- Email: [email protected]
A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations
Chaos, Solitons and Fractals 21 (2004) 445–456 www.elsevier.com/locate/chaos
A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations Yufeng Zhang a b
a,b,*
, Xixiang Xu
c
Institute of Mathematics, Information School, Shandong University of Science and Technology, Taian 271019, PR China Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, PR China c Department of Basic Courses, Shandong University of Science and Technology, Taian 271019, PR China Accepted 10 December 2003 Communicated by Y. Aizawa
Abstract e 2 is first constructed, which has its own special feature. It follows that a new Liouville A subalgebra of loop algebra A integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the e is constructed by making use of a hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Researching new various integrable systems is of an importance and interest work in soliton theory [1–3]. A great host of integrable hierarchies of evolution equations were obtained such as in [4–8] in recent years. Professor Guo Fukui once deduced some interesting integrable Hamiltonian systems with multi-potential functions by use of subalgebras of e 1 , which could reduce to the famous hierarchies, such as AKNS hierarchy, BPT hierarchy and so on loop algebra A e 2 . That is to say, constructing a new subalgebra [9,10]. In this paper, we expand Guo’s idea to the case of loop algebra A e 2 is devoted to deducing a Liouville integrable system and we prove that it possesses a tri-Hamiltonian of loop algebra A structure. As its reduction cases, the NLS–MKdV hierarchy [10] is presented. Especially, the well-known MKdV equation and the heat-conduction equation are obtained, respectively. Hence, we call the hierarchy a MKdV-H system. It is remarkable that three symplectic operators in the MKdV-H system are constructed directly by the recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. In addition, a high-dimension loop e is constructed by employing a proper scalar transformation. Then using G e and Tu scheme deduces a type of algebra G expanding integrable model of the MKdV-H system. Here the expanding integrable model, as a matter of fact, is an integrable coupling of the MKdV-H system. 2. A trick loop algebra and a corresponding integrable system e 2 as follows Consider a subalgebra of loop algebra A *
Corresponding author. E-mail address: [email protected] (Y. Zhang).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.018
446
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
0
0
B e1 ði; nÞ ¼ @ 0 k3nþi 0 0 B e3 ði; nÞ ¼ @ 0 k3nþi ½e1 ði; mÞ; e2 ðj; nÞ ¼
0 k3nþi
1
0
k3nþi
0
0
1
C C B k3nþi 0 A; 0 A; e2 ði; nÞ ¼ @ 0 0 0 0 0 1 0 3nþi 1 0 0 0 k3nþi k C B C 0 0 A; 0 0 A; e4 ði; nÞ ¼ @ 0 0 0 k3nþi 0 0 e3 ði þ j; m þ nÞ; i þ j < 3; 0 0
e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e4 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e3 ðj; nÞ ¼ 2e4 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e3 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e4 ðj; nÞ ¼ 2e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e1 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e3 ðj; nÞ ¼ e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e1 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e4 ðj; nÞ ¼ 2e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ½e2 ði; mÞ; e4 ðj; nÞ ¼ 0; deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4:
In terms of (1), an isospectral problem is in the following 0 k þ uk5 T 0 /x ¼ U /; kt ¼ 0; / ¼ ð/1 ; /2 ; /3 Þ ; U ¼ @ 4 u1 þ u2 þ u3 þu k
ð1Þ
1 4 0 u1 u2 þ u3 u k A k 0 0 uk5
¼ e2 ð1; 0Þ þ u1 e1 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e1 ð2; 1Þ þ u4 e3 ð2; 1Þ þ u5 e4 ð2; 1Þ:
ð2Þ
Let 0
1 cð0Þ þ cð1Þk þ cð2Þk2 0 að0Þ bð0Þ þ ðað1Þ bð1ÞÞk þ ðað2Þ bð2ÞÞk2 A; V ¼@ 0 0 0 að0Þ þ bð0Þ þ ðað1Þ þ bð1ÞÞk þ ðað2Þ þ bð2ÞÞk2 0 cð0Þ cð1Þk cð2Þk2 P P where að0Þ ¼ m P 0 að0; mÞk3m , að1Þ ¼ m P 0 að1; mÞk3m ; . . . Solving the stationary zero-curvature equation Vx ¼ ½U ; V
ð3Þ
yields bð2; m þ 1Þ ¼ ax ð0; mÞ þ 2u2 cð0; mÞ þ 2u4 cð1; mÞ 2u5 bð1; mÞ; að2; m þ 1Þ ¼ bx ð0; mÞ þ 2u1 cð0; mÞ þ 2u3 cð1; mÞ 2u5 að1; mÞ; cx ð0; mÞ ¼ 2u1 bð0; mÞ 2u2 að0; mÞ þ 2u3 bð1; mÞ 2u4 að1; mÞ; bð0; mÞ ¼ ax ð1; mÞ þ 2u2 cð1; mÞ þ 2u4 cð2; mÞ 2u5 bð2; mÞ; að0; mÞ ¼ bx ð1; mÞ þ 2u1 cð1; mÞ þ 2u3 cð2; mÞ 2u5 að2; mÞ; cx ð1; mÞ ¼ 2u1 bð1; mÞ 2u2 að1; mÞ þ 2u3 bð2; mÞ 2u4 að2; mÞ; bð1; m þ 1Þ ¼ ax ð2; m þ 1Þ þ 2u2 cð2; m þ 1Þ þ 2u4 cð0; mÞ 2u5 bð0; mÞ; að1; m þ 1Þ ¼ bx ð2; m þ 1Þ þ 2u1 cð2; m þ 1Þ þ 2u3 cð0; mÞ 2u5 að0; mÞ; cx ð2; m þ 1Þ ¼ 2u1 bð2; m þ 1Þ 2u2 að2; m þ 1Þ þ 2u3 bð0; mÞ 2u4 að0; mÞ; cð0; 0Þ ¼ a; að0; 0Þ ¼ bð0; 0Þ ¼ cð1; 0Þ ¼ bð1; 0Þ ¼ að1; 0Þ ¼ cð2; 0Þ ¼ bð2; 0Þ ¼ að2; 0Þ ¼ 0; bð2; 1Þ ¼ 2au2 ; að2; 1Þ ¼ 2au1 ; cð2; 1Þ ¼ 0; að1; 1Þ ¼ 2aðu2x u3 Þ; bð1; 1Þ ¼ 2aðu1x u4 Þ; cð1; 1Þ ¼ 2aðu21 u22 Þ; að0; 1Þ ¼ 2aðu1xx u4x 2u31 þ 2u1 u22 2u1 u5 Þ; bð0; 1Þ ¼ 2aðu2xx u3x 2u2 u21 þ 2u32 2u2 u5 Þ:
ð4Þ
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
447
Note ðnÞ
Vþ ¼
n X 2 X ðaði; mÞe1 ði; n mÞ þ bði; mÞe3 ði; n mÞ þ cði; mÞe4 ði; n mÞÞ;
ð5Þ
m¼0 i¼0
VðnÞ
3n
¼k V
ðnÞ Vþ ;
ðnÞ
ðnÞ
ðnÞ then we have Vþx þ ½U; Vþ ¼ Vx ½U ; VðnÞ . It is easy to verify that the terms of the left-hand side in (5) are of degree P )1, while the terms of the right-hand side in (5) are of degree 6 0. Therefore, the terms of both sides in (5) are of degree )1,0. It follows that ðnÞ
ðnÞ
Vþx þ ½U ; Vþ ¼ bð2; n þ 1Þe1 ð0; 0Þ þ að2; n þ 1Þe3 ð0; 0Þ þ ½ax ð2; n þ 1Þ þ bð1; n þ 1Þ 2u2 cð2; n þ 1Þe1 ð2; 1Þ þ ½bx ð2; n þ 1Þ þ að1; n þ 1Þ 2u1 cð2; n þ 1Þe3 ð2; 1Þ þ ½cx ð2; n þ 1Þ 2u1 bð2; n þ 1Þ þ 2u2 að2; n þ 1Þe4 ð2; 1Þ: ðnÞ
Taking V ðnÞ ¼ Vþ , then the zero-curvature equation Ut VxðnÞ þ ½U ; V ðnÞ ¼ 0
ð6Þ
admits the Lax integrable system 0
1
u1
0
1
bð2; n þ 1Þ
Bu C B að2; n þ 1Þ B 2C B B C B C B a utn ¼ B u ¼ ð2; n þ 1Þ bð1; n þ 1Þ þ 2u2 cð2; n þ 1Þ x B 3C B B C B @ u4 A @ bx ð2; n þ 1Þ að1; n þ 1Þ þ 2u1 cð2; n þ 1Þ 0
u5
tn
0
0
B 0 B B ¼B B 0 B 1 @2
0 1 2
0
cx ð2; n þ 1Þ þ 2u1 bð2; n þ 1Þ 2u2 að2; n þ 1Þ 1 1 10 0 1 0 2að1; n þ 1Þ 2að1; n þ 1Þ 2 C B B 2bð1; n þ 1Þ C 12 0 0 C C CB 2bð1; n þ 1Þ C B C C CB B o B 2að2; n þ 1Þ C ¼ J1 B 2að2; n þ 1Þ C 2 0 u2 C C C CB B C C CB B o 0 u1 A@ 2bð2; n þ 1Þ A @ 2bð2; n þ 1Þ A 2 0
0 u2
u1
o2 1
2cð2; n þ 1Þ 0 0 12 0 C B B að2; n þ 1Þ C B 12 0 0 B C B B C B ð4ÞB B 2u4 cð0; nÞ þ 2u5 bð0; nÞ C ¼ B 0 0 0 C B B @ 2u3 cð0; nÞ þ 2u5 að0; nÞ A @ 0 0 u5 0
0
C C C C C C A
bð2; n þ 1Þ
0 0 u5 0
2cð2; n þ 1Þ 1 1 10 0 2að2; n þ 1Þ 2að2; n þ 1Þ C B B 2bð2; n þ 1Þ C 0 C C CB 2bð2; n þ 1Þ C B C C CB B B 2að0; nÞ C ¼ J2 B 2að0; nÞ C u4 C C C CB B C C CB B @ 2bð0; nÞ A u3 A@ 2bð0; nÞ A 0
2cð0; nÞ 2cð0; nÞ 2u3 bð0; nÞ þ 2u4 að0; nÞ 0 0 u4 u3 0 1 0 o þ 4u2 o1 u2 að0; nÞ 4u2 o1 u1 bð0; nÞ þ 4u2 o1 u3 þ 2u5 bð1; nÞ þ 4u2 o1 u4 að1; nÞ 2u4 cð1; nÞ C B B o 4u1 o1 u1 bð0; nÞ þ 4u1 o1 u2 að0; nÞ 4u1 o1 u3 bð1; nÞ þ 4u1 o1 u4 þ 2u5 að1; nÞ 2u3 cð1; nÞ C C B C B 1 1 1 1 ð4ÞB C 4u 4 o u1 þ 2u5 bð0; nÞ þ 4u4 o u2 að0; nÞ 4u4 o u3 bð1; nÞ þ 4u4 o u4 að1; nÞ C B C B 1 1 1 1 A @ 4u3 o u1 bð0; nÞ þ 4u3 o u2 þ 2u5 að0; nÞ 4u3 o u3 bð1; nÞ þ 4u3 o u4 að1; nÞ 0
2u3 bð0; nÞ þ 2u4 að0; nÞ o 2
1
þ 2u2 o u2
1
2u2 o u1
B B 2u1 o1 u2 o2 þ 2u1 o1 u1 B B 1 ¼ B 2u4 o u2 2u4 o1 u1 u5 B B 2u3 o1 u2 þ u5 2u3 o1 u1 @ u4 u3 1 0 2að0; nÞ B 2bð0; nÞ C C B C B C: ¼ J3 B 2að1; nÞ C B C B @ 2bð1; nÞ A 2cð1; nÞ
2u2 o1 u4
2u2 o1 u3 u5
2u1 o1 u4 þ u5
2u1 o1 u3
1
2u4 o1 u3
1
2u3 o u4
2u3 o1 u3
0
0
2u4 o u4
10 1 2að0; nÞ CB 2bð0; nÞ C u3 C C CB C CB 2að1; nÞ C 0 CB C B CB C 0 C A@ 2bð1; nÞ A u4
0
2cð1; nÞ
ð7Þ
448
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
A direct calculation reads oU oU ¼ 2ðað0Þ þ að1Þk þ að2Þk2 Þ; ¼ 2ðbð0Þ þ bð1Þk þ abð2Þk2 Þ; V; V; ou1 ou2 oU að0Þ oU bð0Þ þ að1Þ þ að2Þk ; V; þ bð1Þ þ bð2Þk ; V; ¼2 ¼ 2 ou3 k ou4 k oU cð0Þ þ cð1Þ þ cð2Þk ; ¼2 V; ou5 k oU 2u4 bð0Þ u5 cð0Þ 2u3 að0Þ V; ¼ cð0Þ u2 cð2Þ þ 2u4 bð2Þ 2u3 að2Þ þ cð1Þk þ cð2Þk2 þ ok k2 2u4 bð1Þ 2u3 að1Þ u5 cð1Þ : þ k Substituting the above formulae into trace identity leads to E1 0D oU V ; ou 1 EC BD B V ; oU C B C ou 2 BD EC d oU C c o c B oU V; ¼k k B V; C: du ok ok B D ou3 E C B C oU B V ; ou C 4 @D EA oU V ; ou 5 Comparing the coefficients of k3n3 in (8) gives
ð8Þ
0
1 2að1; n þ 1Þ B 2bð1; n þ 1Þ C B C d ðcð0; n þ 1Þ u2 cð2; n þ 1Þ þ 2u4 bð2; n þ 1Þ 2u3 að2; n þ 1ÞÞ ¼ ð3n 2 þ cÞB 2að2; n þ 1Þ C B C: du @ 2bð2; n þ 1Þ A 2cð2; n þ 1Þ
Comparing the coefficients of k3n2 , k3n1 in (8) engenders, respectively, as follows 0 1 2að2; n þ 1Þ B 2bð2; n þ 1Þ C B C d ðcð1; n þ 1Þ u5 cð0; nÞ 2u3 að0; nÞ þ 2u4 bð0; nÞÞ ¼ ð3n 1 þ cÞB 2að0; nÞ C B C; du @ 2bð0; nÞ A 2cð0; nÞ 0 1 2að0; nÞ B 2bð0; nÞ C B C d ðcð2; n þ 1Þ þ 2u4 bð1:nÞ 2u3 að1; nÞ u5 cð1; nÞÞ ¼ ð3n þ cÞB 2að1; nÞ C B C: du @ 2bð1; nÞ A 2cð1; nÞ
ð9Þ
ð10Þ
ð11Þ
Inserting the initial values in (4) into (9)–(11) gives c ¼ 0. Thus, the relations (9)–(11) can determine the following three Hamiltonian functions 0 1 8 2að1; n þ 1Þ > > > B C > > B 2bð1; n þ 1Þ C > > B C > > B C > > > dHð1;3nþ3Þ ¼ B 2að2; n þ 1Þ C; > B C > du < B C B C ð12Þ B C 2bð2; n þ 1Þ > @ A > > > > > > 2cð2; n þ 1Þ > > > > > > > : H ð1; 3n þ 3Þ ¼ cð0; n þ 1Þ u2 cð2; n þ 1Þ þ 2u4 bð2; n þ 1Þ 2u3 að2; n þ 1Þ ; 3n þ 2
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
8 0 1 2að2; n þ 1Þ > > > > B 2bð2; n þ 1Þ C > > B C > > B 2að0; nÞ C; ¼ < dHð2;3nþ2Þ du B C @ 2bð0; nÞ A > > > 2cð0; nÞ > > > > > H ð2; 3n þ 2Þ ¼ cð1; n þ 1Þ u5 cð0; nÞ 2u3 að0; nÞ þ 2u4 bð0; nÞ ; : 3n þ 1 8 0 1 2að0; nÞ > > > > B 2bð0; nÞ C > > B C > > C < dHð3;3nþ1Þ ¼B du B 2að1; nÞ C; @ 2bð1; nÞ A > > > 2cð1; nÞ > > > > u cð1; nÞ þ 2u3 að1; nÞ cð2; n þ 1Þ 2u4 bð1; nÞ > : H ð3; 3n þ 1Þ ¼ 5 : 3n
449
ð13Þ
ð14Þ
Therefore, we have utn ¼ J1
dH ð1; 3n þ 3Þ dH ð2; 3n þ 2Þ dH ð3; 3n þ 1Þ ¼ J2 ¼ J3 ¼ ; du du du
From (4), a recurrence operator is generated as follows 0 4u1 o1 u2 o 4u1 o1 u1 4u1 o1 u4 2u5 B o þ 4u2 o1 u2 4u2 o1 u1 4u2 o1 u4 B L¼B 1 0 0 B @ 0 1 0 2o1 u1 2o1 u4 2o1 u2
ð15Þ
n P 1:
4u1 o1 u3 4u2 o1 u3 2u5 0 0 2o1 u3
1 2u3 2u4 C C 0 C C; 0 A 0
which satisfies the following J1 L ¼ L J1 ¼ J2 ;
J2 L ¼ L J2 ¼ J3 ;
dHð1; 3n þ 3Þ dH ð2; 3n þ 2Þ ¼L ; du du ð16Þ
dHð2; 3n þ 2Þ dH ð3; 3n þ 1Þ ¼L ; du du dHð3; 3n þ 1Þ dH ð1; 3nÞ ¼L : du du
Hence, (16) illustrates that L is a cycle operator. J1 L ¼ L J1 ¼ J2 implies (15) is Liouville integrable. We can prove arbitrary linear combination of fJ1 ; J2 ; J3 g is still a symplectic operator. In fact, let J ¼ c1 J1 þ c2 J2 þ c3 J3 ;
g ¼ ðg1 ; g2 ; g3 ; g4 ; g5 ; g6 ÞT ;
f ¼ ðf1 ; f2 ; f3 ; f4 ; f5 ; f6 ÞT ;
h ¼ ðh1 ; h2 ; h3 ; h4 ; h5 ; h6 ÞT ;
where c1 , c2 , c3 are arbitrary constants. 0 0 1 1 1 1 f f 2 4 2 2 B B C C 12 f3 12 f1 B 1 B C C 1 C þ c2 B u5 f4 u4 f5 C f f þ u f Jf ¼ c1 B 2 3x 2 5 2 B 21 B C C @ f1 þ 1 f4x þ u1 f5 A @ u5 f3 u3 f5 A 2 2 u2 f3 u1 f4 12 f5x u4 f3 þ u3 f4 1 01 1 1 f þ 2u2 o u2 f1 þ 2u2 o u1 f2 þ 2u2 o1 u4 f3 þ 2u2 o1 u3 f4 u5 f4 u4 f5 2 1x B 2u1 o1 u2 f1 1 f2x þ 2u1 o1 u1 f2 þ 2u1 o1 u4 f3 þ u5 f3 þ 2u1 o1 u3 f4 u3 f5 C C B 2 C þ c3 B 2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2u4 o1 u3 f4 C B 1 1 1 1 A @ 2u3 o u2 f1 þ u5 f1 þ 2u3 o u1 f2 þ 2u3 o u4 f3 þ 2u3 o u3 f4 u4 f1 þ u3 f2 ¼ ðF1 ; F2 ; F3 ; F4 ; F5 ÞT ;
450
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
0 B B B B 0 J ½Jf g ¼ B B B @
0 0 c1 F2 g5 c1 F1 g5
1
0 1 0 C C B C 0 C B C C B C þ B c2 F5 g4 c2 F4 g5 C C C @ c2 F5 g3 c2 F3 g5 A C A c2 F4 g3 þ c2 F3 g4
c1 F2 g3 c1 F1 g4 0 1 2c3 F2 o1 F2 g1 þ 2c3 F2 o1 F1 g2 þ 2c3 F2 o1 F4 g3 þ 2c3 F2 o1 F3 g4 F5 g4 c3 F4 g5 B C B 2c3 F1 o1 F2 g1 þ 2c3 F1 o1 F1 g2 þ 2c3 F1 o1 F4 g3 þ c3 F5 g3 þ 2c3 F1 o1 F3 g4 c3 F3 g5 C B C B C þB 2c3 F4 o1 F2 g1 þ 2c3 F4 o1 F1 g2 c3 F5 g2 þ 2c3 F4 o1 F4 g3 þ 2c3 F4 o1 F3 g4 C: B C B C 1 1 1 1 2c3 F3 o F2 g1 þ c3 F5 g1 þ 2c3 F3 o F1 g2 þ 2c3 F3 o F4 g3 þ 2c3 F3 o F3 g4 @ A c3 F4 g1 þ c3 F3 g2 0
ðJ ½Jf g; hÞ ¼ h1 fc3 F5 g4 c3 F4 g5 g þ h2 fc3 F5 g3 c3 F3 g5 g þ h3 fc1 F2 g5 c2 F5 g4 c2 F4 g5 c3 F3 g2 g þ h4 fc1 F1 g5 þ c2 F5 g3 c2 F3 g5 þ c3 F5 g1 g þ h5 fc1 F2 g3 c1 F1 g4 þ c2 F4 g3 þ c2 F3 g4 þ c3 F4 g1 þ c3 F3 g2 g þ 2c3 h1 F2 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h2 F1 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h3 F4 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h4 F3 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ ¼ F1 fc1 ðh4 g5 h5 g4 Þg þ F2 fc1 ðh3 g5 h5 g3 Þg þ F3 fc2 h5 g4 þ c3 h5 g2 c3 h2 g5 c2 h4 g5 g þ F4 fc2 h5 g3 þ c3 h5 g1 c3 h1 g5 c2 h3 g5 g þ F5 fc3 h2 g3 c3 h1 g4 c2 h3 g4 c3 h3 g2 þ c2 h4 g3 þ c3 h4 g1 g þ 2c3 ðh1 F2 þ h2 F1 þ h3 F4 þ h4 F3 Þo1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 g 1 1 1 1 1 f4 ðh4 g5 h5 g4 Þ f3 ðh3 g5 h5 g3 Þ þ c1 c2 f2 ðh4 g5 h5 g4 Þ f1 ðh3 g5 h5 g3 Þ þ ðf2 f3x ¼ c21 2 2 2 2 2 1 1 þ u2 f5 Þðh5 g4 h4 g5 Þ þ ð f1 þ f4x þ u1 f5 Þðh5 g3 h3 g5 Þ þ u2 f3 u1 f4 f5x ðh4 g3 h3 g4 Þ 2 2 1 f2x þ 2u1 o1 u2 f1 þ 2u1 o1 u1 f2 þ 2u1 o1 u4 f3 þ 2u1 o1 u3 f4 þ u5 f3 u3 f5 ðh3 g5 h5 g3 Þ þ c1 c3 2 1 1 1 1 f2 f3x þ u2 f5 ðh5 g2 h2 g5 Þ þ f1 þ f4x þ u1 f5 ðh1 g1 h1 g5 Þ þ u2 f3 u1 f4 þ 2 2 2 2 1 f5x ðh2 g3 h3 g2 þ h4 g1 h1 g4 Þ þ c2 c3 fð2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2 2u4 o1 u3 f4 Þðh5 g4 h4 g5 Þ þ ðu5 f4 u4 f5 Þðh5 g2 h2 g5 Þ þ ðh5 g3 h3 g5 Þð2u4 o1 u2 f1 þ u5 f1 þ 2u3 o1 u1 f2 þ 2u3 o1 u4 f3 þ 2u3 o1 u3 f4 Þ þ ðu5 f3 u3 f5 Þðh5 g1 h1 g5 Þ þ ðu4 f1 þ u3 f2 Þðh4 g3 h3 g4 Þ þ ðu4 f3 þ u3 f4 Þðh2 g3 h3 g2 þ h4 g1 h1 g4 Þg þ c22 fðu5 f4 u4 f5 Þðh5 g4 h4 g5 Þ þ ðu5 f3 u3 f5 Þðh5 g3 h3 g5 Þ þ ðu4 f3 þ u3 f4 Þðh4 g3 h3 g4 Þg þ c23 fðh5 g2 h2 g5 Þð2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2u4 o1 u3 f4 Þ þ ðh5 g1 h1 g5 Þð2u4 o1 u2 f1 þ u5 f1 þ 2u3 o1 u1 f2 þ 2u3 o1 u4 f3 þ 2u3 o1 u3 f4 Þ þ ðu4 f1 þ u3 f2 Þðu2 g3 h3 g2 þ h4 g1 h1 g4 Þg þ fc1 c3 ðh2 f4 h1 f3 2h3 u2 f3 2h3 u1 f4 h3 f5x f1 h4 þ h4 f4x þ h4 u1 f5 Þ þ c2 c3 ðh2 f2 h1 f1 þ 2h3 u4 f3 þ 2h3 u3 f4 þ 2h4 u5 f3 2h4 u3 f5 Þ 1 þ 2c23 ð2h1 u1 o1 u2 f1 h1 f2x þ 2h1 u1 o1 u1 f2 þ 2h1 u1 o1 u4 f3 þ h1 u5 f3 þ 2h1 u1 o1 u3 f4 h1 u3 f5 2 þ 2h3 u3 o1 u2 f1 þ h3 u5 f1 þ 2h3 u3 o1 u1 f2 þ 2h3 u3 o1 u4 f3 þ 2h3 u3 o1 u3 f4 þ 2h4 u4 o1 u2 f1 þ 2h4 u4 o1 u2 f1 þ 2h4 u4 o1 u1 f2 h4 u5 f2 þ 2h4 u4 o1 u4 f3 2h4 u4 o1 u3 f4 Þgo1 fc1 c3 ðg2 f4 g1 f3 2g3 u2 f3 2g3 u1 f4 g3 f5x g4 f1 þ g4 f4x þ g4 u1 f5 Þ þ c2 c3 ðg2 f2 g1 f1 þ 2g3 u4 f3 þ 2g3 u3 f4 þ 2g4 u5 f3 2g4 u3 f5 Þ þ 2c23 ð2g1 u1 o1 u2 f1 ð1=2Þg1 f2x þ 2g1 u1 o1 u1 f2 þ 2g1 u1 o1 u4 f3 þ g1 u5 f3 þ 2g1 u1 o1 u3 f4 g1 u3 f5 þ 2g3 u3 o1 u2 f1 þ g3 u5 f1 þ 2g3 u3 o1 u1 f2 þ 2g3 u3 o1 u4 f3 þ 2g3 u3 o1 u3 f4 þ 2g4 u4 o1 u2 f1 þ 2g4 u4 o1 u1 f2 g4 u5 f2 þ 2g4 u4 o1 u4 f3 2g4 u4 o1 u3 f4 Þg:
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
451
Thus, ðJ 0 ½Jf g; hÞ þ ðJ 0 ½Jgh; f Þ þ ðJ 0 ½Jhf ; gÞ ¼
o ox
1 c1 c2 ðf3 h4 g5 f3 h5 g4 þ f4 h5 g3 f4 h3 g5 þ f5 h3 g4 f5 h4 g3 Þ 2
1 þ c1 c3 ðf2 h5 g3 f2 h3 g5 þ f3 h2 g5 f3 h5 g2 þ f4 h5 g1 f4 h1 g5 þ f5 h3 g2 2 f5 h2 g3 þ f5 h1 g4 f5 h4 g1 Þ ;
which means that fJ1 ; J2 ; J3 g is also a Hamiltonian operator pair. Therefore, (15) is a tri-Hamiltonian structure of the system (7). According to the method in Refs. [8,9], the conjugate operator L of L is a hereditary symmetry.
3. Some reduction cases Case 1: Taking u3 ¼ u4 ¼ u5 ¼ 0 in (7), we have u1
utn ¼
u2
! ¼ tn
o 2
þ 2u2 o1 u2
2u2 o1 u1
2u1 o1 u2
o2 þ 2u1 o1 u1
!
4u1 o1 u2
o 4u1 o1 u1
o þ 4u2 o1 u2
4u2 o1 u1
!
2að0; nÞ
!
2bð0; nÞ
ð17Þ
which is a similar result in Ref. [10]., called NLS–MKdV system. When n ¼ 1 in (17), a nonlinear evolution equation is presented as (
u1t1 ¼ 2aðu1xxx 6u21 u1x þ 6u1x u22 Þ;
ð18Þ
u2t1 ¼ 2aðu2xxx 6u21 u2x þ 6u2x u22 Þ: Especially, taking u1 ¼ iu2 , u2 ¼ T , t1 ¼ t in (18) gives the well-known MKdV equation Tt ¼ 2aðTxxx þ 12T 2 Tx Þ:
ð19Þ
Case 2: Taking u1 ¼ u2 ¼ u5 ¼ 0, n ¼ 1 in (7) engenders (
u3t1 ¼ 2au3xx ; u4t1 ¼ 2au4xx ;
which is just the celebrated heat-conduction equation. Due to cases 1 and 2, we call the system (7) or (15) a MKdV-H system.
4. An expanding integrable system of the MKdV-H system Consider a Lie algebra A1 as follows
1 0 0 1 0 1 ; ~e2 ¼ ; ~e3 ¼ ; ~e4 ¼ 2~e2 ; 0 1 1 0 1 0 ½~e1 ; ~e4 ¼ 4~e3 ; ½~e2 ; ~e3 ¼ 2~e1 ; ½~e3 ; ~e4 ¼ 4~e1 ; ½~e2 ; ~e4 ¼ 0:
~e1 ¼
½~e1 ; ~e2 ¼ 2~e3 ;
½~e1 ; ~e3 ¼ 2~e2 ; ð20Þ
452
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
Let ei ! ai ~ei , i ¼ 1, 2, 3, 4, then a set of algebraic equations are given a3 ¼ 2; a1 a2
2a4 ¼ 1; a1 a3
a1 ¼ 2; a2 a3
a1 ¼2 a3 a4
ð21Þ
which has a solution a1 ¼ a3 ¼ 1, a2 ¼ a4 ¼ 12. Hence, we present a subalgebra of Lie algebra A1 : 1
0
!
e1 ¼
e2 ¼
; 0 1
½e1 ; e2 ¼ e3 ;
0
1 2
1
!
0 ;
1
½e1 ; e3 ¼ 2e4 ;
1
!
e3 ¼
0 1
1
0 ½e1 ; e4 ¼ 2e3 ;
!
e4 ¼
; 0
; 1 0
½e2 ; e3 ¼ e1 ;
½e2 ; e4 ¼ 0;
½e3 ; e4 ¼ 2e1 :
ð22Þ
Taking 0 B e1 ði; nÞ ¼ @
k3nþi 0
0 B e3 ði; nÞ ¼ @
0 k
3nþi
0
k3nþi
k3nþi
0
½e1 ði; mÞ; e2 ðj; nÞ ¼
½e1 ði mÞ; e3 ðj; nÞ ¼
½e1 ði; mÞ; e4 ðj; nÞ ¼
½e2 ði; mÞ; e3 ðj; nÞ ¼
½e3 ði; mÞ; e4 ðj; nÞ ¼
1
0
C A;
1B e2 ði; nÞ ¼ @ 2
1
0
C A;
B e4 ði; nÞ ¼ @
8 > < e3 ði þ j; m þ nÞ;
3nþi
k
0
0
k3nþi
k3nþi
0
i þ j P 3;
i þ j P 3;
ð23Þ
i þ j P 3;
i þ j P 3;
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 1
½e2 ði; mÞ; e4 ðj; nÞ ¼ 0;
C A;
i þ j < 3;
> : e ði þ j 3; m þ n þ 1Þ; 1 8 > < 2e1 ði þ j; m þ nÞ;
1
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 3 8 > < e1 ði þ j; m þ nÞ;
C A;
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 4 8 > < 2e3 ði þ j; m þ nÞ;
1
i þ j < 3;
> : e ði þ j 3; m þ n þ 1Þ; 3 8 > < 2e4 ði þ j; m þ nÞ;
k3nþi
0
i þ j P 3;
deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4:
Remark 1. Here we take e4 ¼ 2e2 to match (23) with loop algebra (1). In what follows, we expand loop algebra (23) into e so that using the approach proposed in Refs. [11–13] obtains integrable couplings, a high-dimensional loop algebra G, of course, a kind of expanding integrable model of the MKdV-H system. For the sake of convenience, the bars in (23) are moved away.
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
453
e From (23), we construct the following high-dimension loop algebra G 0 3nþi 1 0 1 0 k 0 k3nþi 0 0 0 k3nþi 0 1 B C B C B e1 ði; nÞ ¼ @ 0 0 0 A; e3 ði; nÞ ¼ @ k3nþi 0 k3nþi 0 A; e2 ði; nÞ ¼ @ k3nþi 2 0 0 0 0 0 0 0 0 0 1 0 1 0 3nþi 3nþi 1 0 0 0 0 k 0 0 k 0 B B C B C 3nþi C ði; nÞ ¼ ði; nÞ ¼ 0 0 k e4 ði; nÞ ¼ @ k3nþi ; e ; e @ A; @0 0 0 0A 0 A 5 6 0 ½e1 ði; mÞ; e2 ðj; nÞ ¼
0 0 e3 ði þ j; m þ nÞ;
0 0 i þ j < 3;
0
e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e4 ði þ j; m þ nÞ; i þ j < 3;
2e4 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e3 ði þ j; m þ nÞ; i þ j < 3;
2e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e1 ði þ j; m þ nÞ; i þ j < 3;
e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e1 ði þ j; m þ nÞ; i þ j < 3;
½e1 ði; mÞ; e3 ðj; nÞ ¼ ½e1 ði; mÞ; e4 ðj; nÞ ¼ ½e2 ði; mÞ; e3 ðj; nÞ ¼ ½e3 ði; mÞ; e4 ðj; nÞ ¼
2e1 ði þ j 3; m þ n þ 1Þ;
0 0
1
C 0 A; 0
0
i þ j P 3;
½e2 ði; mÞ; e4 ðj; nÞ ¼ 0; e5 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e5 ðj; nÞ ¼ e5 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ( 1 e6 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e5 ðj; nÞ ¼ 21 e ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2 6 e6 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e5 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e6 ði þ j; m þ nÞ; i þ j < 3; ½e4 ði; mÞ; e5 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e6 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e6 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ( 1 e5 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e6 ðj; nÞ ¼ 21 e ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2 5 e5 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e6 ðj; nÞ ¼ ½e4 ði; mÞ; e6 ðj; nÞ ¼ e5 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ½e5 ði; mÞ; e6 ðj; nÞ ¼ 0;
0
ð24Þ
deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4; 5; 6:
e 1 ¼ spanfe1 ði; nÞ; e2 ði; nÞ; e3 ði; nÞ; e4 ði; nÞ ¼ 2e2 ði; nÞg, G e 2 ¼ spanfe5 ði; nÞ; e6 ði; nÞg, then Set G e 2; ½G e ¼G e 1 þ_ G e 1; G e 2 G e2 G
ð25Þ
In terms of (24), let /x ¼ U /; / ¼ ð/1 ; /2 ; /3 ÞT ; U ¼ e2 ð1; 0Þ þ u1 e1 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e1 ð2; 1Þ þ u4 e3 ð2; 1Þ þ u5 e4 ð2; 1Þ þ u6 e5 ð0; 0Þ þ u7 e5 ð2; 1Þ þ u8 e6 ð0; 0Þ þ u9 e6 ð2; 1Þ; V ¼
X mP0
ð26Þ
! 2 X ðaði; mÞe1 ði; mÞ þ bði; mÞe3 ði; mÞ þ cði; mÞe4 ði; mÞ þ dði; mÞe5 ði; mÞ þ f ði; mÞe6 ði; mÞÞ ; i¼0
454
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
solving the stationary zero-curvature equation V x ¼ ½U ; V
ð27Þ
gives bð2; m þ 1Þ ¼ ax ð0; mÞ þ 2u2 cð0; mÞ þ 2u4 cð1; mÞ 2u5 bð1; mÞ; að2; m þ 1Þ ¼ bx ð0; mÞ þ 2u1 cð0; mÞ þ 2u3 cð1; mÞ 2u5 að1; mÞ; cx ð0; mÞ ¼ 2u1 bð0; mÞ 2u2 að0; mÞ þ 2u3 bð1; mÞ 2u4 að1; mÞ; bð0; mÞ ¼ ax ð1; mÞ þ 2u2 cð1; mÞ þ 2u4 cð2; mÞ 2u5 bð2; mÞ; að0; mÞ ¼ bx ð1; mÞ þ 2u1 cð1; mÞ þ 2u3 cð2; mÞ 2u5 að2; mÞ; cx ð1; mÞ ¼ 2u1 bð1; mÞ 2u2 að1; mÞ þ 2u3 bð2; mÞ 2u4 að2; mÞ; bð1; m þ 1Þ ¼ ax ð2; m þ 1Þ þ 2u2 cð2; m þ 1Þ þ 2u4 cð0; mÞ 2u5 bð0; mÞ; að1; m þ 1Þ ¼ bx ð2; m þ 1Þ þ 2u1 cð2; m þ 1Þ þ 2u3 cð0; mÞ 2u5 að0; mÞ; cx ð2; m þ 1Þ ¼ 2u1 bð2; m þ 1Þ 2u2 að2; m þ 1Þ þ 2u3 bð0; mÞ 2u4 að0; mÞ; 1 dx ð0; mÞ ¼ f ð2; m þ 1Þ þ u1 dð0; mÞ þ u2 f ð0; mÞ þ u3 dð1; mÞ þ u4 f ð1; mÞ þ u5 f ð1; mÞ 2 u6 að0; mÞ u7 að1; mÞ u8 cð0; mÞ u8 bð0; mÞ u9 bð1; mÞ u9 cð1; mÞ; 1 fx ð0; mÞ ¼ dð2; m þ 1Þ u1 f ð0; mÞ u2 dð0; mÞ u3 f ð1; mÞ u4 dð1; mÞ þ u5 dð1; mÞ 2 þ u6 bð0; mÞ u6 cð0; mÞ þ u7 bð1; mÞ u7 cð1; mÞ þ u8 að1; mÞ þ u9 að0; mÞ; 1 dx ð1; mÞ ¼ f ð0; mÞ þ u1 dð1; mÞ þ u2 f ð1; mÞ þ u3 dð2; mÞ þ u4 f ð2; mÞ þ u5 f ð2; mÞ 2 u6 að1; mÞ u7 að2; mÞ u8 bð1; mÞ u8 cð1; mÞ u9 bð2; mÞ u9 cð2; mÞ; 1 fx ð1; mÞ ¼ dð0; mÞ u1 f ð1; mÞ u2 dð1; mÞ u3 f ð2; mÞ u4 dð2; mÞ þ u5 dð2; mÞ 2 þ u6 bð1; mÞ u6 cð1; mÞ þ u7 bð2; mÞ u7 cð2; mÞ þ u8 að1; mÞ þ u9 að2; mÞ; 1 dx ð2; m þ 1Þ ¼ f ð1; m þ 1Þ þ u1 dð2; m þ 1Þ þ u2 f ð2; m þ 1Þ þ u3 dð0; mÞ þ u4 f ð0; mÞ 2 þ u5 f ð0; mÞ u6 að2; m þ 1Þ u7 að0; mÞ u8 bð2; m þ 1Þ u8 cð2; m þ 1Þ u9 bð0; mÞ u9 cð0; mÞ; 1 fx ð2; m þ 1Þ ¼ dð1; m þ 1Þ u1 f ð2; m þ 1Þ u2 dð2; m þ 1Þ u3 f ð0; mÞ u4 dð0; mÞ þ u5 dð0; mÞ 2 þ u6 bð2; m þ 1Þ u6 cð2; m þ 1Þ þ u7 bð0; mÞ u7 cð0; mÞ þ u8 að2; m þ 1Þ þ u9 að0; mÞ; cð0; 0Þ ¼ a;
að0; 0Þ ¼ bð0; 0Þ ¼ cð1; 0Þ ¼ bð1; 0Þ ¼ að1; 0Þ ¼ cð2; 0Þ ¼ bð2; 0Þ ¼ að2; 0Þ ¼ dð0; 0Þ ¼ f ð0; 0Þ ¼ dð1; 0Þ ¼ f ð1; 0Þ ¼ 0;
dð2; 1Þ ¼ 2au6 ; f ð1; 1Þ ¼ 4aðu6x þ u9 Þ;
að2; 1Þ ¼ 2au1 ;
bð2; 1Þ ¼ 2au2 ;
cð2; 1Þ ¼ 0;
f ð2; 1Þ ¼ 2au8 ;
dð1; 1Þ ¼ 4aðu8x þ u6 Þ: ð28Þ
Note ðnÞ
Vþ ¼
n X 2 X ðaði; mÞe1 ði; n mÞ þ bði; mÞe3 ði; n mÞ þ cði; mÞe4 ði; n mÞ þ dði; mÞe5 ði; n mÞ m¼0 i¼0
þ f ði; mÞe6 ði; n mÞÞ; ðnÞ
ðnÞ
V ¼ k3n V V þ ; then (27) can be written as ðnÞ
ðnÞ
ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ V x ½U ; V :
ð29Þ
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
455
1 1 ðnÞ ðnÞ ðnÞ ðnÞ V þx þ ½U ; V þ ¼ Vþx þ ½U ; Vþ dð2; n þ 1Þe6 ð0; 0Þ f ð2; n þ 1Þe5 ð0; 0Þ þ ½dx ð2; n þ 1Þ 2 2 1 þ f ð1; n þ 1Þ þ u1 dð2; n þ 1Þ þ u2 f ð2; n þ 1Þ u6 að2; n þ 1Þ u8 bð2; n þ 1Þ 2 1 u8 cð2; n þ 1Þe5 ð2; 1Þ þ ½fx ð2; n þ 1Þ þ dð1; n þ 1Þ þ u1 f ð2; n þ 1Þ þ u2 dð2; n þ 1Þ 2 u6 bð2; n þ 1Þ þ u6 cð2; n þ 1Þ u8 að2; n þ 1Þe6 ð2; 1Þ:
ð30Þ
Similar to (5), we find that
Let V system
ðnÞ
¼V
ðnÞ þ ,
then the zero-curvature equation U t V
ðnÞ x
þ ½U ; V
ðnÞ
¼ 0 brings about the following Lax integrable
1 1 0 bð2;n þ 1Þ u1 Bu C B C að2;n þ 1Þ B 2C B C B C B C B u3 C B C ax ð2; n þ 1Þ bð1; n þ 1Þ þ 2u2 cð2; n þ 1Þ B C B C B C B C B u4 C B C bx ð2; n þ 1Þ að1; n þ 1Þ þ 2u1 cð2; n þ 1Þ B C B C B C B C cx ð2;n þ 1Þ þ 2u1 bð2; n þ 1Þ 2u2 að2; n þ 1Þ ut ¼ B u5 C ¼ B C B C B C 1 Bu C B C f ð2; n þ 1Þ B 6C B C 2 B C B C B u7 C B dx ð2;n þ 1Þ 12 f ð1;n þ 1Þ u1 dð2;n þ 1Þ u2 f ð2; n þ 1Þ þ u6 að2;n þ 1Þ þ u8 bð2; n þ 1Þ þ u8 cð2; n þ 1Þ C B C B C B C B C 1 dð2;n þ 1Þ @ u8 A @ A 2 0
u9 0
0 B 0 B B 0 B 1 B B 2 ¼B B 0 B 0 B B 0 B @ 0 0 0
t
0 0 1 2
0 0 0 0 0 0
fx ð2; n þ 1Þ 12 dð1;n þ 1Þ u1 f ð2; n þ 1Þ þ u2 dð2; n þ 1Þ u6 bð2;n þ 1Þ u6 cð2;n þ 1Þ u8 að2;n þ 1Þ 1 0 1 0 2að1; n þ 1Þ 2að1;n þ 1Þ 1B C C B 1 0 0 0 0 0 B 2bð1;n þ 1Þ C 0 B 2bð1;n þ 1Þ C 2 C B C B 1 C 2 0 0 0 0 0 0 CB 2að2; n þ 1Þ C B 2að2;n þ 1Þ C C B C B o2 0 u2 0 0 0 0C C B C B CB 2bð2;n þ 1Þ C B 2bð2;n þ 1Þ C o C 0 u 0 0 0 0 C B C B 1 2 C C B C B u2 u1 o2 0 0 0 0C B 2cð2; n þ 1Þ C ¼ J1 B 2cð2; n þ 1Þ C C C B C B 1 CB 0 0 0 0 0 0 C C B 2 CB f ð2;n þ 1Þ C B f ð2; n þ 1Þ C u6 u8 u8 1 C 2 2 2 u2 o u1 2 0 CB C C B C B B dð2; n þ 1Þ C 1 0 0 0 0 0 0 AB dð2; n þ 1Þ C C B 2 C B C B u28 u26 u26 o þ u1 u2 0 12 @ f ð1;n þ 1Þ A @ f ð1; n þ 1Þ A dð1; n þ 1Þ bð2;n þ 1Þ
B B að2;n þ 1Þ B B B 2u4 cð0; nÞ þ 2u5 bð0;nÞ B B B 2u3 cð0; nÞ þ 2u5 að0;nÞ B B ¼B 2u 3 bð0;nÞ þ 2u4 að0; nÞ B B 1 B f ð2;n þ 1Þ 2 B B B u3 dð0; nÞ þ u4 f ð0;nÞ þ u5 f ð0; nÞ u7 að0; nÞ u9 bð0; nÞ u9 cð0; nÞ B B 1 B dð2;n þ 1Þ @ 2
1
dð1; n þ 1Þ
C C C C C C C C C C C C C C C C C C C C A
u3 f ð0;nÞ u4 dð0;nÞ þ u5 dð0;nÞ þ u7 bð0;nÞ u7 cð0; nÞ þ u9 að0;nÞ 0 0 1 1 10 1 0 12 0 0 0 0 0 0 0 2að2; n þ 1Þ 2að0;nÞ 2að2;n þ 1Þ B B B 1 C C CB C B 2bð2 n þ 1Þ C B 2bð0; nÞ C B 0 0 B C 0 0 0 0 0 0C B B B 2 C C CB 2bð2 n þ 1Þ C B B B C C CB C B 2að0; nÞ C B 2að1;nÞ C B 0 0 0 u5 u4 0 CB 2að0;nÞ C 0 0 0 B B B C C CB C B B B C C CB C B 2bð0;nÞ C B 2bð1; nÞ C B 0 0 u5 B C 0 u3 0 0 0 0C B B B C C CB 2bð0; nÞ C B B B C C CB C B B C C; C B C 0 0 u u 0 0 0 0 0 ¼B 2cð0; nÞ 2cð1;nÞ ¼ J 2cð0;nÞ ¼ J 4 3 2B 3B B C C CB C B B B C C C B C 1 B B B 0 0 0 C C C B C 0 0 0 0 0 f ð2;n þ 1Þ f ð2; n þ 1Þ f ð2; n þ 1Þ 2 B B B C C CB C B B B C C C B C B B dð0;nÞ C B 0 0 u7 u9 u9 0 C B C u3 u4 þ u5 0 C dð0; nÞ dð0;nÞ B B B C C CB C 2 2 2 B B B C C CB C 1 CB B B dð2; n þ 1Þ C B 0 0 0 C C 0 0 0 0 0 f ð0;nÞ f ð0; nÞ @ @ @ A A A 2 A@ 0 0 u29 u27 u27 0 u5 u4 u3 0 dð2; n þ 1Þ f ð0; nÞ dð2;n þ 1Þ 0
ð31Þ
456
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
where 0
þ 2u2 o1 u2 B 2u o1 u B 1 2 B 2u o1 u B 4 2 B B 2u3 o1 u2 þ u5 B u4 J3 ¼ B B 0 B B u7 B 2 þ u9 o1 u2 B 0 @ u9 u7 o1 u2 2 o 2
2u2 o1 u1 o2 þ 2u1 o1 u1 2u4 o1 u1 u5 2u3 o1 u1 u3 0 u9 1 þ u 9 o u1 2 0 u7 o1 u1
2u2 o1 u4 2u1 o1 u4 þ u5 2u4 o1 u4 2u3 o1 u4 0 0 u9 o1 u4 0 u7 o1 u4
2u2 o1 u3 u5 2u1 o1 u3 2u4 o1 u3 2u3 o1 u3 0 0 u9 o1 u3 0 u7 o1 u3 0
u4 u3 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 1 0 2 0 u3 0 0 0 u5 u4
1 0 0 0 0 C C 0 0 C C C 0 0 C C 0 0 C: C 0 0 C C 0 u4 þ u5 C C 1 0 A 2 0 u3
According to the definition of integrable couplings [14–16], we conclude that (31) is an integrable coupling of the system (7). Of course, (31) is also a type of expanding integrable model of (7), since when taking u6 ¼ u7 ¼ u8 ¼ u9 ¼ 0, (31) reduces to (7). Remark 2. Taking u3 ¼ u4 ¼ u5 ¼ 0 in (31), an integrable coupling of AKNS hierarchy is presented just like the result in Ref. [11]. There exists an open problem. The system (31) is only Lax integrable, how can we construct a proper loop algebra to get Liouville integrable coupling systems? In addition, we may extend loop algebra (1) into a general case, i.e. replacing k3nþi with kNnþi , where N ¼ 4; 5; 6; . . .. It follows Tu scheme that some Liouville integrable hierarchies with N Hamiltonian structures could be worked out. It is special worth pointing out it is a mistake that concluding the integrable coupling was Liouville integrable in Ref. [13], because the Hamiltonian structure of the integrable coupling of BPT hierarchy is wrong. Two J operators in Ref. [17] are not Hamiltonian operators.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Wadati M. J Phys Soc Jpn 1973;34:1289. Tu G. J Math Phys 1989;30(2):330. Gu C et al. Soliton theory and its application. Zhejiang Publishing House of Science and Technology; 1990. Fan E. J Math Phys 2000;41(11):7769. Zhou Z. J Phys Soc Jpn 2002;71(8):1857. Hu X. J Phys A: Math Gen 1994;27:2497. Magri F. J Math Phys 1978;19:1156. Ma W. J Phys A: Math Gen 1990;23:2707. Guo F. J Syst Sci Math Sci 2002;22(1):36. Guo F. Acta Math Sini 1997;40(6):801. Guo F, Zhang Y. Acta Phys Sini 2002;51(5):951. Zhang Y, Zhang H. J Math Phys 2002;43(1):466. Zhang Y et al. Phys Lett A 2002;299(5):543. Lakshmanan, Tamizhamani KM. J Math Phys 1985;26:1189. Ma W, Fuchssteiner B. Phys Lett A 1996;213:49. Ma W, Fuchssteiner B. Chaos, Solitons & Fractals 1996;8:1227. Zhang Y. Phys Lett A 2003;310:19.
A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations Yufeng Zhang a b
a,b,*
, Xixiang Xu
c
Institute of Mathematics, Information School, Shandong University of Science and Technology, Taian 271019, PR China Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, PR China c Department of Basic Courses, Shandong University of Science and Technology, Taian 271019, PR China Accepted 10 December 2003 Communicated by Y. Aizawa
Abstract e 2 is first constructed, which has its own special feature. It follows that a new Liouville A subalgebra of loop algebra A integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the e is constructed by making use of a hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Researching new various integrable systems is of an importance and interest work in soliton theory [1–3]. A great host of integrable hierarchies of evolution equations were obtained such as in [4–8] in recent years. Professor Guo Fukui once deduced some interesting integrable Hamiltonian systems with multi-potential functions by use of subalgebras of e 1 , which could reduce to the famous hierarchies, such as AKNS hierarchy, BPT hierarchy and so on loop algebra A e 2 . That is to say, constructing a new subalgebra [9,10]. In this paper, we expand Guo’s idea to the case of loop algebra A e 2 is devoted to deducing a Liouville integrable system and we prove that it possesses a tri-Hamiltonian of loop algebra A structure. As its reduction cases, the NLS–MKdV hierarchy [10] is presented. Especially, the well-known MKdV equation and the heat-conduction equation are obtained, respectively. Hence, we call the hierarchy a MKdV-H system. It is remarkable that three symplectic operators in the MKdV-H system are constructed directly by the recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. In addition, a high-dimension loop e is constructed by employing a proper scalar transformation. Then using G e and Tu scheme deduces a type of algebra G expanding integrable model of the MKdV-H system. Here the expanding integrable model, as a matter of fact, is an integrable coupling of the MKdV-H system. 2. A trick loop algebra and a corresponding integrable system e 2 as follows Consider a subalgebra of loop algebra A *
Corresponding author. E-mail address: [email protected] (Y. Zhang).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.018
446
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
0
0
B e1 ði; nÞ ¼ @ 0 k3nþi 0 0 B e3 ði; nÞ ¼ @ 0 k3nþi ½e1 ði; mÞ; e2 ðj; nÞ ¼
0 k3nþi
1
0
k3nþi
0
0
1
C C B k3nþi 0 A; 0 A; e2 ði; nÞ ¼ @ 0 0 0 0 0 1 0 3nþi 1 0 0 0 k3nþi k C B C 0 0 A; 0 0 A; e4 ði; nÞ ¼ @ 0 0 0 k3nþi 0 0 e3 ði þ j; m þ nÞ; i þ j < 3; 0 0
e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e4 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e3 ðj; nÞ ¼ 2e4 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e3 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e4 ðj; nÞ ¼ 2e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e1 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e3 ðj; nÞ ¼ e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e1 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e4 ðj; nÞ ¼ 2e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ½e2 ði; mÞ; e4 ðj; nÞ ¼ 0; deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4:
In terms of (1), an isospectral problem is in the following 0 k þ uk5 T 0 /x ¼ U /; kt ¼ 0; / ¼ ð/1 ; /2 ; /3 Þ ; U ¼ @ 4 u1 þ u2 þ u3 þu k
ð1Þ
1 4 0 u1 u2 þ u3 u k A k 0 0 uk5
¼ e2 ð1; 0Þ þ u1 e1 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e1 ð2; 1Þ þ u4 e3 ð2; 1Þ þ u5 e4 ð2; 1Þ:
ð2Þ
Let 0
1 cð0Þ þ cð1Þk þ cð2Þk2 0 að0Þ bð0Þ þ ðað1Þ bð1ÞÞk þ ðað2Þ bð2ÞÞk2 A; V ¼@ 0 0 0 að0Þ þ bð0Þ þ ðað1Þ þ bð1ÞÞk þ ðað2Þ þ bð2ÞÞk2 0 cð0Þ cð1Þk cð2Þk2 P P where að0Þ ¼ m P 0 að0; mÞk3m , að1Þ ¼ m P 0 að1; mÞk3m ; . . . Solving the stationary zero-curvature equation Vx ¼ ½U ; V
ð3Þ
yields bð2; m þ 1Þ ¼ ax ð0; mÞ þ 2u2 cð0; mÞ þ 2u4 cð1; mÞ 2u5 bð1; mÞ; að2; m þ 1Þ ¼ bx ð0; mÞ þ 2u1 cð0; mÞ þ 2u3 cð1; mÞ 2u5 að1; mÞ; cx ð0; mÞ ¼ 2u1 bð0; mÞ 2u2 að0; mÞ þ 2u3 bð1; mÞ 2u4 að1; mÞ; bð0; mÞ ¼ ax ð1; mÞ þ 2u2 cð1; mÞ þ 2u4 cð2; mÞ 2u5 bð2; mÞ; að0; mÞ ¼ bx ð1; mÞ þ 2u1 cð1; mÞ þ 2u3 cð2; mÞ 2u5 að2; mÞ; cx ð1; mÞ ¼ 2u1 bð1; mÞ 2u2 að1; mÞ þ 2u3 bð2; mÞ 2u4 að2; mÞ; bð1; m þ 1Þ ¼ ax ð2; m þ 1Þ þ 2u2 cð2; m þ 1Þ þ 2u4 cð0; mÞ 2u5 bð0; mÞ; að1; m þ 1Þ ¼ bx ð2; m þ 1Þ þ 2u1 cð2; m þ 1Þ þ 2u3 cð0; mÞ 2u5 að0; mÞ; cx ð2; m þ 1Þ ¼ 2u1 bð2; m þ 1Þ 2u2 að2; m þ 1Þ þ 2u3 bð0; mÞ 2u4 að0; mÞ; cð0; 0Þ ¼ a; að0; 0Þ ¼ bð0; 0Þ ¼ cð1; 0Þ ¼ bð1; 0Þ ¼ að1; 0Þ ¼ cð2; 0Þ ¼ bð2; 0Þ ¼ að2; 0Þ ¼ 0; bð2; 1Þ ¼ 2au2 ; að2; 1Þ ¼ 2au1 ; cð2; 1Þ ¼ 0; að1; 1Þ ¼ 2aðu2x u3 Þ; bð1; 1Þ ¼ 2aðu1x u4 Þ; cð1; 1Þ ¼ 2aðu21 u22 Þ; að0; 1Þ ¼ 2aðu1xx u4x 2u31 þ 2u1 u22 2u1 u5 Þ; bð0; 1Þ ¼ 2aðu2xx u3x 2u2 u21 þ 2u32 2u2 u5 Þ:
ð4Þ
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
447
Note ðnÞ
Vþ ¼
n X 2 X ðaði; mÞe1 ði; n mÞ þ bði; mÞe3 ði; n mÞ þ cði; mÞe4 ði; n mÞÞ;
ð5Þ
m¼0 i¼0
VðnÞ
3n
¼k V
ðnÞ Vþ ;
ðnÞ
ðnÞ
ðnÞ then we have Vþx þ ½U; Vþ ¼ Vx ½U ; VðnÞ . It is easy to verify that the terms of the left-hand side in (5) are of degree P )1, while the terms of the right-hand side in (5) are of degree 6 0. Therefore, the terms of both sides in (5) are of degree )1,0. It follows that ðnÞ
ðnÞ
Vþx þ ½U ; Vþ ¼ bð2; n þ 1Þe1 ð0; 0Þ þ að2; n þ 1Þe3 ð0; 0Þ þ ½ax ð2; n þ 1Þ þ bð1; n þ 1Þ 2u2 cð2; n þ 1Þe1 ð2; 1Þ þ ½bx ð2; n þ 1Þ þ að1; n þ 1Þ 2u1 cð2; n þ 1Þe3 ð2; 1Þ þ ½cx ð2; n þ 1Þ 2u1 bð2; n þ 1Þ þ 2u2 að2; n þ 1Þe4 ð2; 1Þ: ðnÞ
Taking V ðnÞ ¼ Vþ , then the zero-curvature equation Ut VxðnÞ þ ½U ; V ðnÞ ¼ 0
ð6Þ
admits the Lax integrable system 0
1
u1
0
1
bð2; n þ 1Þ
Bu C B að2; n þ 1Þ B 2C B B C B C B a utn ¼ B u ¼ ð2; n þ 1Þ bð1; n þ 1Þ þ 2u2 cð2; n þ 1Þ x B 3C B B C B @ u4 A @ bx ð2; n þ 1Þ að1; n þ 1Þ þ 2u1 cð2; n þ 1Þ 0
u5
tn
0
0
B 0 B B ¼B B 0 B 1 @2
0 1 2
0
cx ð2; n þ 1Þ þ 2u1 bð2; n þ 1Þ 2u2 að2; n þ 1Þ 1 1 10 0 1 0 2að1; n þ 1Þ 2að1; n þ 1Þ 2 C B B 2bð1; n þ 1Þ C 12 0 0 C C CB 2bð1; n þ 1Þ C B C C CB B o B 2að2; n þ 1Þ C ¼ J1 B 2að2; n þ 1Þ C 2 0 u2 C C C CB B C C CB B o 0 u1 A@ 2bð2; n þ 1Þ A @ 2bð2; n þ 1Þ A 2 0
0 u2
u1
o2 1
2cð2; n þ 1Þ 0 0 12 0 C B B að2; n þ 1Þ C B 12 0 0 B C B B C B ð4ÞB B 2u4 cð0; nÞ þ 2u5 bð0; nÞ C ¼ B 0 0 0 C B B @ 2u3 cð0; nÞ þ 2u5 að0; nÞ A @ 0 0 u5 0
0
C C C C C C A
bð2; n þ 1Þ
0 0 u5 0
2cð2; n þ 1Þ 1 1 10 0 2að2; n þ 1Þ 2að2; n þ 1Þ C B B 2bð2; n þ 1Þ C 0 C C CB 2bð2; n þ 1Þ C B C C CB B B 2að0; nÞ C ¼ J2 B 2að0; nÞ C u4 C C C CB B C C CB B @ 2bð0; nÞ A u3 A@ 2bð0; nÞ A 0
2cð0; nÞ 2cð0; nÞ 2u3 bð0; nÞ þ 2u4 að0; nÞ 0 0 u4 u3 0 1 0 o þ 4u2 o1 u2 að0; nÞ 4u2 o1 u1 bð0; nÞ þ 4u2 o1 u3 þ 2u5 bð1; nÞ þ 4u2 o1 u4 að1; nÞ 2u4 cð1; nÞ C B B o 4u1 o1 u1 bð0; nÞ þ 4u1 o1 u2 að0; nÞ 4u1 o1 u3 bð1; nÞ þ 4u1 o1 u4 þ 2u5 að1; nÞ 2u3 cð1; nÞ C C B C B 1 1 1 1 ð4ÞB C 4u 4 o u1 þ 2u5 bð0; nÞ þ 4u4 o u2 að0; nÞ 4u4 o u3 bð1; nÞ þ 4u4 o u4 að1; nÞ C B C B 1 1 1 1 A @ 4u3 o u1 bð0; nÞ þ 4u3 o u2 þ 2u5 að0; nÞ 4u3 o u3 bð1; nÞ þ 4u3 o u4 að1; nÞ 0
2u3 bð0; nÞ þ 2u4 að0; nÞ o 2
1
þ 2u2 o u2
1
2u2 o u1
B B 2u1 o1 u2 o2 þ 2u1 o1 u1 B B 1 ¼ B 2u4 o u2 2u4 o1 u1 u5 B B 2u3 o1 u2 þ u5 2u3 o1 u1 @ u4 u3 1 0 2að0; nÞ B 2bð0; nÞ C C B C B C: ¼ J3 B 2að1; nÞ C B C B @ 2bð1; nÞ A 2cð1; nÞ
2u2 o1 u4
2u2 o1 u3 u5
2u1 o1 u4 þ u5
2u1 o1 u3
1
2u4 o1 u3
1
2u3 o u4
2u3 o1 u3
0
0
2u4 o u4
10 1 2að0; nÞ CB 2bð0; nÞ C u3 C C CB C CB 2að1; nÞ C 0 CB C B CB C 0 C A@ 2bð1; nÞ A u4
0
2cð1; nÞ
ð7Þ
448
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
A direct calculation reads oU oU ¼ 2ðað0Þ þ að1Þk þ að2Þk2 Þ; ¼ 2ðbð0Þ þ bð1Þk þ abð2Þk2 Þ; V; V; ou1 ou2 oU að0Þ oU bð0Þ þ að1Þ þ að2Þk ; V; þ bð1Þ þ bð2Þk ; V; ¼2 ¼ 2 ou3 k ou4 k oU cð0Þ þ cð1Þ þ cð2Þk ; ¼2 V; ou5 k oU 2u4 bð0Þ u5 cð0Þ 2u3 að0Þ V; ¼ cð0Þ u2 cð2Þ þ 2u4 bð2Þ 2u3 að2Þ þ cð1Þk þ cð2Þk2 þ ok k2 2u4 bð1Þ 2u3 að1Þ u5 cð1Þ : þ k Substituting the above formulae into trace identity leads to E1 0D oU V ; ou 1 EC BD B V ; oU C B C ou 2 BD EC d oU C c o c B oU V; ¼k k B V; C: du ok ok B D ou3 E C B C oU B V ; ou C 4 @D EA oU V ; ou 5 Comparing the coefficients of k3n3 in (8) gives
ð8Þ
0
1 2að1; n þ 1Þ B 2bð1; n þ 1Þ C B C d ðcð0; n þ 1Þ u2 cð2; n þ 1Þ þ 2u4 bð2; n þ 1Þ 2u3 að2; n þ 1ÞÞ ¼ ð3n 2 þ cÞB 2að2; n þ 1Þ C B C: du @ 2bð2; n þ 1Þ A 2cð2; n þ 1Þ
Comparing the coefficients of k3n2 , k3n1 in (8) engenders, respectively, as follows 0 1 2að2; n þ 1Þ B 2bð2; n þ 1Þ C B C d ðcð1; n þ 1Þ u5 cð0; nÞ 2u3 að0; nÞ þ 2u4 bð0; nÞÞ ¼ ð3n 1 þ cÞB 2að0; nÞ C B C; du @ 2bð0; nÞ A 2cð0; nÞ 0 1 2að0; nÞ B 2bð0; nÞ C B C d ðcð2; n þ 1Þ þ 2u4 bð1:nÞ 2u3 að1; nÞ u5 cð1; nÞÞ ¼ ð3n þ cÞB 2að1; nÞ C B C: du @ 2bð1; nÞ A 2cð1; nÞ
ð9Þ
ð10Þ
ð11Þ
Inserting the initial values in (4) into (9)–(11) gives c ¼ 0. Thus, the relations (9)–(11) can determine the following three Hamiltonian functions 0 1 8 2að1; n þ 1Þ > > > B C > > B 2bð1; n þ 1Þ C > > B C > > B C > > > dHð1;3nþ3Þ ¼ B 2að2; n þ 1Þ C; > B C > du < B C B C ð12Þ B C 2bð2; n þ 1Þ > @ A > > > > > > 2cð2; n þ 1Þ > > > > > > > : H ð1; 3n þ 3Þ ¼ cð0; n þ 1Þ u2 cð2; n þ 1Þ þ 2u4 bð2; n þ 1Þ 2u3 að2; n þ 1Þ ; 3n þ 2
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
8 0 1 2að2; n þ 1Þ > > > > B 2bð2; n þ 1Þ C > > B C > > B 2að0; nÞ C; ¼ < dHð2;3nþ2Þ du B C @ 2bð0; nÞ A > > > 2cð0; nÞ > > > > > H ð2; 3n þ 2Þ ¼ cð1; n þ 1Þ u5 cð0; nÞ 2u3 að0; nÞ þ 2u4 bð0; nÞ ; : 3n þ 1 8 0 1 2að0; nÞ > > > > B 2bð0; nÞ C > > B C > > C < dHð3;3nþ1Þ ¼B du B 2að1; nÞ C; @ 2bð1; nÞ A > > > 2cð1; nÞ > > > > u cð1; nÞ þ 2u3 að1; nÞ cð2; n þ 1Þ 2u4 bð1; nÞ > : H ð3; 3n þ 1Þ ¼ 5 : 3n
449
ð13Þ
ð14Þ
Therefore, we have utn ¼ J1
dH ð1; 3n þ 3Þ dH ð2; 3n þ 2Þ dH ð3; 3n þ 1Þ ¼ J2 ¼ J3 ¼ ; du du du
From (4), a recurrence operator is generated as follows 0 4u1 o1 u2 o 4u1 o1 u1 4u1 o1 u4 2u5 B o þ 4u2 o1 u2 4u2 o1 u1 4u2 o1 u4 B L¼B 1 0 0 B @ 0 1 0 2o1 u1 2o1 u4 2o1 u2
ð15Þ
n P 1:
4u1 o1 u3 4u2 o1 u3 2u5 0 0 2o1 u3
1 2u3 2u4 C C 0 C C; 0 A 0
which satisfies the following J1 L ¼ L J1 ¼ J2 ;
J2 L ¼ L J2 ¼ J3 ;
dHð1; 3n þ 3Þ dH ð2; 3n þ 2Þ ¼L ; du du ð16Þ
dHð2; 3n þ 2Þ dH ð3; 3n þ 1Þ ¼L ; du du dHð3; 3n þ 1Þ dH ð1; 3nÞ ¼L : du du
Hence, (16) illustrates that L is a cycle operator. J1 L ¼ L J1 ¼ J2 implies (15) is Liouville integrable. We can prove arbitrary linear combination of fJ1 ; J2 ; J3 g is still a symplectic operator. In fact, let J ¼ c1 J1 þ c2 J2 þ c3 J3 ;
g ¼ ðg1 ; g2 ; g3 ; g4 ; g5 ; g6 ÞT ;
f ¼ ðf1 ; f2 ; f3 ; f4 ; f5 ; f6 ÞT ;
h ¼ ðh1 ; h2 ; h3 ; h4 ; h5 ; h6 ÞT ;
where c1 , c2 , c3 are arbitrary constants. 0 0 1 1 1 1 f f 2 4 2 2 B B C C 12 f3 12 f1 B 1 B C C 1 C þ c2 B u5 f4 u4 f5 C f f þ u f Jf ¼ c1 B 2 3x 2 5 2 B 21 B C C @ f1 þ 1 f4x þ u1 f5 A @ u5 f3 u3 f5 A 2 2 u2 f3 u1 f4 12 f5x u4 f3 þ u3 f4 1 01 1 1 f þ 2u2 o u2 f1 þ 2u2 o u1 f2 þ 2u2 o1 u4 f3 þ 2u2 o1 u3 f4 u5 f4 u4 f5 2 1x B 2u1 o1 u2 f1 1 f2x þ 2u1 o1 u1 f2 þ 2u1 o1 u4 f3 þ u5 f3 þ 2u1 o1 u3 f4 u3 f5 C C B 2 C þ c3 B 2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2u4 o1 u3 f4 C B 1 1 1 1 A @ 2u3 o u2 f1 þ u5 f1 þ 2u3 o u1 f2 þ 2u3 o u4 f3 þ 2u3 o u3 f4 u4 f1 þ u3 f2 ¼ ðF1 ; F2 ; F3 ; F4 ; F5 ÞT ;
450
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
0 B B B B 0 J ½Jf g ¼ B B B @
0 0 c1 F2 g5 c1 F1 g5
1
0 1 0 C C B C 0 C B C C B C þ B c2 F5 g4 c2 F4 g5 C C C @ c2 F5 g3 c2 F3 g5 A C A c2 F4 g3 þ c2 F3 g4
c1 F2 g3 c1 F1 g4 0 1 2c3 F2 o1 F2 g1 þ 2c3 F2 o1 F1 g2 þ 2c3 F2 o1 F4 g3 þ 2c3 F2 o1 F3 g4 F5 g4 c3 F4 g5 B C B 2c3 F1 o1 F2 g1 þ 2c3 F1 o1 F1 g2 þ 2c3 F1 o1 F4 g3 þ c3 F5 g3 þ 2c3 F1 o1 F3 g4 c3 F3 g5 C B C B C þB 2c3 F4 o1 F2 g1 þ 2c3 F4 o1 F1 g2 c3 F5 g2 þ 2c3 F4 o1 F4 g3 þ 2c3 F4 o1 F3 g4 C: B C B C 1 1 1 1 2c3 F3 o F2 g1 þ c3 F5 g1 þ 2c3 F3 o F1 g2 þ 2c3 F3 o F4 g3 þ 2c3 F3 o F3 g4 @ A c3 F4 g1 þ c3 F3 g2 0
ðJ ½Jf g; hÞ ¼ h1 fc3 F5 g4 c3 F4 g5 g þ h2 fc3 F5 g3 c3 F3 g5 g þ h3 fc1 F2 g5 c2 F5 g4 c2 F4 g5 c3 F3 g2 g þ h4 fc1 F1 g5 þ c2 F5 g3 c2 F3 g5 þ c3 F5 g1 g þ h5 fc1 F2 g3 c1 F1 g4 þ c2 F4 g3 þ c2 F3 g4 þ c3 F4 g1 þ c3 F3 g2 g þ 2c3 h1 F2 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h2 F1 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h3 F4 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ þ 2c3 h4 F3 o1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 Þ ¼ F1 fc1 ðh4 g5 h5 g4 Þg þ F2 fc1 ðh3 g5 h5 g3 Þg þ F3 fc2 h5 g4 þ c3 h5 g2 c3 h2 g5 c2 h4 g5 g þ F4 fc2 h5 g3 þ c3 h5 g1 c3 h1 g5 c2 h3 g5 g þ F5 fc3 h2 g3 c3 h1 g4 c2 h3 g4 c3 h3 g2 þ c2 h4 g3 þ c3 h4 g1 g þ 2c3 ðh1 F2 þ h2 F1 þ h3 F4 þ h4 F3 Þo1 ðF2 g1 þ F1 g2 þ F4 g3 þ F3 g4 g 1 1 1 1 1 f4 ðh4 g5 h5 g4 Þ f3 ðh3 g5 h5 g3 Þ þ c1 c2 f2 ðh4 g5 h5 g4 Þ f1 ðh3 g5 h5 g3 Þ þ ðf2 f3x ¼ c21 2 2 2 2 2 1 1 þ u2 f5 Þðh5 g4 h4 g5 Þ þ ð f1 þ f4x þ u1 f5 Þðh5 g3 h3 g5 Þ þ u2 f3 u1 f4 f5x ðh4 g3 h3 g4 Þ 2 2 1 f2x þ 2u1 o1 u2 f1 þ 2u1 o1 u1 f2 þ 2u1 o1 u4 f3 þ 2u1 o1 u3 f4 þ u5 f3 u3 f5 ðh3 g5 h5 g3 Þ þ c1 c3 2 1 1 1 1 f2 f3x þ u2 f5 ðh5 g2 h2 g5 Þ þ f1 þ f4x þ u1 f5 ðh1 g1 h1 g5 Þ þ u2 f3 u1 f4 þ 2 2 2 2 1 f5x ðh2 g3 h3 g2 þ h4 g1 h1 g4 Þ þ c2 c3 fð2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2 2u4 o1 u3 f4 Þðh5 g4 h4 g5 Þ þ ðu5 f4 u4 f5 Þðh5 g2 h2 g5 Þ þ ðh5 g3 h3 g5 Þð2u4 o1 u2 f1 þ u5 f1 þ 2u3 o1 u1 f2 þ 2u3 o1 u4 f3 þ 2u3 o1 u3 f4 Þ þ ðu5 f3 u3 f5 Þðh5 g1 h1 g5 Þ þ ðu4 f1 þ u3 f2 Þðh4 g3 h3 g4 Þ þ ðu4 f3 þ u3 f4 Þðh2 g3 h3 g2 þ h4 g1 h1 g4 Þg þ c22 fðu5 f4 u4 f5 Þðh5 g4 h4 g5 Þ þ ðu5 f3 u3 f5 Þðh5 g3 h3 g5 Þ þ ðu4 f3 þ u3 f4 Þðh4 g3 h3 g4 Þg þ c23 fðh5 g2 h2 g5 Þð2u4 o1 u2 f1 þ 2u4 o1 u1 f2 u5 f2 þ 2u4 o1 u4 f3 2u4 o1 u3 f4 Þ þ ðh5 g1 h1 g5 Þð2u4 o1 u2 f1 þ u5 f1 þ 2u3 o1 u1 f2 þ 2u3 o1 u4 f3 þ 2u3 o1 u3 f4 Þ þ ðu4 f1 þ u3 f2 Þðu2 g3 h3 g2 þ h4 g1 h1 g4 Þg þ fc1 c3 ðh2 f4 h1 f3 2h3 u2 f3 2h3 u1 f4 h3 f5x f1 h4 þ h4 f4x þ h4 u1 f5 Þ þ c2 c3 ðh2 f2 h1 f1 þ 2h3 u4 f3 þ 2h3 u3 f4 þ 2h4 u5 f3 2h4 u3 f5 Þ 1 þ 2c23 ð2h1 u1 o1 u2 f1 h1 f2x þ 2h1 u1 o1 u1 f2 þ 2h1 u1 o1 u4 f3 þ h1 u5 f3 þ 2h1 u1 o1 u3 f4 h1 u3 f5 2 þ 2h3 u3 o1 u2 f1 þ h3 u5 f1 þ 2h3 u3 o1 u1 f2 þ 2h3 u3 o1 u4 f3 þ 2h3 u3 o1 u3 f4 þ 2h4 u4 o1 u2 f1 þ 2h4 u4 o1 u2 f1 þ 2h4 u4 o1 u1 f2 h4 u5 f2 þ 2h4 u4 o1 u4 f3 2h4 u4 o1 u3 f4 Þgo1 fc1 c3 ðg2 f4 g1 f3 2g3 u2 f3 2g3 u1 f4 g3 f5x g4 f1 þ g4 f4x þ g4 u1 f5 Þ þ c2 c3 ðg2 f2 g1 f1 þ 2g3 u4 f3 þ 2g3 u3 f4 þ 2g4 u5 f3 2g4 u3 f5 Þ þ 2c23 ð2g1 u1 o1 u2 f1 ð1=2Þg1 f2x þ 2g1 u1 o1 u1 f2 þ 2g1 u1 o1 u4 f3 þ g1 u5 f3 þ 2g1 u1 o1 u3 f4 g1 u3 f5 þ 2g3 u3 o1 u2 f1 þ g3 u5 f1 þ 2g3 u3 o1 u1 f2 þ 2g3 u3 o1 u4 f3 þ 2g3 u3 o1 u3 f4 þ 2g4 u4 o1 u2 f1 þ 2g4 u4 o1 u1 f2 g4 u5 f2 þ 2g4 u4 o1 u4 f3 2g4 u4 o1 u3 f4 Þg:
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
451
Thus, ðJ 0 ½Jf g; hÞ þ ðJ 0 ½Jgh; f Þ þ ðJ 0 ½Jhf ; gÞ ¼
o ox
1 c1 c2 ðf3 h4 g5 f3 h5 g4 þ f4 h5 g3 f4 h3 g5 þ f5 h3 g4 f5 h4 g3 Þ 2
1 þ c1 c3 ðf2 h5 g3 f2 h3 g5 þ f3 h2 g5 f3 h5 g2 þ f4 h5 g1 f4 h1 g5 þ f5 h3 g2 2 f5 h2 g3 þ f5 h1 g4 f5 h4 g1 Þ ;
which means that fJ1 ; J2 ; J3 g is also a Hamiltonian operator pair. Therefore, (15) is a tri-Hamiltonian structure of the system (7). According to the method in Refs. [8,9], the conjugate operator L of L is a hereditary symmetry.
3. Some reduction cases Case 1: Taking u3 ¼ u4 ¼ u5 ¼ 0 in (7), we have u1
utn ¼
u2
! ¼ tn
o 2
þ 2u2 o1 u2
2u2 o1 u1
2u1 o1 u2
o2 þ 2u1 o1 u1
!
4u1 o1 u2
o 4u1 o1 u1
o þ 4u2 o1 u2
4u2 o1 u1
!
2að0; nÞ
!
2bð0; nÞ
ð17Þ
which is a similar result in Ref. [10]., called NLS–MKdV system. When n ¼ 1 in (17), a nonlinear evolution equation is presented as (
u1t1 ¼ 2aðu1xxx 6u21 u1x þ 6u1x u22 Þ;
ð18Þ
u2t1 ¼ 2aðu2xxx 6u21 u2x þ 6u2x u22 Þ: Especially, taking u1 ¼ iu2 , u2 ¼ T , t1 ¼ t in (18) gives the well-known MKdV equation Tt ¼ 2aðTxxx þ 12T 2 Tx Þ:
ð19Þ
Case 2: Taking u1 ¼ u2 ¼ u5 ¼ 0, n ¼ 1 in (7) engenders (
u3t1 ¼ 2au3xx ; u4t1 ¼ 2au4xx ;
which is just the celebrated heat-conduction equation. Due to cases 1 and 2, we call the system (7) or (15) a MKdV-H system.
4. An expanding integrable system of the MKdV-H system Consider a Lie algebra A1 as follows
1 0 0 1 0 1 ; ~e2 ¼ ; ~e3 ¼ ; ~e4 ¼ 2~e2 ; 0 1 1 0 1 0 ½~e1 ; ~e4 ¼ 4~e3 ; ½~e2 ; ~e3 ¼ 2~e1 ; ½~e3 ; ~e4 ¼ 4~e1 ; ½~e2 ; ~e4 ¼ 0:
~e1 ¼
½~e1 ; ~e2 ¼ 2~e3 ;
½~e1 ; ~e3 ¼ 2~e2 ; ð20Þ
452
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
Let ei ! ai ~ei , i ¼ 1, 2, 3, 4, then a set of algebraic equations are given a3 ¼ 2; a1 a2
2a4 ¼ 1; a1 a3
a1 ¼ 2; a2 a3
a1 ¼2 a3 a4
ð21Þ
which has a solution a1 ¼ a3 ¼ 1, a2 ¼ a4 ¼ 12. Hence, we present a subalgebra of Lie algebra A1 : 1
0
!
e1 ¼
e2 ¼
; 0 1
½e1 ; e2 ¼ e3 ;
0
1 2
1
!
0 ;
1
½e1 ; e3 ¼ 2e4 ;
1
!
e3 ¼
0 1
1
0 ½e1 ; e4 ¼ 2e3 ;
!
e4 ¼
; 0
; 1 0
½e2 ; e3 ¼ e1 ;
½e2 ; e4 ¼ 0;
½e3 ; e4 ¼ 2e1 :
ð22Þ
Taking 0 B e1 ði; nÞ ¼ @
k3nþi 0
0 B e3 ði; nÞ ¼ @
0 k
3nþi
0
k3nþi
k3nþi
0
½e1 ði; mÞ; e2 ðj; nÞ ¼
½e1 ði mÞ; e3 ðj; nÞ ¼
½e1 ði; mÞ; e4 ðj; nÞ ¼
½e2 ði; mÞ; e3 ðj; nÞ ¼
½e3 ði; mÞ; e4 ðj; nÞ ¼
1
0
C A;
1B e2 ði; nÞ ¼ @ 2
1
0
C A;
B e4 ði; nÞ ¼ @
8 > < e3 ði þ j; m þ nÞ;
3nþi
k
0
0
k3nþi
k3nþi
0
i þ j P 3;
i þ j P 3;
ð23Þ
i þ j P 3;
i þ j P 3;
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 1
½e2 ði; mÞ; e4 ðj; nÞ ¼ 0;
C A;
i þ j < 3;
> : e ði þ j 3; m þ n þ 1Þ; 1 8 > < 2e1 ði þ j; m þ nÞ;
1
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 3 8 > < e1 ði þ j; m þ nÞ;
C A;
i þ j < 3;
> : 2e ði þ j 3; m þ n þ 1Þ; 4 8 > < 2e3 ði þ j; m þ nÞ;
1
i þ j < 3;
> : e ði þ j 3; m þ n þ 1Þ; 3 8 > < 2e4 ði þ j; m þ nÞ;
k3nþi
0
i þ j P 3;
deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4:
Remark 1. Here we take e4 ¼ 2e2 to match (23) with loop algebra (1). In what follows, we expand loop algebra (23) into e so that using the approach proposed in Refs. [11–13] obtains integrable couplings, a high-dimensional loop algebra G, of course, a kind of expanding integrable model of the MKdV-H system. For the sake of convenience, the bars in (23) are moved away.
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
453
e From (23), we construct the following high-dimension loop algebra G 0 3nþi 1 0 1 0 k 0 k3nþi 0 0 0 k3nþi 0 1 B C B C B e1 ði; nÞ ¼ @ 0 0 0 A; e3 ði; nÞ ¼ @ k3nþi 0 k3nþi 0 A; e2 ði; nÞ ¼ @ k3nþi 2 0 0 0 0 0 0 0 0 0 1 0 1 0 3nþi 3nþi 1 0 0 0 0 k 0 0 k 0 B B C B C 3nþi C ði; nÞ ¼ ði; nÞ ¼ 0 0 k e4 ði; nÞ ¼ @ k3nþi ; e ; e @ A; @0 0 0 0A 0 A 5 6 0 ½e1 ði; mÞ; e2 ðj; nÞ ¼
0 0 e3 ði þ j; m þ nÞ;
0 0 i þ j < 3;
0
e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e4 ði þ j; m þ nÞ; i þ j < 3;
2e4 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e3 ði þ j; m þ nÞ; i þ j < 3;
2e3 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e1 ði þ j; m þ nÞ; i þ j < 3;
e1 ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2e1 ði þ j; m þ nÞ; i þ j < 3;
½e1 ði; mÞ; e3 ðj; nÞ ¼ ½e1 ði; mÞ; e4 ðj; nÞ ¼ ½e2 ði; mÞ; e3 ðj; nÞ ¼ ½e3 ði; mÞ; e4 ðj; nÞ ¼
2e1 ði þ j 3; m þ n þ 1Þ;
0 0
1
C 0 A; 0
0
i þ j P 3;
½e2 ði; mÞ; e4 ðj; nÞ ¼ 0; e5 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e5 ðj; nÞ ¼ e5 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ( 1 e6 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e5 ðj; nÞ ¼ 21 e ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2 6 e6 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e5 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e6 ði þ j; m þ nÞ; i þ j < 3; ½e4 ði; mÞ; e5 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; e6 ði þ j; m þ nÞ; i þ j < 3; ½e1 ði; mÞ; e6 ðj; nÞ ¼ e6 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ( 1 e5 ði þ j; m þ nÞ; i þ j < 3; ½e2 ði; mÞ; e6 ðj; nÞ ¼ 21 e ði þ j 3; m þ n þ 1Þ; i þ j P 3; 2 5 e5 ði þ j; m þ nÞ; i þ j < 3; ½e3 ði; mÞ; e6 ðj; nÞ ¼ ½e4 ði; mÞ; e6 ðj; nÞ ¼ e5 ði þ j 3; m þ n þ 1Þ; i þ j P 3; ½e5 ði; mÞ; e6 ðj; nÞ ¼ 0;
0
ð24Þ
deg ek ði; mÞ ¼ 3m þ i; i ¼ 0; 1; 2; k ¼ 1; 2; 3; 4; 5; 6:
e 1 ¼ spanfe1 ði; nÞ; e2 ði; nÞ; e3 ði; nÞ; e4 ði; nÞ ¼ 2e2 ði; nÞg, G e 2 ¼ spanfe5 ði; nÞ; e6 ði; nÞg, then Set G e 2; ½G e ¼G e 1 þ_ G e 1; G e 2 G e2 G
ð25Þ
In terms of (24), let /x ¼ U /; / ¼ ð/1 ; /2 ; /3 ÞT ; U ¼ e2 ð1; 0Þ þ u1 e1 ð0; 0Þ þ u2 e3 ð0; 0Þ þ u3 e1 ð2; 1Þ þ u4 e3 ð2; 1Þ þ u5 e4 ð2; 1Þ þ u6 e5 ð0; 0Þ þ u7 e5 ð2; 1Þ þ u8 e6 ð0; 0Þ þ u9 e6 ð2; 1Þ; V ¼
X mP0
ð26Þ
! 2 X ðaði; mÞe1 ði; mÞ þ bði; mÞe3 ði; mÞ þ cði; mÞe4 ði; mÞ þ dði; mÞe5 ði; mÞ þ f ði; mÞe6 ði; mÞÞ ; i¼0
454
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
solving the stationary zero-curvature equation V x ¼ ½U ; V
ð27Þ
gives bð2; m þ 1Þ ¼ ax ð0; mÞ þ 2u2 cð0; mÞ þ 2u4 cð1; mÞ 2u5 bð1; mÞ; að2; m þ 1Þ ¼ bx ð0; mÞ þ 2u1 cð0; mÞ þ 2u3 cð1; mÞ 2u5 að1; mÞ; cx ð0; mÞ ¼ 2u1 bð0; mÞ 2u2 að0; mÞ þ 2u3 bð1; mÞ 2u4 að1; mÞ; bð0; mÞ ¼ ax ð1; mÞ þ 2u2 cð1; mÞ þ 2u4 cð2; mÞ 2u5 bð2; mÞ; að0; mÞ ¼ bx ð1; mÞ þ 2u1 cð1; mÞ þ 2u3 cð2; mÞ 2u5 að2; mÞ; cx ð1; mÞ ¼ 2u1 bð1; mÞ 2u2 að1; mÞ þ 2u3 bð2; mÞ 2u4 að2; mÞ; bð1; m þ 1Þ ¼ ax ð2; m þ 1Þ þ 2u2 cð2; m þ 1Þ þ 2u4 cð0; mÞ 2u5 bð0; mÞ; að1; m þ 1Þ ¼ bx ð2; m þ 1Þ þ 2u1 cð2; m þ 1Þ þ 2u3 cð0; mÞ 2u5 að0; mÞ; cx ð2; m þ 1Þ ¼ 2u1 bð2; m þ 1Þ 2u2 að2; m þ 1Þ þ 2u3 bð0; mÞ 2u4 að0; mÞ; 1 dx ð0; mÞ ¼ f ð2; m þ 1Þ þ u1 dð0; mÞ þ u2 f ð0; mÞ þ u3 dð1; mÞ þ u4 f ð1; mÞ þ u5 f ð1; mÞ 2 u6 að0; mÞ u7 að1; mÞ u8 cð0; mÞ u8 bð0; mÞ u9 bð1; mÞ u9 cð1; mÞ; 1 fx ð0; mÞ ¼ dð2; m þ 1Þ u1 f ð0; mÞ u2 dð0; mÞ u3 f ð1; mÞ u4 dð1; mÞ þ u5 dð1; mÞ 2 þ u6 bð0; mÞ u6 cð0; mÞ þ u7 bð1; mÞ u7 cð1; mÞ þ u8 að1; mÞ þ u9 að0; mÞ; 1 dx ð1; mÞ ¼ f ð0; mÞ þ u1 dð1; mÞ þ u2 f ð1; mÞ þ u3 dð2; mÞ þ u4 f ð2; mÞ þ u5 f ð2; mÞ 2 u6 að1; mÞ u7 að2; mÞ u8 bð1; mÞ u8 cð1; mÞ u9 bð2; mÞ u9 cð2; mÞ; 1 fx ð1; mÞ ¼ dð0; mÞ u1 f ð1; mÞ u2 dð1; mÞ u3 f ð2; mÞ u4 dð2; mÞ þ u5 dð2; mÞ 2 þ u6 bð1; mÞ u6 cð1; mÞ þ u7 bð2; mÞ u7 cð2; mÞ þ u8 að1; mÞ þ u9 að2; mÞ; 1 dx ð2; m þ 1Þ ¼ f ð1; m þ 1Þ þ u1 dð2; m þ 1Þ þ u2 f ð2; m þ 1Þ þ u3 dð0; mÞ þ u4 f ð0; mÞ 2 þ u5 f ð0; mÞ u6 að2; m þ 1Þ u7 að0; mÞ u8 bð2; m þ 1Þ u8 cð2; m þ 1Þ u9 bð0; mÞ u9 cð0; mÞ; 1 fx ð2; m þ 1Þ ¼ dð1; m þ 1Þ u1 f ð2; m þ 1Þ u2 dð2; m þ 1Þ u3 f ð0; mÞ u4 dð0; mÞ þ u5 dð0; mÞ 2 þ u6 bð2; m þ 1Þ u6 cð2; m þ 1Þ þ u7 bð0; mÞ u7 cð0; mÞ þ u8 að2; m þ 1Þ þ u9 að0; mÞ; cð0; 0Þ ¼ a;
að0; 0Þ ¼ bð0; 0Þ ¼ cð1; 0Þ ¼ bð1; 0Þ ¼ að1; 0Þ ¼ cð2; 0Þ ¼ bð2; 0Þ ¼ að2; 0Þ ¼ dð0; 0Þ ¼ f ð0; 0Þ ¼ dð1; 0Þ ¼ f ð1; 0Þ ¼ 0;
dð2; 1Þ ¼ 2au6 ; f ð1; 1Þ ¼ 4aðu6x þ u9 Þ;
að2; 1Þ ¼ 2au1 ;
bð2; 1Þ ¼ 2au2 ;
cð2; 1Þ ¼ 0;
f ð2; 1Þ ¼ 2au8 ;
dð1; 1Þ ¼ 4aðu8x þ u6 Þ: ð28Þ
Note ðnÞ
Vþ ¼
n X 2 X ðaði; mÞe1 ði; n mÞ þ bði; mÞe3 ði; n mÞ þ cði; mÞe4 ði; n mÞ þ dði; mÞe5 ði; n mÞ m¼0 i¼0
þ f ði; mÞe6 ði; n mÞÞ; ðnÞ
ðnÞ
V ¼ k3n V V þ ; then (27) can be written as ðnÞ
ðnÞ
ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ V x ½U ; V :
ð29Þ
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
455
1 1 ðnÞ ðnÞ ðnÞ ðnÞ V þx þ ½U ; V þ ¼ Vþx þ ½U ; Vþ dð2; n þ 1Þe6 ð0; 0Þ f ð2; n þ 1Þe5 ð0; 0Þ þ ½dx ð2; n þ 1Þ 2 2 1 þ f ð1; n þ 1Þ þ u1 dð2; n þ 1Þ þ u2 f ð2; n þ 1Þ u6 að2; n þ 1Þ u8 bð2; n þ 1Þ 2 1 u8 cð2; n þ 1Þe5 ð2; 1Þ þ ½fx ð2; n þ 1Þ þ dð1; n þ 1Þ þ u1 f ð2; n þ 1Þ þ u2 dð2; n þ 1Þ 2 u6 bð2; n þ 1Þ þ u6 cð2; n þ 1Þ u8 að2; n þ 1Þe6 ð2; 1Þ:
ð30Þ
Similar to (5), we find that
Let V system
ðnÞ
¼V
ðnÞ þ ,
then the zero-curvature equation U t V
ðnÞ x
þ ½U ; V
ðnÞ
¼ 0 brings about the following Lax integrable
1 1 0 bð2;n þ 1Þ u1 Bu C B C að2;n þ 1Þ B 2C B C B C B C B u3 C B C ax ð2; n þ 1Þ bð1; n þ 1Þ þ 2u2 cð2; n þ 1Þ B C B C B C B C B u4 C B C bx ð2; n þ 1Þ að1; n þ 1Þ þ 2u1 cð2; n þ 1Þ B C B C B C B C cx ð2;n þ 1Þ þ 2u1 bð2; n þ 1Þ 2u2 að2; n þ 1Þ ut ¼ B u5 C ¼ B C B C B C 1 Bu C B C f ð2; n þ 1Þ B 6C B C 2 B C B C B u7 C B dx ð2;n þ 1Þ 12 f ð1;n þ 1Þ u1 dð2;n þ 1Þ u2 f ð2; n þ 1Þ þ u6 að2;n þ 1Þ þ u8 bð2; n þ 1Þ þ u8 cð2; n þ 1Þ C B C B C B C B C 1 dð2;n þ 1Þ @ u8 A @ A 2 0
u9 0
0 B 0 B B 0 B 1 B B 2 ¼B B 0 B 0 B B 0 B @ 0 0 0
t
0 0 1 2
0 0 0 0 0 0
fx ð2; n þ 1Þ 12 dð1;n þ 1Þ u1 f ð2; n þ 1Þ þ u2 dð2; n þ 1Þ u6 bð2;n þ 1Þ u6 cð2;n þ 1Þ u8 að2;n þ 1Þ 1 0 1 0 2að1; n þ 1Þ 2að1;n þ 1Þ 1B C C B 1 0 0 0 0 0 B 2bð1;n þ 1Þ C 0 B 2bð1;n þ 1Þ C 2 C B C B 1 C 2 0 0 0 0 0 0 CB 2að2; n þ 1Þ C B 2að2;n þ 1Þ C C B C B o2 0 u2 0 0 0 0C C B C B CB 2bð2;n þ 1Þ C B 2bð2;n þ 1Þ C o C 0 u 0 0 0 0 C B C B 1 2 C C B C B u2 u1 o2 0 0 0 0C B 2cð2; n þ 1Þ C ¼ J1 B 2cð2; n þ 1Þ C C C B C B 1 CB 0 0 0 0 0 0 C C B 2 CB f ð2;n þ 1Þ C B f ð2; n þ 1Þ C u6 u8 u8 1 C 2 2 2 u2 o u1 2 0 CB C C B C B B dð2; n þ 1Þ C 1 0 0 0 0 0 0 AB dð2; n þ 1Þ C C B 2 C B C B u28 u26 u26 o þ u1 u2 0 12 @ f ð1;n þ 1Þ A @ f ð1; n þ 1Þ A dð1; n þ 1Þ bð2;n þ 1Þ
B B að2;n þ 1Þ B B B 2u4 cð0; nÞ þ 2u5 bð0;nÞ B B B 2u3 cð0; nÞ þ 2u5 að0;nÞ B B ¼B 2u 3 bð0;nÞ þ 2u4 að0; nÞ B B 1 B f ð2;n þ 1Þ 2 B B B u3 dð0; nÞ þ u4 f ð0;nÞ þ u5 f ð0; nÞ u7 að0; nÞ u9 bð0; nÞ u9 cð0; nÞ B B 1 B dð2;n þ 1Þ @ 2
1
dð1; n þ 1Þ
C C C C C C C C C C C C C C C C C C C C A
u3 f ð0;nÞ u4 dð0;nÞ þ u5 dð0;nÞ þ u7 bð0;nÞ u7 cð0; nÞ þ u9 að0;nÞ 0 0 1 1 10 1 0 12 0 0 0 0 0 0 0 2að2; n þ 1Þ 2að0;nÞ 2að2;n þ 1Þ B B B 1 C C CB C B 2bð2 n þ 1Þ C B 2bð0; nÞ C B 0 0 B C 0 0 0 0 0 0C B B B 2 C C CB 2bð2 n þ 1Þ C B B B C C CB C B 2að0; nÞ C B 2að1;nÞ C B 0 0 0 u5 u4 0 CB 2að0;nÞ C 0 0 0 B B B C C CB C B B B C C CB C B 2bð0;nÞ C B 2bð1; nÞ C B 0 0 u5 B C 0 u3 0 0 0 0C B B B C C CB 2bð0; nÞ C B B B C C CB C B B C C; C B C 0 0 u u 0 0 0 0 0 ¼B 2cð0; nÞ 2cð1;nÞ ¼ J 2cð0;nÞ ¼ J 4 3 2B 3B B C C CB C B B B C C C B C 1 B B B 0 0 0 C C C B C 0 0 0 0 0 f ð2;n þ 1Þ f ð2; n þ 1Þ f ð2; n þ 1Þ 2 B B B C C CB C B B B C C C B C B B dð0;nÞ C B 0 0 u7 u9 u9 0 C B C u3 u4 þ u5 0 C dð0; nÞ dð0;nÞ B B B C C CB C 2 2 2 B B B C C CB C 1 CB B B dð2; n þ 1Þ C B 0 0 0 C C 0 0 0 0 0 f ð0;nÞ f ð0; nÞ @ @ @ A A A 2 A@ 0 0 u29 u27 u27 0 u5 u4 u3 0 dð2; n þ 1Þ f ð0; nÞ dð2;n þ 1Þ 0
ð31Þ
456
Y. Zhang, X. Xu / Chaos, Solitons and Fractals 21 (2004) 445–456
where 0
þ 2u2 o1 u2 B 2u o1 u B 1 2 B 2u o1 u B 4 2 B B 2u3 o1 u2 þ u5 B u4 J3 ¼ B B 0 B B u7 B 2 þ u9 o1 u2 B 0 @ u9 u7 o1 u2 2 o 2
2u2 o1 u1 o2 þ 2u1 o1 u1 2u4 o1 u1 u5 2u3 o1 u1 u3 0 u9 1 þ u 9 o u1 2 0 u7 o1 u1
2u2 o1 u4 2u1 o1 u4 þ u5 2u4 o1 u4 2u3 o1 u4 0 0 u9 o1 u4 0 u7 o1 u4
2u2 o1 u3 u5 2u1 o1 u3 2u4 o1 u3 2u3 o1 u3 0 0 u9 o1 u3 0 u7 o1 u3 0
u4 u3 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 1 0 2 0 u3 0 0 0 u5 u4
1 0 0 0 0 C C 0 0 C C C 0 0 C C 0 0 C: C 0 0 C C 0 u4 þ u5 C C 1 0 A 2 0 u3
According to the definition of integrable couplings [14–16], we conclude that (31) is an integrable coupling of the system (7). Of course, (31) is also a type of expanding integrable model of (7), since when taking u6 ¼ u7 ¼ u8 ¼ u9 ¼ 0, (31) reduces to (7). Remark 2. Taking u3 ¼ u4 ¼ u5 ¼ 0 in (31), an integrable coupling of AKNS hierarchy is presented just like the result in Ref. [11]. There exists an open problem. The system (31) is only Lax integrable, how can we construct a proper loop algebra to get Liouville integrable coupling systems? In addition, we may extend loop algebra (1) into a general case, i.e. replacing k3nþi with kNnþi , where N ¼ 4; 5; 6; . . .. It follows Tu scheme that some Liouville integrable hierarchies with N Hamiltonian structures could be worked out. It is special worth pointing out it is a mistake that concluding the integrable coupling was Liouville integrable in Ref. [13], because the Hamiltonian structure of the integrable coupling of BPT hierarchy is wrong. Two J operators in Ref. [17] are not Hamiltonian operators.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Wadati M. J Phys Soc Jpn 1973;34:1289. Tu G. J Math Phys 1989;30(2):330. Gu C et al. Soliton theory and its application. Zhejiang Publishing House of Science and Technology; 1990. Fan E. J Math Phys 2000;41(11):7769. Zhou Z. J Phys Soc Jpn 2002;71(8):1857. Hu X. J Phys A: Math Gen 1994;27:2497. Magri F. J Math Phys 1978;19:1156. Ma W. J Phys A: Math Gen 1990;23:2707. Guo F. J Syst Sci Math Sci 2002;22(1):36. Guo F. Acta Math Sini 1997;40(6):801. Guo F, Zhang Y. Acta Phys Sini 2002;51(5):951. Zhang Y, Zhang H. J Math Phys 2002;43(1):466. Zhang Y et al. Phys Lett A 2002;299(5):543. Lakshmanan, Tamizhamani KM. J Math Phys 1985;26:1189. Ma W, Fuchssteiner B. Phys Lett A 1996;213:49. Ma W, Fuchssteiner B. Chaos, Solitons & Fractals 1996;8:1227. Zhang Y. Phys Lett A 2003;310:19.