- Email: [email protected]
A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 39 (2009) 473–478 www.elsevier.com/locate/chaos
A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure Hui Chang *, Yufeng Zhang Information School, Shandong University of Science and Technology, Qingdao 266510, China Accepted 2 January 2007
Abstract A new multi-component matrix loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component integrable couplings of the NLS-MKdV hierarchy is obtained, then the bi-Hamiltonian structure of the above system is given. 2007 Elsevier Ltd. All rights reserved.
1. Introduction It is an important and interesting topic to search for new Hamiltonian hierarchies of soliton equations and their integrable coupling in soliton theory [1–4]. Tu once proposed a simple and efficient method to obtain integrable Hamiltonian hierarchy of soliton equations with infinite dimensions in Refs. [1,5]. In recent years, Ones used Tu scheme to have obtained many interesting results [6–12]. As far as the multi-component integrable hierarchies, there have been developments in Ref. [13–16]. In order to obtain some new integrable hierarchy, we firstly construct a new multi-component e 2 to work out the integrable coumatrix lie algebra G2 and its corresponding multi-component matrix loop algebra G plings of the NLS-MKdV hierarchy. And then the bi-Hamiltonian structure of the above system is presented.
2. A new loop algebra and its application Definition 1. [17]. If a ¼ ða1 ; a2 ; . . . ; aM ÞT ; b ¼ ðb1 ; b2 ; . . . ; bM ÞT are two column vectors, their vector product a b is defined as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM ÞT ;
*
Corresponding author. E-mail address: [email protected] (H. Chang).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.102
474
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
where M stands for an arbitrary positive integer. It follows from Definition 1 that set a ¼ ða1 ; a2 ; . . . ; aM Þ; b ¼ ðb1 ; b2 ; . . . ; bM Þ; we have their product a b as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM Þ: Definition 2. Let a ¼ ða1 ; a2 ; . . . ; aM ÞT ; a ¼ ða1 ; a2 ; ; aM Þ; define a kind of product a and a as follows: 0 1 a1 Ba C B 2 C C aa¼B B .. Cða1 ; a2 ; . . . ; aM Þ ¼ diagðai ai Þ; @. A aM then, we construct a new multi-component matrix Lie algebra G2 by use of Definitions 1 and 2: 8 > > 1 1 0 0 > > > M 0 M M 0 M > > > C C B B > > 2I M 0 A; e1 ¼ @ 0 2I M 0 A; e2 ¼ @ 0 > > > > > M 0 M M 0 M > < 1 1 0 0 0 0 0 I 1M 0 I 1M > C C B > > e3 ¼ B 0 I M 1 A; e4 ¼ @ I M1 0 I M1 A; @ I M 1 > > > > > 0 I 1M 0 0 I 1M 0 > > > > > ½e1 ; e2 ¼ 0; ½e1 ; e3 ¼ 2ðM þ 1Þe4 ; ½e1 ; e4 ¼ 2ðM þ 1Þe3 ; > > > : ½e2 ; e3 ¼ 2e4 ; ½e2 ; e4 ¼ 2e3 ; ½e3 ; e4 ¼ 2e1 : 0 1 1 B1C C IM is M · M unit matrix, I 1M ¼ ð1; 1; . . . ; 1Þ1M ; I M 1 ¼ B @ ... A .
ð1Þ
1 M1 Definition 3. Set a ¼ ða1 ; a2 ; . . . ; aM ÞT , c is a scalar, we define the following operation relations: 1 1 0 0 cM 0 cM 0 aT I 1M 0 C C B B 0 A; ae3 ¼ @ a I M1 ce1 ¼ @ 0 2cI M 0 a I M 1 A; cM 0 cM 0 aT I 1M 0 1 0 1 0 cM 0 cM 0 0 aT I 1M C B C B ce2 ¼ @ 0 2cI M 0 A; ae4 ¼ @ a I M1 0 a I M1 A: T cM 0 cM 0 a I 1M 0 Therefore a loop algebra is presented as 8 > ek ðj; nÞ ¼ ek k2nþj ; k ¼ 1; 2; 3; 4; j ¼ 0; 1; > > > > > ½e1 ði; mÞ; e2 ðj; nÞ ¼ 0; > > > > > ði þ j; m þ nÞ; > > ½e1 ði; mÞ; e3 ðj; nÞ ¼ 2ðM þ 1Þe4 > > > ð0; m þ n þ 1Þ; 2ðM þ 1Þe 4 > > > > > ði þ j; m þ nÞ; 2ðM þ 1Þe 3 > > ½e1 ði; mÞ; e4 ðj; nÞ ¼ > > > 2ðM þ 1Þe 3 ð0; m þ n þ 1Þ; < 2e4 ði þ j; m þ nÞ; > ½e2 ði; mÞ; e3 ði; nÞ ¼ > > > 2e4 ð0; m þ n þ 1Þ; > > > > > 2e 3 ði þ j; m þ nÞ; > > ½e2 ði; mÞ; e4 ði; nÞ ¼ > > > 2e 3 ð0; m þ n þ 1Þ; > > > > 2e1 ði þ j; m þ nÞ; > > > ½e3 ði; mÞ; e4 ði; nÞ ¼ > > > 2e1 ð0; m þ n þ 1Þ; > > : degðek ðj; nÞÞ ¼ 2n þ j; k ¼ 1; 2; 3; 4; i ¼ 0; 1; j ¼ 0; 1:
ð2Þ
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
475
Considering an isospectral problem as follows: ux ¼ U u;
kt ¼ 0;
U ¼ e2 ð1; 0Þ þ u1 e3 ð0; 0Þ þ u2 e4 ð0; 0Þ þ u3 e3 ð1; 1Þ þ u4 e4 ð1; 1Þ þ u5 e1 ð1; 1Þ; 1 0 u1;i B u2;i C C B C ui ¼ B B .. C ; i ¼ 1; 2; 3; 4; u5 is a smooth function: @ . A uM ;i
M1
Taking
! 1 X X ðaði; mÞe2 ði; mÞ þ bði; mÞe3 ði; mÞ þ cði; mÞe4 ði; mÞÞ ; V ¼ mP0
i¼0
0
ðiÞ
bm1
1
B ðiÞ C Bb C B m2 C bði; mÞ ¼ B . C ; C B. A @. ðiÞ bmM M 1
ðiÞ
1
B ðiÞ Bc B m2 cði; mÞ ¼ B . B. @.
C C C C C A
0
cm1
ðiÞ
cmM
;
M1
a(i, m) is a scalar smooth function, i = 0,1. Solving the stationary zero curvature equation, V x ¼ ½U ; V ;
ð3Þ
gives rise to 8 ax ð0; mÞ ¼ 2uT1 cð0; mÞ þ 2uT2 bð0; mÞ 2uT3 cð1; mÞ þ 2uT4 bð1; mÞ; > > > > > ax ð1; m þ 1Þ ¼ 2uT1 cð1; m þ 1Þ þ 2uT2 bð1; m þ 1Þ 2uT3 cð0; mÞ þ 2uT4 bð0; mÞ; > > > > > bx ð0; mÞ ¼ 2cð1; m þ 1Þ 2ðM þ 1Þu2 að0; mÞ 2ðM þ 1Þu4 að1; mÞ þ 2u5 ðM þ 1Þcð1; mÞ; > > > > > > bx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ 2ðM þ 1Þu2 að1; m þ 1Þ 2ðM þ 1Þu4 að0; mÞ þ 2u5 ðM þ 1Þcð0; mÞ; > > > > > > > cx ð0; mÞ ¼ 2bð1; m þ 1Þ 2ðM þ 1Þu1 að0; mÞ 2ðM þ 1Þu3 að1; mÞ þ 2u5 ðM þ 1Þbð1; mÞ; > > > cx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2ðM þ 1Þu1 að1; m þ 1Þ 2ðM þ 1Þu3 að0; mÞ þ 2u5 ðM þ 1Þbð0; mÞ; > > > T > > < að0; 0Þ ¼ a ¼ const;bð0; 0Þ ¼ cð0; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ ð0; 0; . . . ; 0Þ1M ; að1; 0Þ ¼ 0; bð1; 1Þ ¼ aðM þ 1Þu1 ; cð1; 1Þ ¼ aðM þ 1Þu2 ; að1; 1Þ ¼ 0; > > > > bð0; 1Þ ¼ ðMþ1Þa u2x þ ðM þ 1Þau3 ; cð0; 1Þ ¼ ðMþ1Þa u1x þ ðM þ 1Þau4 ; > > 2 2 > > 1 T > 1 T T > að0; 1Þ ¼ ðM þ 1Þ ao ðu u u u Þ þ 2ao ðu2 u3 uT1 u4 Þ 2ao1 ðuT3 u2 uT4 u1 Þ ; > 2x 1x 2 1 > > > > bð1; 2Þ ¼ ðMþ1Þ au ðM þ 1Þ2 u ao1 ðuT u uT u Þ þ 2ao1 ðuT u uT u Þ > > 1xx 1 2 2x 1 1x 2 3 1 4 4 > > > 2 1 T > T > 2ao ðu3 u2 u4 u1 Þ ðM þ 1Þ au5 u1 ; > > > > > cð1; 2Þ ¼ ðMþ1Þ au ðM þ 1Þ2 u o1 ½ao1 ðuT u uT u Þ þ 2ao1 ðuT u uT u Þ > 2xx 2 > 2 2x 1 1x 2 3 1 4 4 > > : 2ao1 ðuT3 u2 uT4 u1 Þ ðM þ 1Þ2 au5 u2 :
ð4Þ
Denoting ðnÞ
Vþ ¼
n X
að0; mÞe1 ð0; n mÞ þ bð0; mÞe3 ð0; n mÞ þ cð0; mÞe4 ð0; n mÞ þ að1; mÞe1 ð1; n mÞ
m¼0
þbð1; mÞe3 ð1; n mÞ þ cð1; mÞe4 ð1; n mÞ ; ðnÞ
2n V ðnÞ ¼ k V Vþ :
ð5Þ
Eq. (3). can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
ð6Þ
476
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
A direct calculation reads: ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2cð1; n þ 1Þe3 ð0; 0Þ bð1; n þ 1Þe4 ð0; 0Þ þ ½bx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2ðM þ 1Þu2 að1; n þ 1Þe3 ð1; 1Þ þ ½cx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2ðM þ 1Þu1 að1; n þ 1Þe4 ð1; 1Þ þ ½ax ð1; n þ 1Þ þ 2uT1 cð1; n þ 1Þ 2bð1; n þ 1ÞuT2 e1 ð1; 1Þ: ðnÞ
Taking V ðnÞ ¼ V þ , then the zero curvature equation ðnÞ ¼0 U t V ðnÞ x þ ½U ; V
admits the following integrable system: 1 0 0 1 0 0 0 0 2cð1; n þ 1Þ u1 C B0 B C B 2 0 2bð1; n þ 1Þ C B B u2 C B M C B B C B 2 o C C B B u t ¼B B u3 C ¼ B bx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2ðM þ 1Þu2 að1; n þ 1Þ C ¼ B 0 M M C B2 B C B 0 0 @ u4 A @ cx ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2ðM þ 1Þu1 að1; n þ 1Þ A @ M 2 T 0 0 u ax ð1; n þ 1Þ 2uT1 cð1; n þ 1Þ þ 2uT2 bð1; n þ 1Þ u5 t M 2 1 1 0 0 2cð1; n þ 1Þ Mbð0; n þ 1Þ C C B B 2bð1; n þ 1Þ Mcð0; n þ 1Þ C C B B C C B B T C C B B B Mbð1; n þ 1Þ C ¼ J 1 Gnþ1 ¼ B 2ðM þ 1Þu4 að0; nÞ 2u5 ðM þ 1Þcð0; nÞ C C C B B T Mcð1; n þ 1Þ A @ @ 2ðM þ 1Þu3 að0; nÞ 2u5 ðM þ 1Þbð0; nÞ A 0
0 B2 BM B B ¼B 0 B B0 @ 0
ðM 2 þ MÞað1; n þ 1Þ 0
0
0
0
0
0
0
2ðMþ1Þ u5 M
0
2ðMþ1Þ u5 M M2 uT4
M2 uT3
From (4), we have 0 L11 B B L21 B Gnþ1 ¼ B B 1 B @ 0 L51
0
L12
L13
L14
L22 0 1 L52
L23 0 0 L53
L24 0 0 L54
u3
0 0 o M 2 M
uT1
0 0
1
C C C M2 u2 C C M2 u1 C A M 2oþM
10
M2
0
2uT3 cð0; nÞ 2uT4 bð0; nÞ 1 0 Mbð1; n þ 1Þ CB 0 CB Mcð1; n þ 1Þ C C CB C CB 2 C ¼ J 2F n: u Mbð0; nÞ C 4 M C B CB C C 2 Mcð0; nÞ A @ u A 3 M ðM 2 þ MÞað0; nÞ 0
M2
10
Mbð1; n þ 1Þ
ð7Þ
1
CB C u4 CB Mcð1; n þ 1Þ C CB C C ¼ LF n ; F n ¼ LGn ; B Mbð0; nÞ 0 C CB C CB C 0 A@ Mcð0; nÞ A 2 ðM þ MÞað0; nÞ 0
Here o L12 ¼ I M þ 2ðM þ 1Þu1 o1 uT1 ; 2 L13 ¼ 2ðM þ 1Þu1 o1 uT4 ðM þ 1Þu5 ; L14 ¼ 2ðM þ 1Þu1 o1 uT3 ; o L21 ¼ I M 2ðM þ 1Þu2 o1 uT2 ; L22 ¼ 2ðM þ 1Þu2 o1 uT1 ; 2 L23 ¼ 2ðM þ 1Þu2 o1 uT4 ; L24 ¼ 2ðM þ 1Þu2 o1 uT3 ðM þ 1Þu5 I M ; L11 ¼ 2ðM þ 1Þu1 o1 uT2 ;
L51 ¼ 2ðM þ 1Þo1 uT2 ;
L52 ¼ 2ðM þ 1Þo1 uT1 ;
1Þo1 uT4 ;
L54 ¼ 2ðM þ 1Þo1 uT3 :
L53 ¼ 2ðM þ
Therefore, the system (6) can be written as 1 0 1 0 Mbð0; 1Þ u1 C B C B Mcð0; 1Þ C B u2 C B C B C B 2 2n B C C: B Mbð1; 1Þ ut ¼ B u3 C ¼ J 1 LF n ¼ J 2 F n ¼ J 2 LGn ¼ J 1 L Gn ¼ J 1 L B C C B C B Mcð1; 1Þ A @ u4 A @ 2 ðM þ MÞað1; 1Þ u5 t
ð8Þ
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
477
A direct calculation reads M M X X oU oU ð0Þ ð1Þ ð0Þ ð1Þ V; ¼4 ¼ 4 ðbmk þ kbmk Þ; V; ðcmk þ kcmk Þ; ou1 ou 2 k¼0 k¼0
M M X X oU 1 ð0Þ oU 1 ð0Þ ð1Þ ð1Þ ¼4 V; ¼ 4 k ¼ 0 bmk þ bmk ; cmk þ cmk ; V; ou3 k ou4 k k¼0 oU 4M V; ¼ að0Þ þ kað1Þ ; ou5 k oU MðM þ 1Þ MðM þ 1Þ að0Þu5 þ 4kMað1Þ 4 V; ¼ 4að0ÞM 4 að1Þu5 ok k k2 2 þ ðbð0ÞT þ cðoÞT Þðu3 u4 Þ þ 2kðbð1ÞT þ cð1ÞT Þðu1 u2 Þ þ 2ðbð0ÞT þ cðoÞT Þðu1 u2 Þ k M 2 X ð0Þ ð1Þ ð0Þ ð1Þ þ 2ðbð1ÞT þ cð1ÞT Þðu3 u4 Þ þ 2 ðbmk þ kbmk cmk kcmk Þðuk3 uk4 Þ; k k¼1 where
X
að0Þ ¼
að0; mÞk2m ;
að1Þ ¼
mP0
X
að1; mÞk2m . . . :
mP0
Substituting the above formulate into the trace identity yields 0D E1 oU V ; ou 1 C B EC BD B V ; oU C C B B D ou2 E C
C B d oU c o c B oU C V; ¼k k B V ; ou3 C: du ok ok B D EC B oU C C B V ; ou 4 C B EA @D oU V ; ou5 Comparison of the coefficients of k2n3 the both side of the above equations gives d ðMðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þ2ðu4 u3 Þ þ 2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu4 u3 Þ du M X ð1Þ ð1Þ 2M ðbmk þ cmk Þðuk3 uk4 Þ ¼ ðc 2n 2ÞGnþ1 ; k¼1
comparison of the coefficients of k2n2 the both side of the above equations leads to d ð2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu1 u2 Þ þ 2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þðu1 u2 Þ du M X ð0Þ ð0Þ 2M ðbmk cmk Þðuk3 uk4 Þ þ 4M 2 að1; n þ 2Þ 4M 2 ðM þ 1Það1; nþÞu5 þ 4M 2 að0; n þ 1Þ k¼1 2
4M ðM þ 1Það0; nÞu5 Þ ¼ ðc 2n 1ÞF n : Taking n ¼ 0, gives c = 0, therefore, dH ð1; nÞ ; du 1 where H ð1; nÞ ¼ 2nþ2 ð2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þ2ðu4 u3 Þ 2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu4 u3 Þþ P ð1Þ ð1Þ 2M M k¼1 ðbmk þ cmk Þðuk3 uk4 Þ; Gnþ1 ¼
Fn ¼
dH ð2; nÞ ; du
1 ð2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu1 u2 Þ 2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þðu1 u2 Þ where H ð2; nÞ ¼ 2nþ1 PM ð0Þ ð0Þ þ2M k¼1 ðbmk cmk Þðuk3 uk4 Þ 4M 2 að1; n þ 2Þ þ 4M 2 ðM þ 1Það1; nþÞu5 4M 2 að0; n þ 1Þ þ 4M 2 ðM þ 1Það0; nÞu5 Þ:
478
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
Thus the bi-Hamiltonian structure of the system is given by 0 1 u1 B C B u2 C B C dH ð1; nÞ dH ð2; nÞ C ¼ J2 : ut ¼ B B u3 C ¼ J 1 du du B C @ u4 A u5 t It is easy to verify that J 1 L ¼ L J 1 ¼ J 2 . Therefore, the system (7) is Liouville integrable. Taking u3 ¼ u4 ¼ u5 ¼ 0; M ¼ 1; n ¼ 1 or n = 2 in (7),we easily give the following equations Case 1: (
Case 2: (
u1t ¼ aðu2xx þ 4u2 ðu21 u22 ÞÞ; u2t ¼ aðu1xx þ 4u1 ðu21 u22 ÞÞ;
u1t ¼ a2 ðu1xxx þ 6au1x ðu22 u21 ÞÞ; u2t ¼ a2 ðu2xxx þ 6au2x ðu22 u21 ÞÞ;
ð9Þ
ð10Þ
which are the NLS-MKdV equations[3]. Taking u3 ¼ u4 ¼ u5 ¼ 0; M > 1, the system (8) reduces to the multi-component NLS-MKdV hierarchy. Hence, the system (8) is the integrable couplings of the multi-component NLS-MKdV hierarchy.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Tu Guizhang. J Math Phys 1989;30(2):330. Hu X. J.Phys. A 1997;30:619. Guo F. Acta Math Sinica 1997;40(6):801–4. Tsuchida T, Wadati M. J Phys Soc Jpn 1998;67:1175–87. Tu G. J Phys A Math Gen 1989(22):2375–92. Zhang Yufeng, Zhang Hongqing. J Math Phys 2002;43(1):466. Fan E. Physica A 2002;301:105. Fan E. J Math Phys 2000;41(11):7769. Guo F. J Syst Sci Math Sci 2002;22(1):36. Zhang Yufeng. Physica A 2003;317:280. Zhang Yufeng, Fan Engui, Tam Honwah. Phys Lett A 2006;359:471–80. Zhang Yufeng, Tam Honwah. Chaos, Soltons & Fractals 2007;32:640. Zhang Yufeng. Phys Lett A 2005;342:82–9. Zhang Y, Guo X, Tam Honwah. Chao, Solitons & Fractals 2006;29:114–24. Ma W, Zhou R. Chin Ann Math B 2002;23(3):373. Zhang Y, Yan Q. Chaos, Soltons & Fractals 2003;16:263. Guo F, Zhang Y. J Math Phys 2003;44(12):5793.
Chaos, Solitons and Fractals 39 (2009) 473–478 www.elsevier.com/locate/chaos
A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure Hui Chang *, Yufeng Zhang Information School, Shandong University of Science and Technology, Qingdao 266510, China Accepted 2 January 2007
Abstract A new multi-component matrix loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component integrable couplings of the NLS-MKdV hierarchy is obtained, then the bi-Hamiltonian structure of the above system is given. 2007 Elsevier Ltd. All rights reserved.
1. Introduction It is an important and interesting topic to search for new Hamiltonian hierarchies of soliton equations and their integrable coupling in soliton theory [1–4]. Tu once proposed a simple and efficient method to obtain integrable Hamiltonian hierarchy of soliton equations with infinite dimensions in Refs. [1,5]. In recent years, Ones used Tu scheme to have obtained many interesting results [6–12]. As far as the multi-component integrable hierarchies, there have been developments in Ref. [13–16]. In order to obtain some new integrable hierarchy, we firstly construct a new multi-component e 2 to work out the integrable coumatrix lie algebra G2 and its corresponding multi-component matrix loop algebra G plings of the NLS-MKdV hierarchy. And then the bi-Hamiltonian structure of the above system is presented.
2. A new loop algebra and its application Definition 1. [17]. If a ¼ ða1 ; a2 ; . . . ; aM ÞT ; b ¼ ðb1 ; b2 ; . . . ; bM ÞT are two column vectors, their vector product a b is defined as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM ÞT ;
*
Corresponding author. E-mail address: [email protected] (H. Chang).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.102
474
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
where M stands for an arbitrary positive integer. It follows from Definition 1 that set a ¼ ða1 ; a2 ; . . . ; aM Þ; b ¼ ðb1 ; b2 ; . . . ; bM Þ; we have their product a b as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM Þ: Definition 2. Let a ¼ ða1 ; a2 ; . . . ; aM ÞT ; a ¼ ða1 ; a2 ; ; aM Þ; define a kind of product a and a as follows: 0 1 a1 Ba C B 2 C C aa¼B B .. Cða1 ; a2 ; . . . ; aM Þ ¼ diagðai ai Þ; @. A aM then, we construct a new multi-component matrix Lie algebra G2 by use of Definitions 1 and 2: 8 > > 1 1 0 0 > > > M 0 M M 0 M > > > C C B B > > 2I M 0 A; e1 ¼ @ 0 2I M 0 A; e2 ¼ @ 0 > > > > > M 0 M M 0 M > < 1 1 0 0 0 0 0 I 1M 0 I 1M > C C B > > e3 ¼ B 0 I M 1 A; e4 ¼ @ I M1 0 I M1 A; @ I M 1 > > > > > 0 I 1M 0 0 I 1M 0 > > > > > ½e1 ; e2 ¼ 0; ½e1 ; e3 ¼ 2ðM þ 1Þe4 ; ½e1 ; e4 ¼ 2ðM þ 1Þe3 ; > > > : ½e2 ; e3 ¼ 2e4 ; ½e2 ; e4 ¼ 2e3 ; ½e3 ; e4 ¼ 2e1 : 0 1 1 B1C C IM is M · M unit matrix, I 1M ¼ ð1; 1; . . . ; 1Þ1M ; I M 1 ¼ B @ ... A .
ð1Þ
1 M1 Definition 3. Set a ¼ ða1 ; a2 ; . . . ; aM ÞT , c is a scalar, we define the following operation relations: 1 1 0 0 cM 0 cM 0 aT I 1M 0 C C B B 0 A; ae3 ¼ @ a I M1 ce1 ¼ @ 0 2cI M 0 a I M 1 A; cM 0 cM 0 aT I 1M 0 1 0 1 0 cM 0 cM 0 0 aT I 1M C B C B ce2 ¼ @ 0 2cI M 0 A; ae4 ¼ @ a I M1 0 a I M1 A: T cM 0 cM 0 a I 1M 0 Therefore a loop algebra is presented as 8 > ek ðj; nÞ ¼ ek k2nþj ; k ¼ 1; 2; 3; 4; j ¼ 0; 1; > > > > > ½e1 ði; mÞ; e2 ðj; nÞ ¼ 0; > > > > > ði þ j; m þ nÞ; > > ½e1 ði; mÞ; e3 ðj; nÞ ¼ 2ðM þ 1Þe4 > > > ð0; m þ n þ 1Þ; 2ðM þ 1Þe 4 > > > > > ði þ j; m þ nÞ; 2ðM þ 1Þe 3 > > ½e1 ði; mÞ; e4 ðj; nÞ ¼ > > > 2ðM þ 1Þe 3 ð0; m þ n þ 1Þ; < 2e4 ði þ j; m þ nÞ; > ½e2 ði; mÞ; e3 ði; nÞ ¼ > > > 2e4 ð0; m þ n þ 1Þ; > > > > > 2e 3 ði þ j; m þ nÞ; > > ½e2 ði; mÞ; e4 ði; nÞ ¼ > > > 2e 3 ð0; m þ n þ 1Þ; > > > > 2e1 ði þ j; m þ nÞ; > > > ½e3 ði; mÞ; e4 ði; nÞ ¼ > > > 2e1 ð0; m þ n þ 1Þ; > > : degðek ðj; nÞÞ ¼ 2n þ j; k ¼ 1; 2; 3; 4; i ¼ 0; 1; j ¼ 0; 1:
ð2Þ
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
475
Considering an isospectral problem as follows: ux ¼ U u;
kt ¼ 0;
U ¼ e2 ð1; 0Þ þ u1 e3 ð0; 0Þ þ u2 e4 ð0; 0Þ þ u3 e3 ð1; 1Þ þ u4 e4 ð1; 1Þ þ u5 e1 ð1; 1Þ; 1 0 u1;i B u2;i C C B C ui ¼ B B .. C ; i ¼ 1; 2; 3; 4; u5 is a smooth function: @ . A uM ;i
M1
Taking
! 1 X X ðaði; mÞe2 ði; mÞ þ bði; mÞe3 ði; mÞ þ cði; mÞe4 ði; mÞÞ ; V ¼ mP0
i¼0
0
ðiÞ
bm1
1
B ðiÞ C Bb C B m2 C bði; mÞ ¼ B . C ; C B. A @. ðiÞ bmM M 1
ðiÞ
1
B ðiÞ Bc B m2 cði; mÞ ¼ B . B. @.
C C C C C A
0
cm1
ðiÞ
cmM
;
M1
a(i, m) is a scalar smooth function, i = 0,1. Solving the stationary zero curvature equation, V x ¼ ½U ; V ;
ð3Þ
gives rise to 8 ax ð0; mÞ ¼ 2uT1 cð0; mÞ þ 2uT2 bð0; mÞ 2uT3 cð1; mÞ þ 2uT4 bð1; mÞ; > > > > > ax ð1; m þ 1Þ ¼ 2uT1 cð1; m þ 1Þ þ 2uT2 bð1; m þ 1Þ 2uT3 cð0; mÞ þ 2uT4 bð0; mÞ; > > > > > bx ð0; mÞ ¼ 2cð1; m þ 1Þ 2ðM þ 1Þu2 að0; mÞ 2ðM þ 1Þu4 að1; mÞ þ 2u5 ðM þ 1Þcð1; mÞ; > > > > > > bx ð1; m þ 1Þ ¼ 2cð0; m þ 1Þ 2ðM þ 1Þu2 að1; m þ 1Þ 2ðM þ 1Þu4 að0; mÞ þ 2u5 ðM þ 1Þcð0; mÞ; > > > > > > > cx ð0; mÞ ¼ 2bð1; m þ 1Þ 2ðM þ 1Þu1 að0; mÞ 2ðM þ 1Þu3 að1; mÞ þ 2u5 ðM þ 1Þbð1; mÞ; > > > cx ð1; m þ 1Þ ¼ 2bð0; m þ 1Þ 2ðM þ 1Þu1 að1; m þ 1Þ 2ðM þ 1Þu3 að0; mÞ þ 2u5 ðM þ 1Þbð0; mÞ; > > > T > > < að0; 0Þ ¼ a ¼ const;bð0; 0Þ ¼ cð0; 0Þ ¼ bð1; 0Þ ¼ cð1; 0Þ ¼ ð0; 0; . . . ; 0Þ1M ; að1; 0Þ ¼ 0; bð1; 1Þ ¼ aðM þ 1Þu1 ; cð1; 1Þ ¼ aðM þ 1Þu2 ; að1; 1Þ ¼ 0; > > > > bð0; 1Þ ¼ ðMþ1Þa u2x þ ðM þ 1Þau3 ; cð0; 1Þ ¼ ðMþ1Þa u1x þ ðM þ 1Þau4 ; > > 2 2 > > 1 T > 1 T T > að0; 1Þ ¼ ðM þ 1Þ ao ðu u u u Þ þ 2ao ðu2 u3 uT1 u4 Þ 2ao1 ðuT3 u2 uT4 u1 Þ ; > 2x 1x 2 1 > > > > bð1; 2Þ ¼ ðMþ1Þ au ðM þ 1Þ2 u ao1 ðuT u uT u Þ þ 2ao1 ðuT u uT u Þ > > 1xx 1 2 2x 1 1x 2 3 1 4 4 > > > 2 1 T > T > 2ao ðu3 u2 u4 u1 Þ ðM þ 1Þ au5 u1 ; > > > > > cð1; 2Þ ¼ ðMþ1Þ au ðM þ 1Þ2 u o1 ½ao1 ðuT u uT u Þ þ 2ao1 ðuT u uT u Þ > 2xx 2 > 2 2x 1 1x 2 3 1 4 4 > > : 2ao1 ðuT3 u2 uT4 u1 Þ ðM þ 1Þ2 au5 u2 :
ð4Þ
Denoting ðnÞ
Vþ ¼
n X
að0; mÞe1 ð0; n mÞ þ bð0; mÞe3 ð0; n mÞ þ cð0; mÞe4 ð0; n mÞ þ að1; mÞe1 ð1; n mÞ
m¼0
þbð1; mÞe3 ð1; n mÞ þ cð1; mÞe4 ð1; n mÞ ; ðnÞ
2n V ðnÞ ¼ k V Vþ :
ð5Þ
Eq. (3). can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
ð6Þ
476
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
A direct calculation reads: ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2cð1; n þ 1Þe3 ð0; 0Þ bð1; n þ 1Þe4 ð0; 0Þ þ ½bx ð1; n þ 1Þ 2cð0; n þ 1Þ þ 2ðM þ 1Þu2 að1; n þ 1Þe3 ð1; 1Þ þ ½cx ð1; n þ 1Þ 2bð0; n þ 1Þ þ 2ðM þ 1Þu1 að1; n þ 1Þe4 ð1; 1Þ þ ½ax ð1; n þ 1Þ þ 2uT1 cð1; n þ 1Þ 2bð1; n þ 1ÞuT2 e1 ð1; 1Þ: ðnÞ
Taking V ðnÞ ¼ V þ , then the zero curvature equation ðnÞ ¼0 U t V ðnÞ x þ ½U ; V
admits the following integrable system: 1 0 0 1 0 0 0 0 2cð1; n þ 1Þ u1 C B0 B C B 2 0 2bð1; n þ 1Þ C B B u2 C B M C B B C B 2 o C C B B u t ¼B B u3 C ¼ B bx ð1; n þ 1Þ þ 2cð0; n þ 1Þ 2ðM þ 1Þu2 að1; n þ 1Þ C ¼ B 0 M M C B2 B C B 0 0 @ u4 A @ cx ð1; n þ 1Þ þ 2bð0; n þ 1Þ 2ðM þ 1Þu1 að1; n þ 1Þ A @ M 2 T 0 0 u ax ð1; n þ 1Þ 2uT1 cð1; n þ 1Þ þ 2uT2 bð1; n þ 1Þ u5 t M 2 1 1 0 0 2cð1; n þ 1Þ Mbð0; n þ 1Þ C C B B 2bð1; n þ 1Þ Mcð0; n þ 1Þ C C B B C C B B T C C B B B Mbð1; n þ 1Þ C ¼ J 1 Gnþ1 ¼ B 2ðM þ 1Þu4 að0; nÞ 2u5 ðM þ 1Þcð0; nÞ C C C B B T Mcð1; n þ 1Þ A @ @ 2ðM þ 1Þu3 að0; nÞ 2u5 ðM þ 1Þbð0; nÞ A 0
0 B2 BM B B ¼B 0 B B0 @ 0
ðM 2 þ MÞað1; n þ 1Þ 0
0
0
0
0
0
0
2ðMþ1Þ u5 M
0
2ðMþ1Þ u5 M M2 uT4
M2 uT3
From (4), we have 0 L11 B B L21 B Gnþ1 ¼ B B 1 B @ 0 L51
0
L12
L13
L14
L22 0 1 L52
L23 0 0 L53
L24 0 0 L54
u3
0 0 o M 2 M
uT1
0 0
1
C C C M2 u2 C C M2 u1 C A M 2oþM
10
M2
0
2uT3 cð0; nÞ 2uT4 bð0; nÞ 1 0 Mbð1; n þ 1Þ CB 0 CB Mcð1; n þ 1Þ C C CB C CB 2 C ¼ J 2F n: u Mbð0; nÞ C 4 M C B CB C C 2 Mcð0; nÞ A @ u A 3 M ðM 2 þ MÞað0; nÞ 0
M2
10
Mbð1; n þ 1Þ
ð7Þ
1
CB C u4 CB Mcð1; n þ 1Þ C CB C C ¼ LF n ; F n ¼ LGn ; B Mbð0; nÞ 0 C CB C CB C 0 A@ Mcð0; nÞ A 2 ðM þ MÞað0; nÞ 0
Here o L12 ¼ I M þ 2ðM þ 1Þu1 o1 uT1 ; 2 L13 ¼ 2ðM þ 1Þu1 o1 uT4 ðM þ 1Þu5 ; L14 ¼ 2ðM þ 1Þu1 o1 uT3 ; o L21 ¼ I M 2ðM þ 1Þu2 o1 uT2 ; L22 ¼ 2ðM þ 1Þu2 o1 uT1 ; 2 L23 ¼ 2ðM þ 1Þu2 o1 uT4 ; L24 ¼ 2ðM þ 1Þu2 o1 uT3 ðM þ 1Þu5 I M ; L11 ¼ 2ðM þ 1Þu1 o1 uT2 ;
L51 ¼ 2ðM þ 1Þo1 uT2 ;
L52 ¼ 2ðM þ 1Þo1 uT1 ;
1Þo1 uT4 ;
L54 ¼ 2ðM þ 1Þo1 uT3 :
L53 ¼ 2ðM þ
Therefore, the system (6) can be written as 1 0 1 0 Mbð0; 1Þ u1 C B C B Mcð0; 1Þ C B u2 C B C B C B 2 2n B C C: B Mbð1; 1Þ ut ¼ B u3 C ¼ J 1 LF n ¼ J 2 F n ¼ J 2 LGn ¼ J 1 L Gn ¼ J 1 L B C C B C B Mcð1; 1Þ A @ u4 A @ 2 ðM þ MÞað1; 1Þ u5 t
ð8Þ
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
477
A direct calculation reads M M X X oU oU ð0Þ ð1Þ ð0Þ ð1Þ V; ¼4 ¼ 4 ðbmk þ kbmk Þ; V; ðcmk þ kcmk Þ; ou1 ou 2 k¼0 k¼0
M M X X oU 1 ð0Þ oU 1 ð0Þ ð1Þ ð1Þ ¼4 V; ¼ 4 k ¼ 0 bmk þ bmk ; cmk þ cmk ; V; ou3 k ou4 k k¼0 oU 4M V; ¼ að0Þ þ kað1Þ ; ou5 k oU MðM þ 1Þ MðM þ 1Þ að0Þu5 þ 4kMað1Þ 4 V; ¼ 4að0ÞM 4 að1Þu5 ok k k2 2 þ ðbð0ÞT þ cðoÞT Þðu3 u4 Þ þ 2kðbð1ÞT þ cð1ÞT Þðu1 u2 Þ þ 2ðbð0ÞT þ cðoÞT Þðu1 u2 Þ k M 2 X ð0Þ ð1Þ ð0Þ ð1Þ þ 2ðbð1ÞT þ cð1ÞT Þðu3 u4 Þ þ 2 ðbmk þ kbmk cmk kcmk Þðuk3 uk4 Þ; k k¼1 where
X
að0Þ ¼
að0; mÞk2m ;
að1Þ ¼
mP0
X
að1; mÞk2m . . . :
mP0
Substituting the above formulate into the trace identity yields 0D E1 oU V ; ou 1 C B EC BD B V ; oU C C B B D ou2 E C
C B d oU c o c B oU C V; ¼k k B V ; ou3 C: du ok ok B D EC B oU C C B V ; ou 4 C B EA @D oU V ; ou5 Comparison of the coefficients of k2n3 the both side of the above equations gives d ðMðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þ2ðu4 u3 Þ þ 2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu4 u3 Þ du M X ð1Þ ð1Þ 2M ðbmk þ cmk Þðuk3 uk4 Þ ¼ ðc 2n 2ÞGnþ1 ; k¼1
comparison of the coefficients of k2n2 the both side of the above equations leads to d ð2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu1 u2 Þ þ 2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þðu1 u2 Þ du M X ð0Þ ð0Þ 2M ðbmk cmk Þðuk3 uk4 Þ þ 4M 2 að1; n þ 2Þ 4M 2 ðM þ 1Það1; nþÞu5 þ 4M 2 að0; n þ 1Þ k¼1 2
4M ðM þ 1Það0; nÞu5 Þ ¼ ðc 2n 1ÞF n : Taking n ¼ 0, gives c = 0, therefore, dH ð1; nÞ ; du 1 where H ð1; nÞ ¼ 2nþ2 ð2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þ2ðu4 u3 Þ 2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu4 u3 Þþ P ð1Þ ð1Þ 2M M k¼1 ðbmk þ cmk Þðuk3 uk4 Þ; Gnþ1 ¼
Fn ¼
dH ð2; nÞ ; du
1 ð2Mðbð0; n þ 1ÞT þ cð0; n þ 1ÞT Þðu1 u2 Þ 2Mðbð1; n þ 2ÞT þ cð1; n þ 2ÞT Þðu1 u2 Þ where H ð2; nÞ ¼ 2nþ1 PM ð0Þ ð0Þ þ2M k¼1 ðbmk cmk Þðuk3 uk4 Þ 4M 2 að1; n þ 2Þ þ 4M 2 ðM þ 1Það1; nþÞu5 4M 2 að0; n þ 1Þ þ 4M 2 ðM þ 1Það0; nÞu5 Þ:
478
H. Chang, Y. Zhang / Chaos, Solitons and Fractals 39 (2009) 473–478
Thus the bi-Hamiltonian structure of the system is given by 0 1 u1 B C B u2 C B C dH ð1; nÞ dH ð2; nÞ C ¼ J2 : ut ¼ B B u3 C ¼ J 1 du du B C @ u4 A u5 t It is easy to verify that J 1 L ¼ L J 1 ¼ J 2 . Therefore, the system (7) is Liouville integrable. Taking u3 ¼ u4 ¼ u5 ¼ 0; M ¼ 1; n ¼ 1 or n = 2 in (7),we easily give the following equations Case 1: (
Case 2: (
u1t ¼ aðu2xx þ 4u2 ðu21 u22 ÞÞ; u2t ¼ aðu1xx þ 4u1 ðu21 u22 ÞÞ;
u1t ¼ a2 ðu1xxx þ 6au1x ðu22 u21 ÞÞ; u2t ¼ a2 ðu2xxx þ 6au2x ðu22 u21 ÞÞ;
ð9Þ
ð10Þ
which are the NLS-MKdV equations[3]. Taking u3 ¼ u4 ¼ u5 ¼ 0; M > 1, the system (8) reduces to the multi-component NLS-MKdV hierarchy. Hence, the system (8) is the integrable couplings of the multi-component NLS-MKdV hierarchy.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Tu Guizhang. J Math Phys 1989;30(2):330. Hu X. J.Phys. A 1997;30:619. Guo F. Acta Math Sinica 1997;40(6):801–4. Tsuchida T, Wadati M. J Phys Soc Jpn 1998;67:1175–87. Tu G. J Phys A Math Gen 1989(22):2375–92. Zhang Yufeng, Zhang Hongqing. J Math Phys 2002;43(1):466. Fan E. Physica A 2002;301:105. Fan E. J Math Phys 2000;41(11):7769. Guo F. J Syst Sci Math Sci 2002;22(1):36. Zhang Yufeng. Physica A 2003;317:280. Zhang Yufeng, Fan Engui, Tam Honwah. Phys Lett A 2006;359:471–80. Zhang Yufeng, Tam Honwah. Chaos, Soltons & Fractals 2007;32:640. Zhang Yufeng. Phys Lett A 2005;342:82–9. Zhang Y, Guo X, Tam Honwah. Chao, Solitons & Fractals 2006;29:114–24. Ma W, Zhou R. Chin Ann Math B 2002;23(3):373. Zhang Y, Yan Q. Chaos, Soltons & Fractals 2003;16:263. Guo F, Zhang Y. J Math Phys 2003;44(12):5793.