ qa3m ra2m ; > > > > > > 2ia2;mþ1 ¼ a2mx þ 2iqra2m þ 2qa1m ; > > > > > > 2ia3;mþ1 ¼ a3mx þ 2iqra3m þ 2ra1m ; > > > > > > a4mx ¼ qa6m ra5m þ u1 a3m u2 a2m ; > < 2ia5;mþ1 ¼ a5mx þ 2iqra5m þ 2qa4m þ 2u1 a1m ; > > > > 2ia6;mþ1 ¼ a6mx þ 2iqra6m þ 2ra4m þ 2u2 a1m ; > > > > > > a7mx ¼ qa9m ra8m þ u1 a6m u2 a5m þ u3 a3m u4 a2m ; > > > > > > 2ia8;mþ1 ¼ a8mx þ 2iqra8m þ 2qa7m þ 2u1 a4m þ 2u3 a1m ; > > > : 2ia9;mþ1 ¼ a9mx þ 2iqra9m þ 2ra7m þ 2u2 a4m þ 2u4 a1m :
ð42Þ ð43Þ
ð44Þ
698
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 ðnÞ
Case 1: Take N þ ¼
Pn
m¼0 ð
P9
e
i¼1 aim T i ðn
ðnÞ mÞÞ ¼ kn N N ; then a direct calculation reads that
ðnÞ ðnÞ N þx þ ½M; N þ ¼ 2ia2;nþ1 Te 2 ð0Þ þ 2ia3;nþ1 Te 3 ð0Þ 2ia5;nþ1 Te 5 ð0Þ þ 2ia6;nþ1 Te 6 ð0Þ 2ia8;nþ1 Te 8 ð0Þ þ 2ia9;nþ1 Te 9 ð0Þ: ðnÞ
Denote N ðnÞ ¼ N þ ; then the (2 + 1)-dimensional partial-differential equation system M t N xðnÞ þ ½M; N ðnÞ M y ¼ 0 admits the following solution for M 8 1 0 1 0 1 0 a2;nþ1 q q > > > > C C C B B B > > B r C B a3;nþ1 C B r C > > C B C B C B > > C B C B C B > < ut ¼ B u1 C ¼ 2iB a5;nþ1 C þ B u1 C ; C B C Bu C B a B 2C B 6;nþ1 C B u2 C > C C B C B B > > @ u3 A @ a8;nþ1 A @ u3 A > > > > > a9;nþ1 u4 t u4 y > > > : ðqrÞt ¼ ðqrÞy :
ð45Þ
ð46Þ
It is easy to see that (qr)t = 2ira2,n+12iqa3,n+1 + (qr)y. Hence, a1,n+1x = 0. Take a1n = 0, then Eq. (44) is still solvable. A (2 + 1)-dimensional integrable solitary hierarchy is obtained 1 0 1 0 a2;nþ1 q B r C B a3;nþ1 C C B C B C B C B B u1 C B a5;nþ1 C C ¼ 2iB C; ~ uz ¼: B ð47Þ C Bu C B a B 2C B 6;nþ1 C C B C B @ u3 A @ a8;nþ1 A u4
ty
a9;nþ1
which is a (2 + 1)-dimensional enlarging AKNS hierarchy. ðnÞ Case 2: Take N ðnÞ ¼ N þ 2a1;nþ1 Te 1 ð0Þ, then Eq. (45) admits the solution 8 1 0 1 0 2ia2;nþ1 4qa1;nþ1 q > > > > C Br C B 2ia > 3;nþ1 þ 4ra1;nþ1 C > B C B > > C C B B > > B u1 C B 2ia5;nþ1 4u1 a1;nþ1 C > <~ C C; B B uz ¼: B C ¼ B C B u2 C B 2ia6;nþ1 þ 4u2 a1;nþ1 C > C B C B > > @ u3 A @ 2ia8;nþ1 4u3 a1;nþ1 A > > > > > 2ia9;nþ1 þ 4u4 a1;nþ1 u4 ty > > > : ðqrÞty ¼ 2ia1;nþ1x :
ð48Þ
In the following, we deduce the Hamiltonian expression of (47). e 9 ¼ spanfcðnÞ ¼ ckn ; c 2 R9 g along with the commutator [X(m), Y(n)]1 = Consider the loop algebra R m+n 9 [X, Y]1k , X, Y 2 R . Let ¼ ðik iqr; q; r; 0; u1 ; u2 ; 0; u3 ; u4 ÞT ; M n X T T ðnÞ w þ wy ; N ðnÞ ¼ wt ¼ N ða1m ; a2m ; . . . ; a9m Þ knm þ ð2a1;nþ1 ; 0; 0; 0; 0; 0; 0; 0; 0Þ :
wx ¼ Mw;
m¼0
Then the zero curvature equation t N ðnÞ þ ½M; N ðnÞ M y ¼ 0 M x
ð49Þ ð50Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
699
also gives rise to the system (47). Hence, it can be expressed as 10 1 0 0 1 a6;nþ1 þ a9;nþ1 q 0 0 0 0 0 1 C B 0 0 0 B r C B 0 1 0 C CB a5;nþ1 þ a8;nþ1 C B B C CB C B B C B 0 0 0 B u1 C 1 0 1 CB a3;nþ1 þ a6;nþ1 C CB C ¼ 2iB C ¼: JP nþ1 ; ~ uz ¼ B C B 0 0 1 0 Bu C B 1 0 C CB a2;nþ1 þ a5;nþ1 C B B 2C CB C B B C A @ 0 1 0 1 0 @ u3 A a3;nþ1 1 A@ a2;nþ1 1 0 1 0 1 0 u4 ty where J is a Hamiltonian operator. A direct calculation gives 1 0 2irða4;nþ1 þ a7;nþ1 Þ þ a3;nþ1 þ a6;nþ1 þ a9;nþ1 C B 2iqða 4;nþ1 þ a7;nþ1 Þ þ a2;nþ1 þ a5;nþ1 þ a8;nþ1 C B C B C B a3;nþ1 þ a6;nþ1 C ¼: Rnþ1 B C B a2;nþ1 þ a5;nþ1 C B C B A @ a3;nþ1 a2;nþ1
0
1
2iro q þ 1 B 2iqo1 q B B B 0 ¼B B 0 B B @ 0 0
0
2iro1 r 2iqo1 r þ 1
2iro1 u1 2iqo1 u1
2iro1 u2 2iqo1 u2
2iro1 u3 þ 1 2iqo1 u3
0
1
0
0
0 0
0 0
1 0
0 1
0
0
0
0
10
a6;nþ1 þ a9;nþ1
1
1 2iro1 u4 2iqo1 u4 þ 1 C C C C 0 C C 0 C C A 0 1
a6;nþ1 þ a9;nþ1 C Ba B 5;nþ1 þ a8;nþ1 C C B B a3;nþ1 þ a6;nþ1 C C ¼: QP nþ1 : B B C B a2;nþ1 þ a5;nþ1 C C B A @ a3;nþ1 a2;nþ1 Thus, we have 0 0 1 q 0 B 0 B r C B B C B B C B 0 B u1 C C ¼ 2iB ~ uz ¼ B B 0 Bu C B B 2C B B C @ 0 @ u3 A 1 u4 ty
0
0
0
0
0
0
0
1
0 0
0 1
1 0
0 1
1 0
0 1
1 0
0 1
1
1
C B 0 C CB a5;nþ1 þ a8;nþ1 C CB C 1 CB a3;nþ1 þ a6;nþ1 C CB C ¼: JP nþ1 ¼ 2iJQ1 Rnþ1 : C B 0 C CB a2;nþ1 þ a5;nþ1 C CB C A a3;nþ1 1 A@ a2;nþ1 0
From the formula (17) and F, a direct computation gives ; oM ¼ 2irða4 þ a7 Þ þ a3 þ a6 þ a9 ; N oq o ; M ¼ 2iqða4 þ a7 Þ þ a2 þ a5 þ a8 ; N or
ð51Þ
700
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
oM oM N; N; ¼ a3 þ a6 ; ¼ a2 þ a5 ; ou1 ou2 oM oM oM N; N; N; ¼ 2iða4 þ a7 Þ: ¼ a3 ; ¼ a2 ; ou3 ou4 ok Substituting the above results into the quadratic-form identity yields 1 0 2irða4 þ a7 Þ þ a3 þ a6 þ a9 C B B 2iqða4 þ a7 Þ þ a2 þ a5 þ a8 C C B C B a3 þ a6 d C c o c B 2iða4 þ a7 Þ ¼ k kB C: C d~ u ok B a þ a 2 5 C B C B a3 A @
ð52Þ
a2 Comparing the coefficients of k
n1
in (52) gives rise to
d 2iða4;nþ1 þ a7;nþ1 Þ ¼ ðn þ cÞRn d~ u 2iða
þa7;nþ1 Þ
n . It is easy to see that c = 0. Thus, Rn ¼ dH ; H n ¼ 4;nþ1n d~ u structure of the higher-dimensional integrable hierarchy (47)
~ uz ¼ 2iJQ1 Rnþ1 ¼ 2iJQ1
. Hence, we obtain the quasi-Hamiltonian
dH nþ1 : d~ u
ð53Þ
Remark 2. Because Q is not a sky-symmetric matrix, JQ1 is not Hamiltonian. Hence, we call (53) the quasiHamiltonian structure. Of course, if we omit the term qr Te 1 ð0Þ in (42), it is easy to obtain the Hamiltonian structure of the corresponding enlarging higher-dimensional AKNS hierarchy. Example 5. Consider the isospectral problem ux ¼ Mu; M ¼ h2 ð1Þ þ uh3 ð0Þ þ vh1 ð0Þ þ u1 h4 ð0Þ þ u2 h5 ð0Þ þ u3 h8 ð0Þ þ u4 h9 ð0Þ: ð54Þ P Set N ¼ mP0 ðb1m h1 ðmÞ þ b2m h2 ðmÞ þ b3m h3 ðmÞ þ b4m h4 ðmÞ þ b5m h5 ðmÞ þ b6m h6 ðmÞ þ b7m h7 ðmÞþ b8m h8 ðmÞ þ b9m h9 ðmÞÞ, a solution of the linear equation for N N x ¼ ½M; N is given by 8 b3;mþ1 ¼ b1mx ub2m ; > > > > > > b2mx ¼ ub1m þ vb3m ; > > > > > b1;mþ1 ¼ b3mx vb2m ; > > > > > > b5;mþ1 ¼ b4mx þ ub6m u2 b2m ; > > > > > b4;mþ1 ¼ b5mx þ vb6m u1 b2m ; > > > > < b6mx ¼ ub4m vb5m u1 b3m þ u2 b1m ; > b7mx ¼ ub9m þ vb8m þ u1 b5m u2 b4m u3 b1m þ u4 b3m ; > > > > > > b9;mþ1 ¼ b8mx vb7m þ u1 b6m u4 b2m ; > > > >b > 8;mþ1 ¼ b9mx ub7m þ u2 b6m u3 b2m ; > > > > > b2;0 ¼ a ¼ const:; b1;0 ¼ b3;0 ¼ 0; b1;1 ¼ av; b3;1 ¼ au; > > > > > > b2;1 ¼ 0; b4;0 ¼ b5;0 ¼ b6;0 ¼ b7;0 ¼ b8;0 ¼ b9;0 ¼ 0; > > > : b4;1 ¼ au1 ; b5;1 ¼ au2 ; b6;1 ¼ b7;1 ¼ 0; b8;1 ¼ au3 ; b9;1 ¼ au4 ; . . .
ð55Þ
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702 ðnÞ
Let N þ ¼
Pn
m¼0 ð
P9
ðnÞ
i¼1 bim hi ðn
701
mÞÞ ¼ kn N N ðnÞ ; then
ðnÞ
N þx þ ½M; N þ ¼ b1;nþ1 h3 ð0Þ þ b3;nþ1 h1 ð0Þ b4;nþ1 h5 ð0Þ þ b5;nþ1 h4 ð0Þ þ b8;nþ1 h9 ð0Þ b9;nþ1 h8 ð0Þ: A solution of the (2 + 1)-dimensional partial-differential equation system for M ðnÞ
ðnÞ
M t N þx þ ½M; N þ M y ¼ 0 is obtained 0
u
1
0
0
1
0
C B B b B 3;nþ1 C B 0 C B B B b5;nþ1 C B 0 C B B ¼B C¼B B b4;nþ1 C B 0 C B B C B B @ b9;nþ1 A @ 0
B C BvC B C B u1 C B C ~ uz ¼ B C B u2 C B C B C @ u3 A u4
b1;nþ1
ty
b8;nþ1
0 1
10
b3;nþ1 þ b5;nþ1 b8;nþ1
1
0
0
0
0
0
0
0
0
1
0
1
0
1 1
0
CB C 0 CB b1;nþ1 þ b4;nþ1 b9;nþ1 C CB C C B b1;nþ1 b4;nþ1 1 0 C CB C CB C ¼: JW nþ1 ; C B b b 0 1 C 3;nþ1 5;nþ1 CB C CB C b3;nþ1 0 0 A@ A
1
0
1
0
0
1
0
b1;nþ1 ð56Þ
where J is a Hamiltonian operator. In order to get Hamiltonian structure of (56), a Lax pair is constructed by e 9 ¼ spanfxðnÞ ¼ xkn ; x 2 R9 g with the commuting operation [x(m), y(n)]2 = using the loop algebra R m+n 9 [x, y]2k , x, y 2 R : 8 e u; M e ¼ ðv; k; u; u1 ; u2 ; 0; 0; u3 ; u4 ÞT ; > < ux ¼ M n P T e ðnÞ u þ uy ; N e ðnÞ ¼ > ðb1m ; b2m ; . . . ; b9m Þ knm ; : ut ¼ N m¼0
etN e ðnÞ þ ½M; N e ðnÞ M e y ¼ 0 also admits the solution (56). Taking use of F , a direct whose compatibility M x computation gives ( ) ( ) e e oM oM e e N; N; ¼ b3 þ b 5 b 8 ; ¼ b1 þ b4 b9 ; ou ov ( ) ( ) e e o M o M e; e; N N ¼ b1 b4 ; ¼ b3 b5 ; ou1 ou2 ( ) ( ) ( ) e e e o M o M o M e; e; e; N N N ¼ b2 b 6 b7 ; ¼ b3 ; ¼ b1 ; ou3 ou4 ok e ¼ ðb1 ; b2 ; . . . ; b9 Þ ; bi ¼ where N form identity gives T
P
mP0 bim k
0
m
; i ¼ 1; 2; . . . ; 9. Inserting the above results into the quadratic-
b3 þ b5 b8
1
C B B b1 þ b4 b9 C C B B b1 b4 C d C c o c B ðb2 b6 b7 Þ ¼ k kB C: d~ u ok B b3 b5 C C B C B b3 A @
ð57Þ
b1 Comparing the coefficients of kn1 in (57) leads to ( ; W nþ1 ¼ dHd~nþ1 u 1 H nþ1 ¼ nþc ðb2;nþ2 b6;nþ2 b7;nþ2 Þ;
d ðb2;nþ1 d~ u
b6;nþ1 b7;nþ1 Þ ¼ ðn þ cÞW n , ð58Þ
702
Y. Zhang et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 682–702
From the recurrence relations (55), we have c = 0. Hence, the Hamiltonian structure of the (2 + 1)-dimensional soliton equation hierarchy is produced ~ uz ¼ JW nþ1 ¼ J
dH nþ1 : d~ u
ð59Þ
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