- Email: [email protected]
A new Lie algebra, a corresponding multi-component integrable hierarchy and an integrable coupling
Chaos, Solitons and Fractals 29 (2006) 114–124 www.elsevier.com/locate/chaos
A new Lie algebra, a corresponding multi-component integrable hierarchy and an integrable coupling q Yufeng Zhang a, Xiurong Guo b
b,*
, Honwah Tam
c
a Mathematical School, Liaoning Normal University, Dalian 116029, PR China Science School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China c Department of Computer Science, Hong Kong Baptist University, Hong Kong, China
Accepted 30 March 2005
Abstract A new simple loop algebra is constructed, which is devote to establishing an isospectral problem. By making use of Tu scheme, a new multi-component integrable hierarchy is obtained. Again via expanding the loop algebra above, another higher-dimensional loop algebra is presented. It follows that the binary integrable coupling systems are given. This method proposed in this paper can be used to other soliton hierarchies. 2005 Published by Elsevier Ltd.
1. Introduction The Lax pair method [1] is a general technique for generating single component integrable hierarchies of soliton equations. In terms of the Lax equation Lt ¼ AL LA ½A; L;
ð1Þ
many interesting integrable soliton equations with physical backgrounds, such as the KdV equation, the Burgers equations and so on [2,3] have been worked out. Tu Guizhang further developed the Lax pair method and presented a simple and efficient approach for generating integrable Hamiltonian hierarchies [4], Ma Wenxiu called it Tu scheme [5]. By taking advantage of the scheme, the celebrated hierarchies, such as AKNS hierarchy, KN hierarchy, WKI hierarchy, e 2 and obtained some interetc. were given [4–9]. Hu Xingbiao extended the Tu scheme in the frame of the loop algebra A esting results [6,7]. Guo Fukui again presented some integrable Hamiltonian systems of soliton equations with multipotential functions, in general, less than 8 functions, by changing the power times of the spectral parameters in the e 1 [10,11]. By using Guos idea, a few interesting results were also obtained in [12,13]. As far frame of loop algebra A as the multi-component integrable hierarchies of soliton equations are concerned, there have been developments in [14,15]. Ma Wenxiu and Zhou Ruguang gave the multi-component AKNS hierarchy by use of the Lax pair method [16]. In order to conveniently produce some interesting multi-component integrable hierarchies, a simple method was proposed in [17]. As its application, the multi-component system M-AKNS-KN hierarchy was generated. In this q *
This work was supported by The National Science Foundation of China (10471139). Corresponding author. E-mail address: [email protected] (X. Guo).
0960-0779/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.03.034
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
115
paper, a simple and convenient way for obtaining multi-component integrable systems is presented. Firstly, a new Lie algebra is constructed, whose commuting operation is just the same with that in [17]. Secondly, a corresponding highere M is showed. It follows that an isospectral problem is established. By employing the Tu dimensional loop algebra G scheme, a new multi- component integrable hierarchy is obtained. In addition, as we know, integrable couplings are a quite important aspect in the field of soliton theory [18]. A general method is the perturbation approach. In [19], a simple way for producing integrable couplings were once given. By employing this approach, the integrable couplings of some known integrable systems were obtained [19–22]. Therefore, in this paper, we also construct another higher-dimensional loop algebra to deduce the integrable coupling of the hierarchy (17) presented below.
2. A new Lie algebra and its corresponding loop algebra In [17], we presented the following definition: Set a = (a1, a2, . . ., aM)T, b = ( b1, b2, . . ., bM)T to be two vectors, define their product a*b as follows a b ¼ b a ¼ ða1 b1 ; . . . ; aM bM ÞT . Thus, a Lie algebra was constructed by GM ¼ fa ¼ ðaij ÞM3 ¼ ða1 ; a2 ; a3 Þg;
ð2Þ
with a commuting operation defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ; 8a;b 2 GM .
ð3Þ
A corresponding loop algebra is given by e M ¼ fakm ; a 2 GM ; m ¼ 0; 1; 2; . . . ; g G
ð4Þ
with a commuting operation expressed as ½akm ; bkn ¼ ½a; bkmþn ; 8a; b 2 GM .
ð5Þ
e M to work out multi-component integrable hierarchies. In We find that it is not easy to directly use the loop algebra G order to overcome the shortcoming, we construct another Lie algebra. 0 1 1 B1C C Definition 1. Let I M ¼ B be a matrix, and set @ ... A 1 M1 h ¼ ðI M ; 0; 0Þ; e ¼ ð0; I M ; 0Þ; f ¼ ð0; 0; I M Þ;
ð6Þ
where M is a positive integer. A commuting relation among them is defined as ½h; e ¼ ½e; h ¼ 2e; ½h; f ¼ ½f ; h ¼ 2f ; ½e; f ¼ ½f ; e ¼ h.
ð7Þ
Then {h, e, f} along 0 with 1 (7) constitutes a Lie algebra, and we denote it as GM. a1 B a2 C B C Definition 2. If a ¼ B .. C is a column vector, A = (0, . . ., 0, IM,0, . . ., 0)M·N is a M · N matrix, where IM is in the ith @ . A aM column of the matrix a. Then a commuting relation between them is defined as ð8Þ
a A ¼ A a ¼ ð0; . . . ; a I M ; 0; . . . ; 0Þ.
Let a = a1 Æ h + a2 Æ e + a3 Æ f and b = b1 Æ h + b2 Æ e + b3 Æ f, then from the loop algebra (6) and the definition (8), we have ½a; b ¼ ða2 b3 a3 b2 Þ h þ 2ða1 b2 a2 b1 Þ e þ 2ða3 b1 a1 b3 Þ f ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ; ðiÞ
ðiÞ
ðiÞ
ðiÞ
ð9Þ ðiÞ
ðiÞ
which is just the formula (3). where ai ¼ ðam1 ; am2 ; . . . ; amM ÞT ; bi ¼ ðbm1 ; bm2 ; . . . ; bmM ÞT ; i ¼ 1; 2; 3. In what follows, we shall find that the Lie algebra (6) is more convenient than GM in the aspect of deducing multi-component integrable hierarchies.
116
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
e e M as follows In terms of the Lie alegbra (6), we establish a new loop algebra G 8 hði; nÞ ¼ h k2nþi hk2nþi ; eði; nÞ ¼ ek2nþi ; f ði; nÞ ¼ f k2nþi ; > > > ( > > > 2eði þ j; m þ nÞ; i þ j < 2; > > > ½hði; mÞ; eðj; nÞ ¼ > > > 2eð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > ( > < 2f ði þ j; m þ nÞ; i þ j < 2; ½hði; mÞ; f ðj; nÞ ¼ > > 2f ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > ( > > > hði þ j; m þ nÞ; i þ j < 2; > > > ½eði; mÞ; f ðj; nÞ ¼ > > > hð0; m þ n þ 1Þ; i þ j ¼ 2; > > > : degðhði; nÞÞ ¼ degðeði; nÞÞ ¼ degðf ði; nÞÞ ¼ 2n þ i; i ¼ 0; 1; j ¼ 0; 1.
ð10Þ
3. A new multi-component integrable system e e M to work out a new multi-component integrable hierarchy, an isospectral problem Next, we use the loop algebra G is given by ux ¼ ½U ; u; kt ¼ 0; ð11Þ U ¼ ðkI M þ u5 k I M ; u1 I M þ u3 Ik M ; u2 I M þ u4 Ik M Þ; ðiÞ
ðiÞ
ðiÞ
where ui ¼ ðu1 ; u2 ; . . . ; uM ÞT ; i ¼ 1; 2; 3; 4; 5. Set X V ¼ ðað0; mÞ hð0; mÞ þ að1; mÞ hð1; mÞ þ bð0; mÞ eð0; mÞ þ bð1; mÞ eð1; mÞ þ cð0; mÞ f ð0; mÞ mP0
þ cð1; mÞ f ð1; mÞÞ; ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
here að0; mÞ ¼ ðam1 ; am2 ; . . . ; amM ÞT ; að1; mÞ ¼ ðam1 ; am2 ; . . . ; amM ÞT ; . . .From the stationary zero curvature equation V x ¼ ½U ; V ;
ð12Þ
one arrives at ax ð0; mÞ ¼ u1 cð0; mÞ u2 bð0; mÞ þ u3 cð1; mÞ u4 bð1; mÞ; ax ð1; m þ 1Þ ¼ u1 cð1; m þ 1Þ u2 bð1; m þ 1Þ þ u3 cð0; mÞ u4 bð0; mÞ; bx ð0; mÞ ¼ 2bð1; m þ 1Þ 2u1 að0; mÞ 2u3 að1; mÞ þ 2u5 bð1; mÞ; bx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2u1 að1; m þ 1Þ 2u3 að0; mÞ þ 2u5 bð0; mÞ; cx ð0; mÞ ¼ 2cð1; m þ 1Þ þ 2u2 að0; mÞ þ 2u4 að1; mÞ 2u5 cð1; mÞ; cx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ þ 2u2 að1; m þ 1Þ þ 2u4 að0; mÞ 2u5 cð0; mÞ;
ð13Þ
að0; 0Þ ¼ a ¼ ða1 ; a2 . . . ; aM ÞT ; cð0; 0Þ ¼ bð0; 0Þ ¼ að1; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ 0; a bð1; 1Þ ¼ a u1 ; cð1; 1Þ ¼ a u2 ; að1; 1Þ ¼ 0; cð0; 1Þ ¼ u2x þ a u4 ; 2 a a bð0; 1Þ ¼ u1x þ a u3 ; að0; 1Þ ¼ u1 u2 ; 2 2 where ai are constants, i = 1,2, . . ., M. Note that n X ðnÞ Vþ ¼ ðað0; mÞ hð0; n mÞ þ að1; mÞ hð1; n mÞ þ bð0; mÞ eð0; n mÞ þ bð1; mÞ eð1; n mÞ m¼0
þ cð0; mÞ f ð0; n mÞ þ cð1; mÞ f ð1; n mÞÞ; ðnÞ
2n V ðnÞ ¼k V V þ .
Then Eq. (12) can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V .
ð14Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
117
A direct calculation reads that ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ ðnÞ ðnÞ V þx þ ½U ; V þ ¼ ; 2bð1; n þ 1Þ k bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ ; 2cð1; n þ 1Þ þ k cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ þ k ¼ 2bð1; n þ 1Þ eð0; 0Þ þ 2cð1; n þ 1Þ f ð0; 0Þ þ ½bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ eð1; 1Þ þ ½cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ f ð1; 1Þ þ ½ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ hð1; 1Þ. Thus, the zero curvature equation ðnÞ
ðnÞ
U t V þx þ ½U ; V þ ¼ 0
ð15Þ
leads to the following Lax integrable system 0 1 0 1 u1 2bð1; n þ 1Þ B C B C B u2 C B C 2cð1; n þ 1Þ B C B C B C B C B C C ut ¼ B u ¼ b ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2u að1; n þ 1Þ x 1 B 3C B C B C B C B u4 C B cx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2u2 að1; n þ 1Þ C @ A @ A u5 t ax ð1; n þ 1Þ þ u1 cð1; n þ 1Þ u2 bð1; n þ 1Þ 0 0 1 1 10 0 cð0; n þ 1Þ cð0; n þ 1Þ 0 0 0 2I M B B C C CB B B bð0; n þ 1Þ C B 0 C 0 0 C 0 2I M B B C CB bð0; n þ 1Þ C B B C C CB B B B C C C 0 @ u 0 2I cð1; n þ 1Þ cð1; n þ 1Þ ¼B M 1 CB C ¼ J 1B C B B C C CB B B B 2I M C C C 0 @ 0 u2 A@ bð1; n þ 1Þ A @ bð1; n þ 1Þ A @ u2 @2 0 0 u1 12 að1; n þ 1Þ 12 að1; n þ 1Þ 0 1 bx ð0; nÞ þ 2u1 að0; nÞ þ 2u3 að1; nÞ 2u5 bð1; nÞ B C B cx ð0; nÞ 2u2 að0; nÞ 2u4 að1; nÞ þ 2u5 cð1; nÞ C B C B C C að0; nÞ þ 2u bð0; nÞ 2u ¼B 3 5 B C B C B C að0; nÞ þ 2u cð0; nÞ 2u 4 5 @ A u3 cð0; nÞ þ u4 bð0; nÞ 0 @ 2u1 @ 1 u2 2u1 @ 1 u3 2u5 2u1 @ 1 u4 2u1 @ 1 u1 B B @ 2u2 @ 1 u1 2u2 @ 1 u2 2u5 2u2 @ 1 u3 2u2 @ 1 u4 B B ¼B 2u5 2u3 @ 1 u2 2u3 @ 1 u3 2u3 @ 1 u4 2u3 @ 1 u1 B B B 2u 2u @ 1 u 2u4 @ 1 u2 2u4 @ 1 u3 2u4 @ 1 u4 4 1 @ 5 u3 1 cð0; nÞ B C B bð0; nÞ C B C B C B C; cð1; nÞ ¼ J 2B C B C B bð1; nÞ C @ A 1 að1; nÞ 2 0
u4
where J1 and J2 are all the Hamiltonian operators.
0
0
ð16Þ
u3
1
C u4 C C C 0 C C C 0 C A 0
118
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
From (13), a recurrence operator is given by L = (lij)5·5, where 1 1 1 l11 ¼ @ 2 u2 @ 1 u1 @ þ u2 @ 1 u3 u5 þ u4 @u2 @ 1 u1 ; 4 2 2 1 1 l12 ¼ u2 @ 1 u2 @ u2 @ 1 u4 þ @u2 u4 @ 1 u4 ; 2 2 @ @ l13 ¼ u5 u2 @ 1 u1 u5 þ u4 u2 @ 1 u3 ; 2 2 @ l14 ¼ u2 @ 1 u2 u5 þ u2 u4 @ 1 u4 ; 2 l15 ¼ @u4 þ2u2 @ 1 ðu1 u4 u2 u3 Þ; 1 1 l21 ¼ u1 @ 1 u1 @ þ u1 @ 1 u3 þ u1x þu3 þu1 @ @ 1 u1 ; 2 2 2 @ 1 1 l22 ¼ u1 @ 1 u2 @ u1 @ 1 u4 u5 u1x þu3 þu1 @ @ 1 u2 ; 4 2 2 1 1 l23 ¼ u1 @ u1 u5 þ u1x þu3 þu1 @ @ 1 u3 ; 2 1 1 l24 ¼ @u5 þu1 @ 1 u2 u5 u1x þu3 þu1 @ @ 1 u4 ; 2 2 l25 ¼ @u3 þ2u1 @ 1 ðu1 u4 u2 u3 Þ; @ l31 ¼ þ u2 @ 1 u1 ; l32 ¼ u2 ; @ 1 u2 ; l33 ¼ u5 þu2 @ 1 u3 ; 2 @ l34 ¼ u2 @ 1 u4 ; l35 ¼ 2u4 ; l41 ¼ u1 @ 1 u1 ; l42 ¼ u1 @ 1 u2 ; 2 l43 ¼ u1 @ 1 u3 ; l44 ¼ u5 u1 @ 1 u4 ; l45 ¼ 2u3 ; 1 1 1 1 l51 ¼ @ 1 u3 u4 @ ; l52 ¼ @ 1 u4 þ u2 @ ; 2 4 2 4 1 1 l53 ¼ @ 1 u1 u5 ; l54 ¼ @ 1 u2 u5 ; l55 ¼ @ 1 ðu1 u4 u2 u3 Þ. 2 2 And L satisfies that 0 1 0 1 cð0; nÞ cð0; n þ 1Þ B bð0; n þ 1Þ C B bð0; nÞ C B B C C B B C C B cð1; n þ 1Þ C ¼ LB cð1; nÞ C B B C C B C B C @ bð1; n þ 1Þ A @ bð1; nÞ A 1 1 að1; n þ 1Þ að1; nÞ 2 2 Hence, the system (16) can be written as 1 1 1 0 a 0 0 0 1 cð0; nÞ cð0; nÞ u1 2 u2x þ a u4 C B a B bð0; nÞ C B bð0; nÞ C B C C C B 2 u1x þ a u3 C B B B u2 C C C C B B B B C n C. C C C B B B ut ¼ B a u2 C B u3 C ¼ J 1 LB cð1; nÞ C ¼ J 2 B cð1; nÞ C ¼ J 1 L B C C C B B B B C bð1; nÞ bð1; nÞ a u1 A A A @ @ @ @ u4 A 1 1 að1; nÞ að1; nÞ u5 t 0 2 2
ð17Þ
When M > 1, the hierarchy (17) is a multi-component integrable system. A reduction case, a coupled nonlinear multi-component Schro¨dinger equation is presented by taking u3 = u4 = u5 = 0 in the hierarchy (17) as follows: u1t ¼ a2 u1xx a u1 u1 u2 ; u2t ¼ a2 u2xx þ a u1 u2 u2 .
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
119
4. The binary multi-component integrable coupling systems Next, we look for the binary multi-component integrable coupling systems of the hierarchy (17). We expand the Lie algebra (6) into the following loop algebra FM 8 > < e1 ¼ ðI M ; 0; 0; 0; 0Þ; e2 ¼ ð0; I M ; 0; 0; 0Þ; e3 ¼ ð0; 0; I M ; 0; 0Þ; e4 ¼ ð0; 0; 0; I M ; 0Þ; e5 ¼ ð0; 0; 0; 0; I M Þ; ½e1 ; e2 ¼ 2e2 ; ½e1 ; e3 ¼ 2e3 ; ½e2 ; e3 ¼ e1 ; ½e1 ; e4 ¼ e4 ; ½e1 ; e5 ¼ e5 ; ð18Þ > : ½e2 ; e4 ¼ 0; ½e3 ; e4 ¼ e5 ; ½e2 ; e5 ¼ e4 ; ½e3 ; e5 ¼ 0; ½e4 ; e5 ¼ 0. P P Let a ¼ 5i¼1 ai ei ; b ¼ 5i¼1 bi ei 2 F M ,define their commutative operation as ½a; b ¼ ða2 b3 a3 b2 ; 2a1 b2 2a2 b1 ; 2a3 b1 2a1 b3 ; a1 b4 a4 b1 þ a2 b5 a5 b2 ; a5 b1 a1 b5 þ a3 b4 a4 b3 Þ.
ð19Þ
A corresponding loop algebra Fe M is given as Fe M ¼ fakn ; a 2 F M ; n ¼ 0; 1; 2; . . .g;
ð20Þ
with a commuting operation bejing defined by ½akm ; bkn ¼ ½a; bkmþn ; 8a; b 2 F M . In terms of the Lie algebra FM, we give a concrete loop algebra Fe M as follows: 8 e ðj; nÞ ¼ ei k2nþj ; i ¼ 1; 2; 3; 4; 5; j ¼ 0; 1; > > > i > > 2ei ði þ j; m þ nÞ; i þ j < 2; > > > ½e1 ði; mÞ; e2 ðj; nÞ ¼ > > > 2e1 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > 2e 3 ði þ j; m þ nÞ; i þ j < 2; > > ½e1 ði; mÞ; e3 ðj; nÞ ¼ > > > 2e3 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e 1 ði þ j; m þ nÞ; i þ j < 2; > > > ½e2 ði; mÞ; e3 ðj; nÞ ¼ > > e > 1 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > < ½e ði; mÞ; e ðj; nÞ ¼ e4 ði þ j; m þ nÞ; i þ j < 2; 1 4 e4 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > e > 5 ði þ j; m þ nÞ; i þ j < 2; > > ½e1 ði; mÞ; e5 ðj; nÞ ¼ > > > e5 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e 4 ði þ j; m þ nÞ; i þ j < 2; > > ½e2 ði; mÞ; e5 ðj; nÞ ¼ > > > e4 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e > 5 ði þ j; m þ nÞ; i þ j < 2; > > > > ½e3 ði; mÞ; e4 ðj; nÞ ¼ e5 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > > ½e2 ði; mÞ; e4 ðj; nÞ ¼ ½e3 ði; mÞ; e5 ðj; nÞ ¼ ½e4 ði; mÞ; e5 ðj; nÞ ¼ 0; > > : degðej ði; mÞÞ ¼ 2m þ i.
ð21Þ
ð22Þ
Let Fe M ð1Þ ¼ spanfe1 ðnÞ; e2 ðnÞ; e3 ðnÞg; Fe M ð2Þ ¼ spanfe4 ðnÞ; e5 ðnÞg; then we find e e M ; ½ Fe ð1Þ; Fe M ð2Þ Fe M ð2Þ; Fe M ¼ Fe M ð1Þ Fe M ð2Þ; Fe M ð1Þw G
ð23Þ
from which the binary integrable couplings of the hierarchy (17) can be worked out. In terms of the relations (22), an isospectral problem is constructed 8 > < ux ¼ ½U ; u; kt ¼ 0; U ¼ e1 ð1; 0Þ þ u1 e2 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e2 ð1; 1Þ þ u4 e3 ð1; 1Þ þ u5 e1 ð1; 1Þ ð24Þ > : þu6 e4 ð0; 0Þ þ u7 e5 ð0; 0Þ þ u8 e4 ð1; 1Þ þ u9 e5 ð1; 1Þ; ðiÞ
ðiÞ
ðiÞ
where ui ¼ ðu1 ; u2 ; . . . ; u9 ÞT ; i ¼ 1; . . . ; 9.
120
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
Taking V ¼
1 X ðað0; mÞ e1 ð0; mÞ þ að1; mÞ e1 ð1; mÞ þ bð0; mÞ e2 ð0; mÞ þ bð1; mÞ e2 ð1; mÞÞ þ cð0; mÞ e3 ð0; mÞ m¼0
þ cð1; mÞ e3 ð1; mÞ þ dð0; mÞ e4 ð0; mÞ þ dð1; mÞ e4 ð1; mÞ þ f ð0; mÞ e5 ð0; mÞ þ f ð1; mÞ e5 ð1; mÞ. Solving the equation similar to Eq. (12) yields 8 ax ð0; mÞ ¼ u1 cð0; mÞ u2 bð0; mÞ þ u3 cð1; mÞ u4 bð1; mÞ; > > > > > > ax ð1; m þ 1Þ ¼ u1 cð1; m þ 1Þ u2 bð1; m þ 1Þ þ u3 cð0; mÞ u4 bð0; mÞ; > > > > > > > bx ð0; mÞ ¼ 2bð1; m þ 1Þ 2u1 að0; mÞ 2u3 að1; mÞ þ 2u5 bð1; mÞ; > > > > > > > bx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2u1 að1; m þ 1Þ 2u3 að0; mÞ þ 2u5 bð0; mÞ; > > > > > > cx ð0; mÞ ¼ 2cð1; m þ 1Þ þ 2u2 að0; mÞ þ 2u4 að1; mÞ 2u5 cð1; mÞ; > > > > > > > cx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ þ 2u2 að1; m þ 1Þ þ 2u4 að0; mÞ 2u5 cð0; mÞ; > > > > > > > d x ð0; mÞ ¼ dð1; m þ 1Þ þ u1 f ð0; mÞ þ u3 f ð1; mÞ þ u5 dð1; mÞ u6 að0; mÞ u7 bð0; mÞ > > > > > > > u8 að1; mÞ u9 bð1; mÞ; > > > > > < d x ð1; m þ 1Þ ¼ dð0; m þ 1Þ þ u1 f ð1; m þ 1Þ þ u3 f ð0; mÞ þ u5 dð0; mÞ u6 að1; m þ 1Þ > u7 að1; m þ 1Þ u7 bð1; m þ 1Þ u8 að0; mÞ u9 ð0; mÞ; > > > > > > > fx ð0; mÞ ¼ f ð1; m þ 1Þ þ u2 dð0; mÞ þ u4 dð1; mÞ u5 f ð1; mÞ u6 cð0; mÞ þ u7 að0; mÞ > > > > > > > u8 cð1; mÞ þ u9 að1; mÞ; > > > > > > fx ð1; m þ 1Þ ¼ f ð0; m þ 1Þ þ u2 dð1; m þ 1Þ þ u4 dð0; mÞ u5 f ð0; mÞ u6 cð1; m þ 1Þ > > > > > > > > > þu7 að1; m þ 1Þ u8 cð0; mÞ þ u9 að0; mÞ; > > > > > cð0; 0Þ ¼ bð0; 0Þ ¼ að1; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ dð0; 0Þ ¼ f ð0; 0Þ ¼ dð1; 0Þ ¼ f ð1; 0Þ ¼ 0; > > > > > > > að0; 0Þ ¼ a ¼ ða1 ; ; aM ÞT ; að1; 1Þ ¼ 0; bð1; 1Þ ¼ a u1 ; cð1; 1Þ ¼ a u2 ; > > > > > a a > > > dð1; 1Þ ¼ a u6 ; f ð1; 1Þ ¼ a u7 ; að0; 1Þ ¼ 2 u1 u2 ; bð0; 1Þ ¼ 2 u1x þ a u3 ; > > : cð0; 1Þ ¼ a2 u2x þ a u4 ; f ð0; 1Þ ¼ a u9 a u7x ; dð0; 1Þ ¼ a u6x .
ð25Þ
Denoting ðnÞ
Vþ ¼
n X ðað0; mÞ e1 ð0; n mÞ þ að1; mÞ e1 ð1; n mÞ þ bð0; mÞ e2 ð0; n mÞ m¼0
þ bð1; mÞ e2 ð1; n mÞ þ cð0; mÞ e3 ð0; n mÞ þ cð1; mÞ e3 ð1; n mÞ þ dð0; mÞ e4 ð0; n mÞ þ dð1; mÞ e4 ð1; n mÞ þ f ð0; mÞ e5 ð0; n mÞ þ f ð1; mÞ e5 ð1; n mÞÞ; V
ðnÞ
ðnÞ
¼ k2n V V þ ;
then a direct calculation gives ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2bð1; n þ 1Þ eð0; 0Þ þ 2cð1; n þ 1Þ f ð0; 0Þ þ ½bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ eð1; 1Þ þ ½cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ f ð1; 1Þ þ ½ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ hð1; 1Þ þ ½d x ð1; n þ 1Þ dð0; n þ 1Þ u2 f ð1; n þ 1Þ þ u6 að1; n þ 1Þ þ u7 bð1; n þ 1Þ e4 ð1; 1Þ dð1; n þ 1Þ e4 ð0; 0Þ þ f ð1; n þ 1Þ e5 ð0; 0Þ þ ½fx ð1; n þ 1Þ þ f ð0; n þ 1Þ u2 dð1; n þ 1Þ þ u6 cð1; n þ 1Þ u7 að1; n þ 1Þ e5 ð1; 1Þ. Thus, the zero curvature equation ðnÞ
ðnÞ
U t V þx þ ½U ; V þ ¼ 0;
ð26Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
leads to
0
u1
1
0
2bð1; n þ 1Þ
121
1
B C B C B u2 C B C 2cð1; n þ 1Þ B C B C B C B C Bu C B C bx ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2u1 að1; n þ 1Þ B 3C B C B C B C Bu C B C cx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2u2 að1; n þ 1Þ B 4C B C B C B C B C B C ax ð1; n þ 1Þ þ u1 ð1; n þ 1Þ u2 bð1; n þ 1Þ ut ¼ B u5 C ¼ B C B C B C B C B C B u6 C B C dð1; n þ 1Þ B C B C B C B C B u7 C B C f ð1; n þ 1Þ B C B C B C B C B u8 C B d x ð1; n þ 1Þ þ dð0; n þ 1Þ þ u2 f ð1; n þ 1Þ u6 að1; n þ 1Þ u7 bð1; n þ 1Þ C @ A @ A fx ð1; n þ 1Þ f ð0; n þ 1Þ þ u2 dð1; n þ 1Þ u6 cð1; n þ 1Þ þ u7 að1; n þ 1Þ u9 t 0 0 1 1 10 0 0 0 0 0 cð0; n þ 1Þ cð0; n þ 1Þ 0 0 0 2I M B B C C CB B B bð0; n þ 1Þ C B 0 C 0 2I M 0 0 0 0 0 0 C B B C CB bð0; n þ 1Þ C B B C C CB B cð1; n þ 1Þ C B 0 CB cð1; n þ 1Þ C 2I 0 @ u 0 0 0 0 M 1 B B C C CB B B C C CB B bð1; n þ 1Þ C B 2I CB bð1; n þ 1Þ C 0 @ 0 u 0 0 0 0 B B C C CB M 2 B B C C CB B C e B1 C CB 1 @ 0 u1 u2 2 0 0 0 0 CB 2 að1; n þ 1Þ C ¼ J 1 B 2 að1; n þ 1Þ C ¼B 0 B B C C CB B B C C CB B dð0; n þ 1Þ C B 0 0 0 0 0 IM 0 0 0 CB dð0; n þ 1Þ C B B C C CB B B C C CB B f ð0; n þ 1Þ C B 0 CB f ð0; n þ 1Þ C 0 0 0 0 0 I 0 0 M B B C C CB B B C C CB B dð1; n þ 1Þ C B 0 CB dð1; n þ 1Þ C 0 0 u 2u I 0 @ u 7 6 M 2 @ @ A A A@ f ð1; n þ 1Þ f ð1; n þ 1Þ 0 0 u6 0 2u7 0 I M u2 @ 0 1 bx ð0; nÞ þ 2u1 að0; nÞ þ 2u3 að1; nÞ 2u5 bð1; nÞ B C B C cx ð0; nÞ 2u2 að0; nÞ að0; nÞ 2u4 að1; nÞ þ 2u5 cð1; nÞ B C B C B C 2u3 að1; nÞ 2u5 bð0; nÞ B C B C B C 2u4 að0; nÞ þ 2u5 cð0; nÞ B C B C B C u3 cð0; nÞ þ u4 bð0; nÞ ¼B C B C B C B d x ð0; nÞ u2 f ð0; nÞ u3 f ð1; nÞ u5 dð1; nÞ þ u6 að0; nÞ þ u7 bð0; nÞ þ u8 að1; nÞ þ u9 bð1; nÞ C B C B C B fx ð0; nÞ u2 dð0; nÞ u4 dð1; nÞ þ u5 f ð1; nÞ þ u6 cð0; nÞ u7 að0; nÞ þ u8 cð1; nÞ u9 að1; nÞ C B C B C B C u f ð0; nÞ u dð0; nÞ þ u að0; nÞ þ u bð0; nÞ 3 5 8 9 @ A u4 dð0; nÞ þ u5 f ð0; nÞ þ u8 cð0; nÞ þ u9 að0; nÞ 0 1 cð0; nÞ cð0; nÞ B B C C B bð0; nÞ C B bð0; nÞ C B B C C B B C C B cð1; nÞ C B cð1; nÞ C B B C C B B C C B bð1; nÞ C B bð1; nÞ C B B C C B B C C B1 C e B1 C ¼ ðM N ÞB 2 að1; nÞ C ¼ J 2 B 2 að1; nÞ C B B C C B B C C B dð0; nÞ C B dð0; nÞ C B B C C B B C C B f ð0; nÞ C B f ð0; nÞ C B B C C B B C C B dð1; nÞ C B dð1; nÞ C @ @ A A f ð1; nÞ f ð1; nÞ 0
1
ð27Þ
122
with
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
0
2u1 @ 1 u1
@ 2u1 @ 1 u2
u8 u9 @ 1 u1
u9 @ 1 u2
B B @ 2u2 @ 1 u1 2u2 @ 1 u2 B B 2u3 @ 1 u1 2u5 2u3 @ 1 u2 B B B 2u5 2u4 @ 1 u1 2u4 @ 1 u2 B B M ¼B u3 u4 B B u6 @ 1 u1 u7 u6 @ 1 u2 B B B u6 u7 @ 1 u1 u7 @ 1 u2 B B u8 @ 1 u1 u9 u8 @ 1 u2 @ 0
u3 0 0 2u5 2u1 @ 1 u4 B 2u2 @ 1 u4 u4 0 0 B B B 2u3 @ 1 u4 0 0 0 B B B 2u4 @ 1 u4 0 0 0 B N ¼B 0 0 0 0 B B 1 B u9 u6 @ u4 2u8 @ u2 B 1 B u @ u 2u u @ 7 4 9 2 B B @ u8 @ 1 u4 0 u5 u3 u9 @ 1 u4 0 u4 u5
2u1 @ 1 u3
1
C 2u5 2u2 @ 1 u3 C C C 2u3 @ 1 u3 C C 1 2u4 @ u3 C C C C 0 C 1 C u6 @ u3 C C 1 u8 u7 @ u3 C C C u8 @ 1 u3 A 1 u9 @ u3 0 0 0 0 0
0 0 0 0 0
1
C C C C C C C C C C C u5 u3 C C u4 u5 C C C 0 0 A 0 0
In terms of (25), a recurrence operator is given by L ¼ ðA
B Þ;
where 0
M1
N1
P1
u2 @ 1 u2 u5 þð@2 u2 u4 Þ@ 1 u4
B B B M2 N2 P2 P3 B B 1 1 1 @ B þ u2 @ u1 u2 @ u2 u5 þu2 @ u3 u2 @ 1 u4 B 2 B B @ B u1 @ 1 u2 u1 @ 1 u3 u5 u1 @ 1 u4 u1 @ 1 u1 2 B B 1 1 A¼B 12 @ 1 u1 u5 12 @ 1 u2 u5 B @ ð12 u3 14 u1 @Þ @ ð 14 u2 @ 12 u4 Þ B B B M3 N3 P4 Q1 B B B M4 N4 P5 Q2 B B B 1 1 1 u6 @ u1 u7 u6 @ u2 u6 @ u3 u9 u6 @ 1 u4 B @ u7 @ 1 u1 u6 0 u7 @ 1 u3 u8 u7 @ 1 u4 0 1 0 0 0 0 B C B C 0 0 0 0 B C B C B C B C 0 0 0 0 B C B C B C 0 0 0 0 B C B C B C 0 0 0 0 B¼B C B C B 2 C B @ u2 u2 u5 @u2 þu2 @ u3 @u5 u2 u4 @u3 þu2 u5 C B C B C B @u þu @ þ u @ 2 u u u @u u u @u u u C 2 2 4 2 2 5 4 2 5 5 2 3 B C B C B C B C @ u1 u5 u3 @ A u2 @ u4 u5
W1
1
C C C C C C 2u4 C C C C 2u3 C C 1 @ ðu1 u4 u2 u3 Þ C C C C C R1 C C C R2 C C C 2u8 C A 0 W2
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
123
with W 1 ¼ @u4 þ2u2 @ 1 ðu1 u4 u2 u3 Þ; W 2 ¼ @u3 þ2u1 @ 1 ðu1 u4 u2 u3 Þ; 1 M 3 ¼ @u6 @ 1 u1 þu2 u6 u6 @ 1 u1 @ þ u6 @ 1 u3 þu8 @ 1 u1 ; 2 1 @ 1 u1 @ 1 u2 N 3 ¼ @ðu7 u6 @ u2 Þ u6 @u2 @ u6 @ 1 u4 þu7 2 2 u8 @ 1 u2 þu9 ; P 4 ¼ @u6 @ 1 u3 þu2 u8 u6 @ 1 u1 u5 þu8 @ 1 u3 ; Q1 ¼ @ðu9 u6 @ 1 u4 Þ þ u2 u7 @ 1 u4 þu6 @ 1 u2 u5 u5 u7 u1 u7 @ 1 u4 u8 @ 1 u4 ; R1 ¼ 2@u8 2u2 u9 þ2u6 @ 1 ðu1 u4 u2 u3 Þ þ2u3 u7 ; @ 1 u2 @ 1 u1 u7 @ 1 u1 @ M 4 ¼ @u6 @u7 @ 1 u1 þu6 @ 1 u1 þu6 2 2 þ u7 @ 1 u3 u8 þu9 @ 1 u1 ; 1 N 4 ¼ u2 u7 u7 @ 1 u2 @ u7 @ 1 u4 u9 @ 1 u2 ; 2 P 5 ¼ @u8 @u7 @ 1 u3 þu2 u6 @ 1 u3 þu6 u5 u2 u6 @ 1 u3 u7 @ 1 u1 u5 þ u9 @ 1 u3 ; Q2 ¼ @u7 @ 1 u4 þu2 u9 þu7 @ 1 u2 u5 u9 @ 1 u4 ; R2 ¼ @u9 þu2 u8 u4 u6 þu7 @ 1 ðu1 u4 u2 u3 Þ; while M1, N1, P1, M2, N2, P2, P3 are the same with those of Lin (17). Therefore, the system (27) can be written as 0 0 1 1 cð0; 1Þ u1 B bð0; 1Þ C Bu C B B 2C C B B C C B cð1; 1Þ C B u3 C B B C C B B C C B bð1; 1Þ C B u4 C B1 B C C n C C e B ut ¼ B B u5 C ¼ J 1 L B 2 að1; 1Þ C B B C C B dð0; 1Þ C B u6 C B B C C B f ð0; 1Þ C B u7 C B B C C B B C C @ dð1; 1Þ A @ u8 A f ð1; 1Þ u9 t 0 1 cð0; 1Þ B bð0; 1Þ C B C B C B cð1; 1Þ C B C B C B bð1; 1Þ C B1 C n1 C ¼e J 2L B B 2 að1; 1Þ C B C B dð0; 1Þ C B C B f ð0; 1Þ C B C B C @ dð1; 1Þ A f ð1; 1Þ
ð28Þ
ð29Þ
The systems (28) and (29) are the binary multi-component integrable couplings of the multi-component hierarchy (17) according to the definition of the integrable couplings. Acknowledgement The second author (Y. Zhang) is grateful to Professor M.Wadati for his enthusiastic guidance and help.
124
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. PA: SIAM; 1991. Magri F. J Math Phys 1978;19:1156. Li Y. Soliton and Integrable System. Shanghai: Shanghai Scientific and Technological Education Publishing House; 1999. Tu G. J Math Phys 1989;30(2):330. Ma W. Chin Ann Math Sin 1992;12A(1):115. Hu X. J Phys A 1994;27:2497. Hu X. J Phys A 1997;30:619. Fan E. J Math Phys 2000;41(11):7769. Fan E. Physica A 2002;301:105. Guo F. J Syst Sci Math Sci 2000;22(1):36. Zhang Y. Phys Lett A 2003;317:280. Guo F, Zhang Y. Acta Phys Sin 2002;51(5):951. Zhang Y, Zhang H. J Math Phys 2002;43(1):466. Tsuchida T, Wadati M. Phys Lett A 1999;257:53. Tsuchida T, Wadati M. J Phys Soc Jpn 1999;69:2241. Ma W, Zhou R. Chin Ann Math 2002;23B(3):373. Guo F, Zhang Y. J Math Phys 2003;44(12):5793. Ma W, Fuchssteiner B. Chaos, Solitons & Fractals 1996;7:1227. Guo F. Acta Math Phys Sin 1999;19(5):507. Zhang Y et al. Chaos, Solitons & Fractals 2004;19:563. Zhang Y et al. Phys Lett A 2002;299:543. Zhang Y. Chaos, Solitons & Fractals 2003;18:855.
A new Lie algebra, a corresponding multi-component integrable hierarchy and an integrable coupling q Yufeng Zhang a, Xiurong Guo b
b,*
, Honwah Tam
c
a Mathematical School, Liaoning Normal University, Dalian 116029, PR China Science School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China c Department of Computer Science, Hong Kong Baptist University, Hong Kong, China
Accepted 30 March 2005
Abstract A new simple loop algebra is constructed, which is devote to establishing an isospectral problem. By making use of Tu scheme, a new multi-component integrable hierarchy is obtained. Again via expanding the loop algebra above, another higher-dimensional loop algebra is presented. It follows that the binary integrable coupling systems are given. This method proposed in this paper can be used to other soliton hierarchies. 2005 Published by Elsevier Ltd.
1. Introduction The Lax pair method [1] is a general technique for generating single component integrable hierarchies of soliton equations. In terms of the Lax equation Lt ¼ AL LA ½A; L;
ð1Þ
many interesting integrable soliton equations with physical backgrounds, such as the KdV equation, the Burgers equations and so on [2,3] have been worked out. Tu Guizhang further developed the Lax pair method and presented a simple and efficient approach for generating integrable Hamiltonian hierarchies [4], Ma Wenxiu called it Tu scheme [5]. By taking advantage of the scheme, the celebrated hierarchies, such as AKNS hierarchy, KN hierarchy, WKI hierarchy, e 2 and obtained some interetc. were given [4–9]. Hu Xingbiao extended the Tu scheme in the frame of the loop algebra A esting results [6,7]. Guo Fukui again presented some integrable Hamiltonian systems of soliton equations with multipotential functions, in general, less than 8 functions, by changing the power times of the spectral parameters in the e 1 [10,11]. By using Guos idea, a few interesting results were also obtained in [12,13]. As far frame of loop algebra A as the multi-component integrable hierarchies of soliton equations are concerned, there have been developments in [14,15]. Ma Wenxiu and Zhou Ruguang gave the multi-component AKNS hierarchy by use of the Lax pair method [16]. In order to conveniently produce some interesting multi-component integrable hierarchies, a simple method was proposed in [17]. As its application, the multi-component system M-AKNS-KN hierarchy was generated. In this q *
This work was supported by The National Science Foundation of China (10471139). Corresponding author. E-mail address: [email protected] (X. Guo).
0960-0779/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.03.034
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
115
paper, a simple and convenient way for obtaining multi-component integrable systems is presented. Firstly, a new Lie algebra is constructed, whose commuting operation is just the same with that in [17]. Secondly, a corresponding highere M is showed. It follows that an isospectral problem is established. By employing the Tu dimensional loop algebra G scheme, a new multi- component integrable hierarchy is obtained. In addition, as we know, integrable couplings are a quite important aspect in the field of soliton theory [18]. A general method is the perturbation approach. In [19], a simple way for producing integrable couplings were once given. By employing this approach, the integrable couplings of some known integrable systems were obtained [19–22]. Therefore, in this paper, we also construct another higher-dimensional loop algebra to deduce the integrable coupling of the hierarchy (17) presented below.
2. A new Lie algebra and its corresponding loop algebra In [17], we presented the following definition: Set a = (a1, a2, . . ., aM)T, b = ( b1, b2, . . ., bM)T to be two vectors, define their product a*b as follows a b ¼ b a ¼ ða1 b1 ; . . . ; aM bM ÞT . Thus, a Lie algebra was constructed by GM ¼ fa ¼ ðaij ÞM3 ¼ ða1 ; a2 ; a3 Þg;
ð2Þ
with a commuting operation defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ; 8a;b 2 GM .
ð3Þ
A corresponding loop algebra is given by e M ¼ fakm ; a 2 GM ; m ¼ 0; 1; 2; . . . ; g G
ð4Þ
with a commuting operation expressed as ½akm ; bkn ¼ ½a; bkmþn ; 8a; b 2 GM .
ð5Þ
e M to work out multi-component integrable hierarchies. In We find that it is not easy to directly use the loop algebra G order to overcome the shortcoming, we construct another Lie algebra. 0 1 1 B1C C Definition 1. Let I M ¼ B be a matrix, and set @ ... A 1 M1 h ¼ ðI M ; 0; 0Þ; e ¼ ð0; I M ; 0Þ; f ¼ ð0; 0; I M Þ;
ð6Þ
where M is a positive integer. A commuting relation among them is defined as ½h; e ¼ ½e; h ¼ 2e; ½h; f ¼ ½f ; h ¼ 2f ; ½e; f ¼ ½f ; e ¼ h.
ð7Þ
Then {h, e, f} along 0 with 1 (7) constitutes a Lie algebra, and we denote it as GM. a1 B a2 C B C Definition 2. If a ¼ B .. C is a column vector, A = (0, . . ., 0, IM,0, . . ., 0)M·N is a M · N matrix, where IM is in the ith @ . A aM column of the matrix a. Then a commuting relation between them is defined as ð8Þ
a A ¼ A a ¼ ð0; . . . ; a I M ; 0; . . . ; 0Þ.
Let a = a1 Æ h + a2 Æ e + a3 Æ f and b = b1 Æ h + b2 Æ e + b3 Æ f, then from the loop algebra (6) and the definition (8), we have ½a; b ¼ ða2 b3 a3 b2 Þ h þ 2ða1 b2 a2 b1 Þ e þ 2ða3 b1 a1 b3 Þ f ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ; ðiÞ
ðiÞ
ðiÞ
ðiÞ
ð9Þ ðiÞ
ðiÞ
which is just the formula (3). where ai ¼ ðam1 ; am2 ; . . . ; amM ÞT ; bi ¼ ðbm1 ; bm2 ; . . . ; bmM ÞT ; i ¼ 1; 2; 3. In what follows, we shall find that the Lie algebra (6) is more convenient than GM in the aspect of deducing multi-component integrable hierarchies.
116
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
e e M as follows In terms of the Lie alegbra (6), we establish a new loop algebra G 8 hði; nÞ ¼ h k2nþi hk2nþi ; eði; nÞ ¼ ek2nþi ; f ði; nÞ ¼ f k2nþi ; > > > ( > > > 2eði þ j; m þ nÞ; i þ j < 2; > > > ½hði; mÞ; eðj; nÞ ¼ > > > 2eð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > ( > < 2f ði þ j; m þ nÞ; i þ j < 2; ½hði; mÞ; f ðj; nÞ ¼ > > 2f ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > ( > > > hði þ j; m þ nÞ; i þ j < 2; > > > ½eði; mÞ; f ðj; nÞ ¼ > > > hð0; m þ n þ 1Þ; i þ j ¼ 2; > > > : degðhði; nÞÞ ¼ degðeði; nÞÞ ¼ degðf ði; nÞÞ ¼ 2n þ i; i ¼ 0; 1; j ¼ 0; 1.
ð10Þ
3. A new multi-component integrable system e e M to work out a new multi-component integrable hierarchy, an isospectral problem Next, we use the loop algebra G is given by ux ¼ ½U ; u; kt ¼ 0; ð11Þ U ¼ ðkI M þ u5 k I M ; u1 I M þ u3 Ik M ; u2 I M þ u4 Ik M Þ; ðiÞ
ðiÞ
ðiÞ
where ui ¼ ðu1 ; u2 ; . . . ; uM ÞT ; i ¼ 1; 2; 3; 4; 5. Set X V ¼ ðað0; mÞ hð0; mÞ þ að1; mÞ hð1; mÞ þ bð0; mÞ eð0; mÞ þ bð1; mÞ eð1; mÞ þ cð0; mÞ f ð0; mÞ mP0
þ cð1; mÞ f ð1; mÞÞ; ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
here að0; mÞ ¼ ðam1 ; am2 ; . . . ; amM ÞT ; að1; mÞ ¼ ðam1 ; am2 ; . . . ; amM ÞT ; . . .From the stationary zero curvature equation V x ¼ ½U ; V ;
ð12Þ
one arrives at ax ð0; mÞ ¼ u1 cð0; mÞ u2 bð0; mÞ þ u3 cð1; mÞ u4 bð1; mÞ; ax ð1; m þ 1Þ ¼ u1 cð1; m þ 1Þ u2 bð1; m þ 1Þ þ u3 cð0; mÞ u4 bð0; mÞ; bx ð0; mÞ ¼ 2bð1; m þ 1Þ 2u1 að0; mÞ 2u3 að1; mÞ þ 2u5 bð1; mÞ; bx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2u1 að1; m þ 1Þ 2u3 að0; mÞ þ 2u5 bð0; mÞ; cx ð0; mÞ ¼ 2cð1; m þ 1Þ þ 2u2 að0; mÞ þ 2u4 að1; mÞ 2u5 cð1; mÞ; cx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ þ 2u2 að1; m þ 1Þ þ 2u4 að0; mÞ 2u5 cð0; mÞ;
ð13Þ
að0; 0Þ ¼ a ¼ ða1 ; a2 . . . ; aM ÞT ; cð0; 0Þ ¼ bð0; 0Þ ¼ að1; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ 0; a bð1; 1Þ ¼ a u1 ; cð1; 1Þ ¼ a u2 ; að1; 1Þ ¼ 0; cð0; 1Þ ¼ u2x þ a u4 ; 2 a a bð0; 1Þ ¼ u1x þ a u3 ; að0; 1Þ ¼ u1 u2 ; 2 2 where ai are constants, i = 1,2, . . ., M. Note that n X ðnÞ Vþ ¼ ðað0; mÞ hð0; n mÞ þ að1; mÞ hð1; n mÞ þ bð0; mÞ eð0; n mÞ þ bð1; mÞ eð1; n mÞ m¼0
þ cð0; mÞ f ð0; n mÞ þ cð1; mÞ f ð1; n mÞÞ; ðnÞ
2n V ðnÞ ¼k V V þ .
Then Eq. (12) can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V .
ð14Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
117
A direct calculation reads that ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ ðnÞ ðnÞ V þx þ ½U ; V þ ¼ ; 2bð1; n þ 1Þ k bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ ; 2cð1; n þ 1Þ þ k cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ þ k ¼ 2bð1; n þ 1Þ eð0; 0Þ þ 2cð1; n þ 1Þ f ð0; 0Þ þ ½bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ eð1; 1Þ þ ½cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ f ð1; 1Þ þ ½ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ hð1; 1Þ. Thus, the zero curvature equation ðnÞ
ðnÞ
U t V þx þ ½U ; V þ ¼ 0
ð15Þ
leads to the following Lax integrable system 0 1 0 1 u1 2bð1; n þ 1Þ B C B C B u2 C B C 2cð1; n þ 1Þ B C B C B C B C B C C ut ¼ B u ¼ b ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2u að1; n þ 1Þ x 1 B 3C B C B C B C B u4 C B cx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2u2 að1; n þ 1Þ C @ A @ A u5 t ax ð1; n þ 1Þ þ u1 cð1; n þ 1Þ u2 bð1; n þ 1Þ 0 0 1 1 10 0 cð0; n þ 1Þ cð0; n þ 1Þ 0 0 0 2I M B B C C CB B B bð0; n þ 1Þ C B 0 C 0 0 C 0 2I M B B C CB bð0; n þ 1Þ C B B C C CB B B B C C C 0 @ u 0 2I cð1; n þ 1Þ cð1; n þ 1Þ ¼B M 1 CB C ¼ J 1B C B B C C CB B B B 2I M C C C 0 @ 0 u2 A@ bð1; n þ 1Þ A @ bð1; n þ 1Þ A @ u2 @2 0 0 u1 12 að1; n þ 1Þ 12 að1; n þ 1Þ 0 1 bx ð0; nÞ þ 2u1 að0; nÞ þ 2u3 að1; nÞ 2u5 bð1; nÞ B C B cx ð0; nÞ 2u2 að0; nÞ 2u4 að1; nÞ þ 2u5 cð1; nÞ C B C B C C að0; nÞ þ 2u bð0; nÞ 2u ¼B 3 5 B C B C B C að0; nÞ þ 2u cð0; nÞ 2u 4 5 @ A u3 cð0; nÞ þ u4 bð0; nÞ 0 @ 2u1 @ 1 u2 2u1 @ 1 u3 2u5 2u1 @ 1 u4 2u1 @ 1 u1 B B @ 2u2 @ 1 u1 2u2 @ 1 u2 2u5 2u2 @ 1 u3 2u2 @ 1 u4 B B ¼B 2u5 2u3 @ 1 u2 2u3 @ 1 u3 2u3 @ 1 u4 2u3 @ 1 u1 B B B 2u 2u @ 1 u 2u4 @ 1 u2 2u4 @ 1 u3 2u4 @ 1 u4 4 1 @ 5 u3 1 cð0; nÞ B C B bð0; nÞ C B C B C B C; cð1; nÞ ¼ J 2B C B C B bð1; nÞ C @ A 1 að1; nÞ 2 0
u4
where J1 and J2 are all the Hamiltonian operators.
0
0
ð16Þ
u3
1
C u4 C C C 0 C C C 0 C A 0
118
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
From (13), a recurrence operator is given by L = (lij)5·5, where 1 1 1 l11 ¼ @ 2 u2 @ 1 u1 @ þ u2 @ 1 u3 u5 þ u4 @u2 @ 1 u1 ; 4 2 2 1 1 l12 ¼ u2 @ 1 u2 @ u2 @ 1 u4 þ @u2 u4 @ 1 u4 ; 2 2 @ @ l13 ¼ u5 u2 @ 1 u1 u5 þ u4 u2 @ 1 u3 ; 2 2 @ l14 ¼ u2 @ 1 u2 u5 þ u2 u4 @ 1 u4 ; 2 l15 ¼ @u4 þ2u2 @ 1 ðu1 u4 u2 u3 Þ; 1 1 l21 ¼ u1 @ 1 u1 @ þ u1 @ 1 u3 þ u1x þu3 þu1 @ @ 1 u1 ; 2 2 2 @ 1 1 l22 ¼ u1 @ 1 u2 @ u1 @ 1 u4 u5 u1x þu3 þu1 @ @ 1 u2 ; 4 2 2 1 1 l23 ¼ u1 @ u1 u5 þ u1x þu3 þu1 @ @ 1 u3 ; 2 1 1 l24 ¼ @u5 þu1 @ 1 u2 u5 u1x þu3 þu1 @ @ 1 u4 ; 2 2 l25 ¼ @u3 þ2u1 @ 1 ðu1 u4 u2 u3 Þ; @ l31 ¼ þ u2 @ 1 u1 ; l32 ¼ u2 ; @ 1 u2 ; l33 ¼ u5 þu2 @ 1 u3 ; 2 @ l34 ¼ u2 @ 1 u4 ; l35 ¼ 2u4 ; l41 ¼ u1 @ 1 u1 ; l42 ¼ u1 @ 1 u2 ; 2 l43 ¼ u1 @ 1 u3 ; l44 ¼ u5 u1 @ 1 u4 ; l45 ¼ 2u3 ; 1 1 1 1 l51 ¼ @ 1 u3 u4 @ ; l52 ¼ @ 1 u4 þ u2 @ ; 2 4 2 4 1 1 l53 ¼ @ 1 u1 u5 ; l54 ¼ @ 1 u2 u5 ; l55 ¼ @ 1 ðu1 u4 u2 u3 Þ. 2 2 And L satisfies that 0 1 0 1 cð0; nÞ cð0; n þ 1Þ B bð0; n þ 1Þ C B bð0; nÞ C B B C C B B C C B cð1; n þ 1Þ C ¼ LB cð1; nÞ C B B C C B C B C @ bð1; n þ 1Þ A @ bð1; nÞ A 1 1 að1; n þ 1Þ að1; nÞ 2 2 Hence, the system (16) can be written as 1 1 1 0 a 0 0 0 1 cð0; nÞ cð0; nÞ u1 2 u2x þ a u4 C B a B bð0; nÞ C B bð0; nÞ C B C C C B 2 u1x þ a u3 C B B B u2 C C C C B B B B C n C. C C C B B B ut ¼ B a u2 C B u3 C ¼ J 1 LB cð1; nÞ C ¼ J 2 B cð1; nÞ C ¼ J 1 L B C C C B B B B C bð1; nÞ bð1; nÞ a u1 A A A @ @ @ @ u4 A 1 1 að1; nÞ að1; nÞ u5 t 0 2 2
ð17Þ
When M > 1, the hierarchy (17) is a multi-component integrable system. A reduction case, a coupled nonlinear multi-component Schro¨dinger equation is presented by taking u3 = u4 = u5 = 0 in the hierarchy (17) as follows: u1t ¼ a2 u1xx a u1 u1 u2 ; u2t ¼ a2 u2xx þ a u1 u2 u2 .
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
119
4. The binary multi-component integrable coupling systems Next, we look for the binary multi-component integrable coupling systems of the hierarchy (17). We expand the Lie algebra (6) into the following loop algebra FM 8 > < e1 ¼ ðI M ; 0; 0; 0; 0Þ; e2 ¼ ð0; I M ; 0; 0; 0Þ; e3 ¼ ð0; 0; I M ; 0; 0Þ; e4 ¼ ð0; 0; 0; I M ; 0Þ; e5 ¼ ð0; 0; 0; 0; I M Þ; ½e1 ; e2 ¼ 2e2 ; ½e1 ; e3 ¼ 2e3 ; ½e2 ; e3 ¼ e1 ; ½e1 ; e4 ¼ e4 ; ½e1 ; e5 ¼ e5 ; ð18Þ > : ½e2 ; e4 ¼ 0; ½e3 ; e4 ¼ e5 ; ½e2 ; e5 ¼ e4 ; ½e3 ; e5 ¼ 0; ½e4 ; e5 ¼ 0. P P Let a ¼ 5i¼1 ai ei ; b ¼ 5i¼1 bi ei 2 F M ,define their commutative operation as ½a; b ¼ ða2 b3 a3 b2 ; 2a1 b2 2a2 b1 ; 2a3 b1 2a1 b3 ; a1 b4 a4 b1 þ a2 b5 a5 b2 ; a5 b1 a1 b5 þ a3 b4 a4 b3 Þ.
ð19Þ
A corresponding loop algebra Fe M is given as Fe M ¼ fakn ; a 2 F M ; n ¼ 0; 1; 2; . . .g;
ð20Þ
with a commuting operation bejing defined by ½akm ; bkn ¼ ½a; bkmþn ; 8a; b 2 F M . In terms of the Lie algebra FM, we give a concrete loop algebra Fe M as follows: 8 e ðj; nÞ ¼ ei k2nþj ; i ¼ 1; 2; 3; 4; 5; j ¼ 0; 1; > > > i > > 2ei ði þ j; m þ nÞ; i þ j < 2; > > > ½e1 ði; mÞ; e2 ðj; nÞ ¼ > > > 2e1 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > 2e 3 ði þ j; m þ nÞ; i þ j < 2; > > ½e1 ði; mÞ; e3 ðj; nÞ ¼ > > > 2e3 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e 1 ði þ j; m þ nÞ; i þ j < 2; > > > ½e2 ði; mÞ; e3 ðj; nÞ ¼ > > e > 1 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > < ½e ði; mÞ; e ðj; nÞ ¼ e4 ði þ j; m þ nÞ; i þ j < 2; 1 4 e4 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > e > 5 ði þ j; m þ nÞ; i þ j < 2; > > ½e1 ði; mÞ; e5 ðj; nÞ ¼ > > > e5 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e 4 ði þ j; m þ nÞ; i þ j < 2; > > ½e2 ði; mÞ; e5 ðj; nÞ ¼ > > > e4 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > e > 5 ði þ j; m þ nÞ; i þ j < 2; > > > > ½e3 ði; mÞ; e4 ðj; nÞ ¼ e5 ð0; m þ n þ 1Þ; i þ j ¼ 2; > > > > > > ½e2 ði; mÞ; e4 ðj; nÞ ¼ ½e3 ði; mÞ; e5 ðj; nÞ ¼ ½e4 ði; mÞ; e5 ðj; nÞ ¼ 0; > > : degðej ði; mÞÞ ¼ 2m þ i.
ð21Þ
ð22Þ
Let Fe M ð1Þ ¼ spanfe1 ðnÞ; e2 ðnÞ; e3 ðnÞg; Fe M ð2Þ ¼ spanfe4 ðnÞ; e5 ðnÞg; then we find e e M ; ½ Fe ð1Þ; Fe M ð2Þ Fe M ð2Þ; Fe M ¼ Fe M ð1Þ Fe M ð2Þ; Fe M ð1Þw G
ð23Þ
from which the binary integrable couplings of the hierarchy (17) can be worked out. In terms of the relations (22), an isospectral problem is constructed 8 > < ux ¼ ½U ; u; kt ¼ 0; U ¼ e1 ð1; 0Þ þ u1 e2 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e2 ð1; 1Þ þ u4 e3 ð1; 1Þ þ u5 e1 ð1; 1Þ ð24Þ > : þu6 e4 ð0; 0Þ þ u7 e5 ð0; 0Þ þ u8 e4 ð1; 1Þ þ u9 e5 ð1; 1Þ; ðiÞ
ðiÞ
ðiÞ
where ui ¼ ðu1 ; u2 ; . . . ; u9 ÞT ; i ¼ 1; . . . ; 9.
120
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
Taking V ¼
1 X ðað0; mÞ e1 ð0; mÞ þ að1; mÞ e1 ð1; mÞ þ bð0; mÞ e2 ð0; mÞ þ bð1; mÞ e2 ð1; mÞÞ þ cð0; mÞ e3 ð0; mÞ m¼0
þ cð1; mÞ e3 ð1; mÞ þ dð0; mÞ e4 ð0; mÞ þ dð1; mÞ e4 ð1; mÞ þ f ð0; mÞ e5 ð0; mÞ þ f ð1; mÞ e5 ð1; mÞ. Solving the equation similar to Eq. (12) yields 8 ax ð0; mÞ ¼ u1 cð0; mÞ u2 bð0; mÞ þ u3 cð1; mÞ u4 bð1; mÞ; > > > > > > ax ð1; m þ 1Þ ¼ u1 cð1; m þ 1Þ u2 bð1; m þ 1Þ þ u3 cð0; mÞ u4 bð0; mÞ; > > > > > > > bx ð0; mÞ ¼ 2bð1; m þ 1Þ 2u1 að0; mÞ 2u3 að1; mÞ þ 2u5 bð1; mÞ; > > > > > > > bx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2u1 að1; m þ 1Þ 2u3 að0; mÞ þ 2u5 bð0; mÞ; > > > > > > cx ð0; mÞ ¼ 2cð1; m þ 1Þ þ 2u2 að0; mÞ þ 2u4 að1; mÞ 2u5 cð1; mÞ; > > > > > > > cx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ þ 2u2 að1; m þ 1Þ þ 2u4 að0; mÞ 2u5 cð0; mÞ; > > > > > > > d x ð0; mÞ ¼ dð1; m þ 1Þ þ u1 f ð0; mÞ þ u3 f ð1; mÞ þ u5 dð1; mÞ u6 að0; mÞ u7 bð0; mÞ > > > > > > > u8 að1; mÞ u9 bð1; mÞ; > > > > > < d x ð1; m þ 1Þ ¼ dð0; m þ 1Þ þ u1 f ð1; m þ 1Þ þ u3 f ð0; mÞ þ u5 dð0; mÞ u6 að1; m þ 1Þ > u7 að1; m þ 1Þ u7 bð1; m þ 1Þ u8 að0; mÞ u9 ð0; mÞ; > > > > > > > fx ð0; mÞ ¼ f ð1; m þ 1Þ þ u2 dð0; mÞ þ u4 dð1; mÞ u5 f ð1; mÞ u6 cð0; mÞ þ u7 að0; mÞ > > > > > > > u8 cð1; mÞ þ u9 að1; mÞ; > > > > > > fx ð1; m þ 1Þ ¼ f ð0; m þ 1Þ þ u2 dð1; m þ 1Þ þ u4 dð0; mÞ u5 f ð0; mÞ u6 cð1; m þ 1Þ > > > > > > > > > þu7 að1; m þ 1Þ u8 cð0; mÞ þ u9 að0; mÞ; > > > > > cð0; 0Þ ¼ bð0; 0Þ ¼ að1; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ dð0; 0Þ ¼ f ð0; 0Þ ¼ dð1; 0Þ ¼ f ð1; 0Þ ¼ 0; > > > > > > > að0; 0Þ ¼ a ¼ ða1 ; ; aM ÞT ; að1; 1Þ ¼ 0; bð1; 1Þ ¼ a u1 ; cð1; 1Þ ¼ a u2 ; > > > > > a a > > > dð1; 1Þ ¼ a u6 ; f ð1; 1Þ ¼ a u7 ; að0; 1Þ ¼ 2 u1 u2 ; bð0; 1Þ ¼ 2 u1x þ a u3 ; > > : cð0; 1Þ ¼ a2 u2x þ a u4 ; f ð0; 1Þ ¼ a u9 a u7x ; dð0; 1Þ ¼ a u6x .
ð25Þ
Denoting ðnÞ
Vþ ¼
n X ðað0; mÞ e1 ð0; n mÞ þ að1; mÞ e1 ð1; n mÞ þ bð0; mÞ e2 ð0; n mÞ m¼0
þ bð1; mÞ e2 ð1; n mÞ þ cð0; mÞ e3 ð0; n mÞ þ cð1; mÞ e3 ð1; n mÞ þ dð0; mÞ e4 ð0; n mÞ þ dð1; mÞ e4 ð1; n mÞ þ f ð0; mÞ e5 ð0; n mÞ þ f ð1; mÞ e5 ð1; n mÞÞ; V
ðnÞ
ðnÞ
¼ k2n V V þ ;
then a direct calculation gives ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2bð1; n þ 1Þ eð0; 0Þ þ 2cð1; n þ 1Þ f ð0; 0Þ þ ½bx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2u1 að1; n þ 1Þ eð1; 1Þ þ ½cx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2u2 að1; n þ 1Þ f ð1; 1Þ þ ½ax ð1; n þ 1Þ u1 cð1; n þ 1Þ þ u2 bð1; n þ 1Þ hð1; 1Þ þ ½d x ð1; n þ 1Þ dð0; n þ 1Þ u2 f ð1; n þ 1Þ þ u6 að1; n þ 1Þ þ u7 bð1; n þ 1Þ e4 ð1; 1Þ dð1; n þ 1Þ e4 ð0; 0Þ þ f ð1; n þ 1Þ e5 ð0; 0Þ þ ½fx ð1; n þ 1Þ þ f ð0; n þ 1Þ u2 dð1; n þ 1Þ þ u6 cð1; n þ 1Þ u7 að1; n þ 1Þ e5 ð1; 1Þ. Thus, the zero curvature equation ðnÞ
ðnÞ
U t V þx þ ½U ; V þ ¼ 0;
ð26Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
leads to
0
u1
1
0
2bð1; n þ 1Þ
121
1
B C B C B u2 C B C 2cð1; n þ 1Þ B C B C B C B C Bu C B C bx ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2u1 að1; n þ 1Þ B 3C B C B C B C Bu C B C cx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2u2 að1; n þ 1Þ B 4C B C B C B C B C B C ax ð1; n þ 1Þ þ u1 ð1; n þ 1Þ u2 bð1; n þ 1Þ ut ¼ B u5 C ¼ B C B C B C B C B C B u6 C B C dð1; n þ 1Þ B C B C B C B C B u7 C B C f ð1; n þ 1Þ B C B C B C B C B u8 C B d x ð1; n þ 1Þ þ dð0; n þ 1Þ þ u2 f ð1; n þ 1Þ u6 að1; n þ 1Þ u7 bð1; n þ 1Þ C @ A @ A fx ð1; n þ 1Þ f ð0; n þ 1Þ þ u2 dð1; n þ 1Þ u6 cð1; n þ 1Þ þ u7 að1; n þ 1Þ u9 t 0 0 1 1 10 0 0 0 0 0 cð0; n þ 1Þ cð0; n þ 1Þ 0 0 0 2I M B B C C CB B B bð0; n þ 1Þ C B 0 C 0 2I M 0 0 0 0 0 0 C B B C CB bð0; n þ 1Þ C B B C C CB B cð1; n þ 1Þ C B 0 CB cð1; n þ 1Þ C 2I 0 @ u 0 0 0 0 M 1 B B C C CB B B C C CB B bð1; n þ 1Þ C B 2I CB bð1; n þ 1Þ C 0 @ 0 u 0 0 0 0 B B C C CB M 2 B B C C CB B C e B1 C CB 1 @ 0 u1 u2 2 0 0 0 0 CB 2 að1; n þ 1Þ C ¼ J 1 B 2 að1; n þ 1Þ C ¼B 0 B B C C CB B B C C CB B dð0; n þ 1Þ C B 0 0 0 0 0 IM 0 0 0 CB dð0; n þ 1Þ C B B C C CB B B C C CB B f ð0; n þ 1Þ C B 0 CB f ð0; n þ 1Þ C 0 0 0 0 0 I 0 0 M B B C C CB B B C C CB B dð1; n þ 1Þ C B 0 CB dð1; n þ 1Þ C 0 0 u 2u I 0 @ u 7 6 M 2 @ @ A A A@ f ð1; n þ 1Þ f ð1; n þ 1Þ 0 0 u6 0 2u7 0 I M u2 @ 0 1 bx ð0; nÞ þ 2u1 að0; nÞ þ 2u3 að1; nÞ 2u5 bð1; nÞ B C B C cx ð0; nÞ 2u2 að0; nÞ að0; nÞ 2u4 að1; nÞ þ 2u5 cð1; nÞ B C B C B C 2u3 að1; nÞ 2u5 bð0; nÞ B C B C B C 2u4 að0; nÞ þ 2u5 cð0; nÞ B C B C B C u3 cð0; nÞ þ u4 bð0; nÞ ¼B C B C B C B d x ð0; nÞ u2 f ð0; nÞ u3 f ð1; nÞ u5 dð1; nÞ þ u6 að0; nÞ þ u7 bð0; nÞ þ u8 að1; nÞ þ u9 bð1; nÞ C B C B C B fx ð0; nÞ u2 dð0; nÞ u4 dð1; nÞ þ u5 f ð1; nÞ þ u6 cð0; nÞ u7 að0; nÞ þ u8 cð1; nÞ u9 að1; nÞ C B C B C B C u f ð0; nÞ u dð0; nÞ þ u að0; nÞ þ u bð0; nÞ 3 5 8 9 @ A u4 dð0; nÞ þ u5 f ð0; nÞ þ u8 cð0; nÞ þ u9 að0; nÞ 0 1 cð0; nÞ cð0; nÞ B B C C B bð0; nÞ C B bð0; nÞ C B B C C B B C C B cð1; nÞ C B cð1; nÞ C B B C C B B C C B bð1; nÞ C B bð1; nÞ C B B C C B B C C B1 C e B1 C ¼ ðM N ÞB 2 að1; nÞ C ¼ J 2 B 2 að1; nÞ C B B C C B B C C B dð0; nÞ C B dð0; nÞ C B B C C B B C C B f ð0; nÞ C B f ð0; nÞ C B B C C B B C C B dð1; nÞ C B dð1; nÞ C @ @ A A f ð1; nÞ f ð1; nÞ 0
1
ð27Þ
122
with
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
0
2u1 @ 1 u1
@ 2u1 @ 1 u2
u8 u9 @ 1 u1
u9 @ 1 u2
B B @ 2u2 @ 1 u1 2u2 @ 1 u2 B B 2u3 @ 1 u1 2u5 2u3 @ 1 u2 B B B 2u5 2u4 @ 1 u1 2u4 @ 1 u2 B B M ¼B u3 u4 B B u6 @ 1 u1 u7 u6 @ 1 u2 B B B u6 u7 @ 1 u1 u7 @ 1 u2 B B u8 @ 1 u1 u9 u8 @ 1 u2 @ 0
u3 0 0 2u5 2u1 @ 1 u4 B 2u2 @ 1 u4 u4 0 0 B B B 2u3 @ 1 u4 0 0 0 B B B 2u4 @ 1 u4 0 0 0 B N ¼B 0 0 0 0 B B 1 B u9 u6 @ u4 2u8 @ u2 B 1 B u @ u 2u u @ 7 4 9 2 B B @ u8 @ 1 u4 0 u5 u3 u9 @ 1 u4 0 u4 u5
2u1 @ 1 u3
1
C 2u5 2u2 @ 1 u3 C C C 2u3 @ 1 u3 C C 1 2u4 @ u3 C C C C 0 C 1 C u6 @ u3 C C 1 u8 u7 @ u3 C C C u8 @ 1 u3 A 1 u9 @ u3 0 0 0 0 0
0 0 0 0 0
1
C C C C C C C C C C C u5 u3 C C u4 u5 C C C 0 0 A 0 0
In terms of (25), a recurrence operator is given by L ¼ ðA
B Þ;
where 0
M1
N1
P1
u2 @ 1 u2 u5 þð@2 u2 u4 Þ@ 1 u4
B B B M2 N2 P2 P3 B B 1 1 1 @ B þ u2 @ u1 u2 @ u2 u5 þu2 @ u3 u2 @ 1 u4 B 2 B B @ B u1 @ 1 u2 u1 @ 1 u3 u5 u1 @ 1 u4 u1 @ 1 u1 2 B B 1 1 A¼B 12 @ 1 u1 u5 12 @ 1 u2 u5 B @ ð12 u3 14 u1 @Þ @ ð 14 u2 @ 12 u4 Þ B B B M3 N3 P4 Q1 B B B M4 N4 P5 Q2 B B B 1 1 1 u6 @ u1 u7 u6 @ u2 u6 @ u3 u9 u6 @ 1 u4 B @ u7 @ 1 u1 u6 0 u7 @ 1 u3 u8 u7 @ 1 u4 0 1 0 0 0 0 B C B C 0 0 0 0 B C B C B C B C 0 0 0 0 B C B C B C 0 0 0 0 B C B C B C 0 0 0 0 B¼B C B C B 2 C B @ u2 u2 u5 @u2 þu2 @ u3 @u5 u2 u4 @u3 þu2 u5 C B C B C B @u þu @ þ u @ 2 u u u @u u u @u u u C 2 2 4 2 2 5 4 2 5 5 2 3 B C B C B C B C @ u1 u5 u3 @ A u2 @ u4 u5
W1
1
C C C C C C 2u4 C C C C 2u3 C C 1 @ ðu1 u4 u2 u3 Þ C C C C C R1 C C C R2 C C C 2u8 C A 0 W2
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
123
with W 1 ¼ @u4 þ2u2 @ 1 ðu1 u4 u2 u3 Þ; W 2 ¼ @u3 þ2u1 @ 1 ðu1 u4 u2 u3 Þ; 1 M 3 ¼ @u6 @ 1 u1 þu2 u6 u6 @ 1 u1 @ þ u6 @ 1 u3 þu8 @ 1 u1 ; 2 1 @ 1 u1 @ 1 u2 N 3 ¼ @ðu7 u6 @ u2 Þ u6 @u2 @ u6 @ 1 u4 þu7 2 2 u8 @ 1 u2 þu9 ; P 4 ¼ @u6 @ 1 u3 þu2 u8 u6 @ 1 u1 u5 þu8 @ 1 u3 ; Q1 ¼ @ðu9 u6 @ 1 u4 Þ þ u2 u7 @ 1 u4 þu6 @ 1 u2 u5 u5 u7 u1 u7 @ 1 u4 u8 @ 1 u4 ; R1 ¼ 2@u8 2u2 u9 þ2u6 @ 1 ðu1 u4 u2 u3 Þ þ2u3 u7 ; @ 1 u2 @ 1 u1 u7 @ 1 u1 @ M 4 ¼ @u6 @u7 @ 1 u1 þu6 @ 1 u1 þu6 2 2 þ u7 @ 1 u3 u8 þu9 @ 1 u1 ; 1 N 4 ¼ u2 u7 u7 @ 1 u2 @ u7 @ 1 u4 u9 @ 1 u2 ; 2 P 5 ¼ @u8 @u7 @ 1 u3 þu2 u6 @ 1 u3 þu6 u5 u2 u6 @ 1 u3 u7 @ 1 u1 u5 þ u9 @ 1 u3 ; Q2 ¼ @u7 @ 1 u4 þu2 u9 þu7 @ 1 u2 u5 u9 @ 1 u4 ; R2 ¼ @u9 þu2 u8 u4 u6 þu7 @ 1 ðu1 u4 u2 u3 Þ; while M1, N1, P1, M2, N2, P2, P3 are the same with those of Lin (17). Therefore, the system (27) can be written as 0 0 1 1 cð0; 1Þ u1 B bð0; 1Þ C Bu C B B 2C C B B C C B cð1; 1Þ C B u3 C B B C C B B C C B bð1; 1Þ C B u4 C B1 B C C n C C e B ut ¼ B B u5 C ¼ J 1 L B 2 að1; 1Þ C B B C C B dð0; 1Þ C B u6 C B B C C B f ð0; 1Þ C B u7 C B B C C B B C C @ dð1; 1Þ A @ u8 A f ð1; 1Þ u9 t 0 1 cð0; 1Þ B bð0; 1Þ C B C B C B cð1; 1Þ C B C B C B bð1; 1Þ C B1 C n1 C ¼e J 2L B B 2 að1; 1Þ C B C B dð0; 1Þ C B C B f ð0; 1Þ C B C B C @ dð1; 1Þ A f ð1; 1Þ
ð28Þ
ð29Þ
The systems (28) and (29) are the binary multi-component integrable couplings of the multi-component hierarchy (17) according to the definition of the integrable couplings. Acknowledgement The second author (Y. Zhang) is grateful to Professor M.Wadati for his enthusiastic guidance and help.
124
Y. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 114–124
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. PA: SIAM; 1991. Magri F. J Math Phys 1978;19:1156. Li Y. Soliton and Integrable System. Shanghai: Shanghai Scientific and Technological Education Publishing House; 1999. Tu G. J Math Phys 1989;30(2):330. Ma W. Chin Ann Math Sin 1992;12A(1):115. Hu X. J Phys A 1994;27:2497. Hu X. J Phys A 1997;30:619. Fan E. J Math Phys 2000;41(11):7769. Fan E. Physica A 2002;301:105. Guo F. J Syst Sci Math Sci 2000;22(1):36. Zhang Y. Phys Lett A 2003;317:280. Guo F, Zhang Y. Acta Phys Sin 2002;51(5):951. Zhang Y, Zhang H. J Math Phys 2002;43(1):466. Tsuchida T, Wadati M. Phys Lett A 1999;257:53. Tsuchida T, Wadati M. J Phys Soc Jpn 1999;69:2241. Ma W, Zhou R. Chin Ann Math 2002;23B(3):373. Guo F, Zhang Y. J Math Phys 2003;44(12):5793. Ma W, Fuchssteiner B. Chaos, Solitons & Fractals 1996;7:1227. Guo F. Acta Math Phys Sin 1999;19(5):507. Zhang Y et al. Chaos, Solitons & Fractals 2004;19:563. Zhang Y et al. Phys Lett A 2002;299:543. Zhang Y. Chaos, Solitons & Fractals 2003;18:855.