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The multi-component classical-Boussinesq hierarchy of soliton equations and its multi-component integrable coupling system
Chaos, Solitons and Fractals 23 (2005) 1163–1167 www.elsevier.com/locate/chaos
The multi-component classical-Boussinesq hierarchy of soliton equations and its multi-component integrable coupling system Tiecheng Xia b
a,b,*
, Fa-Jun Yu a, Dengyuan Chen
b
a Department of Mathematics, Bohai University, Jinzhou 121000, PR China Department of Mathematics, Shanghai University, Shanghai 200436, PR China
Accepted 7 June 2004 Communicated by Prof. M. Wadati
Abstract e M is constructed, which is devoted to establishing an isospectral problem. By making A new simple loop algebra G use of Tu scheme, the multi-component classical-Boussinesq hierarchy is obtained. Furthermore, an expanding loop e M is presented. Based on Fe M , the multi-component integrable coupling system of algebra Fe M of the loop algebra G the multi-component classical-Boussinesq hierarchy is worked out. The method can be applied to other nonlinear evolution equations hierarchy. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Searching for new integrable hierarchies of soliton equations is an important and interesting topic in soliton theory. One took various efficient approaches to have obtained many integrable systems such as AKNS hierarchy, KN hierarchy, Schro¨dinger system, and so on [1–10]. As far as the multi-component integrable hierarchies are concerned, there have been developments such as in Refs. [11,12]. In Refs. [13,14], a simple and efficient method for generating multicomponent integrable hierarchies was proposed. Constructing a simple new loop algebra becomes a key step in this method. Although using the loop algebra in Refs. [13,14] can produce many multi-component integrable hierarchies, it is not suitable for seeking the multi-component classical-Boussinesq hierarchy [15]. In this paper, a new loop algebra e M is first constructed, and then an isospectral problem is designed. By employing Tu scheme [6,16], the multi-compoG eM nent classical-Boussinesq hierarchy is worked out. In addition, an expanding loop algebra Fe M of the loop algebra G is presented, which is devoted to deducing the integrable coupling of the multi-component classical-Boussinesq hierarchy.
* Corresponding author. Address: Department of Mathematics, Bohai University, Jinzhou 121000, PR China. Tel.: +86 416 3157323. E-mail address: [email protected] (Tiecheng Xia).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.005
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Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
2. A new loop algebra If GM = {a = (aij)M·3 = (a1, a2, a3)} denotes a set of matrices, where M is a positive integer, ai (i = 1, 2, 3) is the ith column of the matrix a. Then GM is a linear space. Let a = (a1, a2, . . . , aM)T, b = (b1, b2, . . ., bM)T, and define their product and sum a * b = b * a = (a1T * b = b * a = (a1b1,a2b2, . . . , aMbM) , a + b = (a1 + b1, a2 + b2, . . . , aM + bM), respectively. If a ¼ ða1 ; a2 ; a3 Þ; e b ¼ ðb1 ; b2 ; b3 Þ 2 G M , a commutation operation for GM is defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ:
ð1Þ
It is easy to verify that the operation (1) is linear and antisymmetric. For "a, b, c 2 GM, we can also verify that ½½a; b; c þ ½½b; c; a þ ½½c; a; b ¼ 0;
ð2Þ
i.e., Jacobian identity holds. Therefore, GM with the operation (1) becomes a Lie algebra. A corresponding loop algebra e M is defined as G e M ¼ fakn ; a 2 GM ; n ¼ 0; 1; 2; . . .g G
ð3Þ
with a commutation operation defined as ½akm ; bkn ¼ ½a; bkmþn
8a; b 2 GM :
ð4Þ
e 1 is equal to A e 1 in Refs. [17,18], we conclude that G e M is an extension of the loop algebra A e 1 . We also find that Since G e M has two features. (i) The commutation operation is the simple and straightforward, as that in the loop algebra A e 1. G e M , we proceed to simple calculation to be able to obtain various multi-component integrable systems. (ii) By means of G Consider linear isospectral problem as follows: ( eM; /x ¼ ½U ; /; kt ¼ 0; /; U ; V 2 G ð5Þ /t ¼ ½V ; /: Whose compatibility gives rise to /xt ¼ ½U t ; / þ ½U ; ½V ; / ¼ /tx ¼ ½V x ; / þ ½V ; ½U ; /; that is ½U t ; / ½V x ; / þ ½U ; ½V ; / þ ½V ; ½/; U ¼ 0:
ð6Þ
By employing (2), the formula (6) can be written as ½U t ; / ½V x ; / þ ½½U ; V ; / ¼ 0:
ð7Þ
Since / is arbitrary, a condition of (7) holds if and only if the following equation does: U t V x þ ½U ; V ¼ 0:
ð8Þ
Hence, the compatibility of (5) leads to the zero-curvature equation (8).
3. The multi-component classical-Boussinesq hierarchy We consider an isospectral problem q /x ¼ ½U ; /; kt ¼ 0; U ¼ kI M I M ; r I M ; I M ; 4 where
IM
0 1 1 B1C B C C ¼B B .. C ; @.A 1 M1
q ¼ ðq1 ; q2 ; . . . ; qM ÞT ;
r ¼ ðr1 ; r2 ; . . . ; rM ÞT :
ð9Þ
Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
1165
Let V ¼
1 X ðam ; bm ; cm Þkm ; m¼0
where am = (am1, am2, . . . , amM)T, bm = (bm1, bm2, . . . , bmM)T, cm = (cm1, cm2, . . . , cmM)T. Solving the stationary zero curvature equation V x ¼ ½U ; V
ð10Þ
gives amx ¼ r cm þ bm ;
ð11:1Þ
q bmx ¼ 2bmþ1 bm 2r am ; 2
ð11:2Þ
q cmx ¼ 2cmþ1 2am þ cm ; 2
ð11:3Þ
b0 ¼ c0 ¼ 0;
a0 ¼ k ¼ ðk 1 ; k 2 ; . . . ; k m Þ 6¼ 0;
a1 ¼ 0;
b1 ¼ k r;
c1 ¼ k;
kr kq ; c2 ¼ : 2 4 P ðnÞ n ¼ nm¼0 ðam ; bm ; cm Þknm , V ðnÞ ¼ k V V þ , then Eq. (6) can be written as
a2 ¼ ðnÞ
Denote V þ
ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
Again taking Dn ¼ ðcnþ1 ; 0; 0Þ, V Ut V
ðnÞ x
þ ½U ; V
ðnÞ
ðnÞ
¼V
þ Dn , the zero curvature equation
¼0
ð13Þ
with
4o 0 ; 0 o
L¼
ð11:5Þ
ð12Þ ðnÞ þ
gives rise to the Lax integrable system ! kr 0 q 2 n1 n ¼ JL ¼ JL ut ¼ k r t kq 4
J¼
ð11:4Þ
1 4
2o o1 q o 8I M
ð14Þ
! o1 r o þ r : 2o q
When M = 1, the system (14) is just the classical-Boussinesq hierarchy [15]. When M > 1, the system (14) is the multicomponent classical-Boussinesq hierarchy.
4. The multi-component integrable coupling system Set F M ¼ fa ¼ ðaij ÞM 5 ¼ ða1 ; a2 ; a3 ; a4 ; a5 Þg
ð15Þ
with a commutation operation defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 Þ; a1 b4 a4 b1 þ a2 b5 a5 b2 ; a3 b4 a4 b3 þ a5 b1 a1 b5 Þ:
ð16Þ
Then FM is a Lie algebra. A corresponding loop algebra Fe M is defined Fe M ¼ fakn ; a 2 F M ; n ¼ 0; 1; 2; . . .g
ð17Þ
with a commutation operation defined as ½akm ; bkn ¼ ½a; bkmþn
8a; b 2 F M :
ð18Þ
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Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
Denote Fe M ð1Þ ¼ fða1 ; a2 ; a3 ; 0; 0Þkn g; Fe M ð2Þ ¼ fð0; 0; 0; a4 ; a5 Þkn g; e M ; (ii) ½ Fe M ð1Þ; Fe M ð2Þ Fe M ð2Þ. Based on Fe M , we consider an isospectral then (i) Fe M ¼ Fe M ð1Þ Fe M ð2Þ, Fe M ð1Þ ffi G problem u1 ð19Þ /x ¼ ½U ; /; kt ¼ 0; U ¼ kI M I M ; u2 I M ; I M ; u3 I M ; u4 I M ; 4 P where ui = (ui1, ui2, . . . , uiM)T, i = 1, 2, 3, 4. Let V ¼ nm¼0 ðam ; bm ; cm ; d m ; fm Þkm . Solving the equation similar to (10) gives amx ¼ u2 cm þ bm ; bmx ¼ 2bmþ1
u1 bm 2u2 am ; 2
cmx ¼ 2cmþ1 2am þ d mx ¼ d mþ1 fmx ¼ fmþ1 þ
ð20:1Þ
u1 cm ; 2
ð20:3Þ
u1 d m þ u2 fm u3 am u4 bm ; 4
ð20:4Þ
u1 fm d m u3 cm þ u4 am ; 4
b0 ¼ c0 ¼ d 0 ¼ f0 ¼ ð0; 0; . . . ; 0Þ; b1 ¼ a u1 ;
ð20:2Þ
c1 ¼ a u2 ;
a1 ¼ 0;
ð20:5Þ
a0 ¼ a ¼ ða1 ; a2 ; . . . ; am Þ ¼ const: 6¼ 0; d 1 ¼ a u3 ;
f 1 ¼ a u4 :
ð20:6Þ ð20:7Þ
ðnÞ
Taking Dn ¼ ðcnþ1 ; 0; 0; 0; 0Þ, V ðnÞ ¼ V þ þ Dn , the zero curvature equation similar to (13) admits the following: 0 1 0 anþ1 1 0 10 anþ1 1 u1 4o 0 0 0 2 2 Bu C B C Bc C B 0 o 0 0 C B 2C B nþ1 C B CB cnþ1 C ut ¼ B C ¼ B C ¼ JB C: CB @ u3 A @ fnþ1 A @ 0 u3 0 I M A@ fnþ1 A 0 u4 I M u4 t 0 d nþ1 d nþ1
ð21Þ
From (20), a recurrence operator is presented 0 1 0 0 14 o1 u1 o 12 o 14 o1 u2 o u42 B C o B C 2I M u41 0 0 2 C: L¼B u1 B u3 o 4 IM C 2u4 @ A 2u4 o þ 2u3 u2 u4 u2 o u41 Thus, Eq. (21) can be written as 0 1 0 1 u1 0 Bu C B au C 2 C B 2C B ut ¼ B C ¼ JLn B C: @ u3 A @ a u4 A u4 t a u3
ð22Þ
When M = 1, the system (22) is the integrable coupling of the system (14). When M > 1, the system (22) is the multicomponent integrable coupling of the system (14). Remarks: The loop algebra presented here can be used to other known integrable hierarchies of soliton equation for eM generating the multi-component systems. But there exist two open problems. How do we improve the loop algebra G to work out the Hamiltonian structures, infinite conserved laws of the multi-component hierarchies? It is another open problem how to construct multi-component integrable coupling of nonisospectral problem and corresponding evolution equation hierarchy. These problems are worthwhile studying in future.
Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
1167
Acknowledgments One of the authors Tiecheng Xia would like to express his sincere thanks to Profs. Y.F. Zhang, F.K. Guo and M. Wadati for theirs enthusiastic guidance and help. This work was supported by the National Nature Science Foundation of China (10371070), the Special Funds for Major Specialties of Shanghai Educational Committee and Educational Committee of Liaoning Province.
References [1] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. [2] Newell AC. Soliton in mathematics and physics. Philadelphia: SIAM; 1985. [3] Wadati M. J Phys Soc Jpn 1972;32:1681. [4] Wadati M. J Phys Soc Jpn 1973;34:1289–96. [5] Wadati M, Sanuki H, Konno K. Prog Theor Phys 1975;53:419–36. [6] Tu GZ. J Math Phys 1989;30:330–7. [7] Hu XB. J Phys A: Math Gen 1994;27:2497–502. [8] Hu XB. J Phys A: Math Gen 1997;30:619–27. [9] Fan EG. Physica A 2001;301:105–10. [10] Fan EG. J Math Phys 2000;41:7769–78. [11] Tsuchida T, Wadati M. Phys Lett A 1999;257:53–64. [12] Tsuchida T, Wadati M. J Phys Soc Jpn 1999;68:2241–5. [13] Guo FK, Zhang YF. J Math Phys 2003;44:5793–803. [14] Zhang YF. A generalized multi-component Glachette–Johnson (GJ) hierarchy and its integrable coupling system. Chaos, Solitons & Fractals 2004;44:305–10. [15] Tu GZ, Meng DZ. Acta Math Appl Sin 1989;5:89–99. [16] Zeng YB. Chin Sci Bull 1992;37:769–73. [17] Guo FK. Acta Math Appl Sin 2000;23:181–8 [in Chinese]. [18] Guo FK. J Sys Sci Math Scis 2002;22:36–43 [in Chinese].
The multi-component classical-Boussinesq hierarchy of soliton equations and its multi-component integrable coupling system Tiecheng Xia b
a,b,*
, Fa-Jun Yu a, Dengyuan Chen
b
a Department of Mathematics, Bohai University, Jinzhou 121000, PR China Department of Mathematics, Shanghai University, Shanghai 200436, PR China
Accepted 7 June 2004 Communicated by Prof. M. Wadati
Abstract e M is constructed, which is devoted to establishing an isospectral problem. By making A new simple loop algebra G use of Tu scheme, the multi-component classical-Boussinesq hierarchy is obtained. Furthermore, an expanding loop e M is presented. Based on Fe M , the multi-component integrable coupling system of algebra Fe M of the loop algebra G the multi-component classical-Boussinesq hierarchy is worked out. The method can be applied to other nonlinear evolution equations hierarchy. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Searching for new integrable hierarchies of soliton equations is an important and interesting topic in soliton theory. One took various efficient approaches to have obtained many integrable systems such as AKNS hierarchy, KN hierarchy, Schro¨dinger system, and so on [1–10]. As far as the multi-component integrable hierarchies are concerned, there have been developments such as in Refs. [11,12]. In Refs. [13,14], a simple and efficient method for generating multicomponent integrable hierarchies was proposed. Constructing a simple new loop algebra becomes a key step in this method. Although using the loop algebra in Refs. [13,14] can produce many multi-component integrable hierarchies, it is not suitable for seeking the multi-component classical-Boussinesq hierarchy [15]. In this paper, a new loop algebra e M is first constructed, and then an isospectral problem is designed. By employing Tu scheme [6,16], the multi-compoG eM nent classical-Boussinesq hierarchy is worked out. In addition, an expanding loop algebra Fe M of the loop algebra G is presented, which is devoted to deducing the integrable coupling of the multi-component classical-Boussinesq hierarchy.
* Corresponding author. Address: Department of Mathematics, Bohai University, Jinzhou 121000, PR China. Tel.: +86 416 3157323. E-mail address: [email protected] (Tiecheng Xia).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.005
1164
Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
2. A new loop algebra If GM = {a = (aij)M·3 = (a1, a2, a3)} denotes a set of matrices, where M is a positive integer, ai (i = 1, 2, 3) is the ith column of the matrix a. Then GM is a linear space. Let a = (a1, a2, . . . , aM)T, b = (b1, b2, . . ., bM)T, and define their product and sum a * b = b * a = (a1T * b = b * a = (a1b1,a2b2, . . . , aMbM) , a + b = (a1 + b1, a2 + b2, . . . , aM + bM), respectively. If a ¼ ða1 ; a2 ; a3 Þ; e b ¼ ðb1 ; b2 ; b3 Þ 2 G M , a commutation operation for GM is defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞ:
ð1Þ
It is easy to verify that the operation (1) is linear and antisymmetric. For "a, b, c 2 GM, we can also verify that ½½a; b; c þ ½½b; c; a þ ½½c; a; b ¼ 0;
ð2Þ
i.e., Jacobian identity holds. Therefore, GM with the operation (1) becomes a Lie algebra. A corresponding loop algebra e M is defined as G e M ¼ fakn ; a 2 GM ; n ¼ 0; 1; 2; . . .g G
ð3Þ
with a commutation operation defined as ½akm ; bkn ¼ ½a; bkmþn
8a; b 2 GM :
ð4Þ
e 1 is equal to A e 1 in Refs. [17,18], we conclude that G e M is an extension of the loop algebra A e 1 . We also find that Since G e M has two features. (i) The commutation operation is the simple and straightforward, as that in the loop algebra A e 1. G e M , we proceed to simple calculation to be able to obtain various multi-component integrable systems. (ii) By means of G Consider linear isospectral problem as follows: ( eM; /x ¼ ½U ; /; kt ¼ 0; /; U ; V 2 G ð5Þ /t ¼ ½V ; /: Whose compatibility gives rise to /xt ¼ ½U t ; / þ ½U ; ½V ; / ¼ /tx ¼ ½V x ; / þ ½V ; ½U ; /; that is ½U t ; / ½V x ; / þ ½U ; ½V ; / þ ½V ; ½/; U ¼ 0:
ð6Þ
By employing (2), the formula (6) can be written as ½U t ; / ½V x ; / þ ½½U ; V ; / ¼ 0:
ð7Þ
Since / is arbitrary, a condition of (7) holds if and only if the following equation does: U t V x þ ½U ; V ¼ 0:
ð8Þ
Hence, the compatibility of (5) leads to the zero-curvature equation (8).
3. The multi-component classical-Boussinesq hierarchy We consider an isospectral problem q /x ¼ ½U ; /; kt ¼ 0; U ¼ kI M I M ; r I M ; I M ; 4 where
IM
0 1 1 B1C B C C ¼B B .. C ; @.A 1 M1
q ¼ ðq1 ; q2 ; . . . ; qM ÞT ;
r ¼ ðr1 ; r2 ; . . . ; rM ÞT :
ð9Þ
Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
1165
Let V ¼
1 X ðam ; bm ; cm Þkm ; m¼0
where am = (am1, am2, . . . , amM)T, bm = (bm1, bm2, . . . , bmM)T, cm = (cm1, cm2, . . . , cmM)T. Solving the stationary zero curvature equation V x ¼ ½U ; V
ð10Þ
gives amx ¼ r cm þ bm ;
ð11:1Þ
q bmx ¼ 2bmþ1 bm 2r am ; 2
ð11:2Þ
q cmx ¼ 2cmþ1 2am þ cm ; 2
ð11:3Þ
b0 ¼ c0 ¼ 0;
a0 ¼ k ¼ ðk 1 ; k 2 ; . . . ; k m Þ 6¼ 0;
a1 ¼ 0;
b1 ¼ k r;
c1 ¼ k;
kr kq ; c2 ¼ : 2 4 P ðnÞ n ¼ nm¼0 ðam ; bm ; cm Þknm , V ðnÞ ¼ k V V þ , then Eq. (6) can be written as
a2 ¼ ðnÞ
Denote V þ
ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
Again taking Dn ¼ ðcnþ1 ; 0; 0Þ, V Ut V
ðnÞ x
þ ½U ; V
ðnÞ
ðnÞ
¼V
þ Dn , the zero curvature equation
¼0
ð13Þ
with
4o 0 ; 0 o
L¼
ð11:5Þ
ð12Þ ðnÞ þ
gives rise to the Lax integrable system ! kr 0 q 2 n1 n ¼ JL ¼ JL ut ¼ k r t kq 4
J¼
ð11:4Þ
1 4
2o o1 q o 8I M
ð14Þ
! o1 r o þ r : 2o q
When M = 1, the system (14) is just the classical-Boussinesq hierarchy [15]. When M > 1, the system (14) is the multicomponent classical-Boussinesq hierarchy.
4. The multi-component integrable coupling system Set F M ¼ fa ¼ ðaij ÞM 5 ¼ ða1 ; a2 ; a3 ; a4 ; a5 Þg
ð15Þ
with a commutation operation defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 Þ; a1 b4 a4 b1 þ a2 b5 a5 b2 ; a3 b4 a4 b3 þ a5 b1 a1 b5 Þ:
ð16Þ
Then FM is a Lie algebra. A corresponding loop algebra Fe M is defined Fe M ¼ fakn ; a 2 F M ; n ¼ 0; 1; 2; . . .g
ð17Þ
with a commutation operation defined as ½akm ; bkn ¼ ½a; bkmþn
8a; b 2 F M :
ð18Þ
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Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
Denote Fe M ð1Þ ¼ fða1 ; a2 ; a3 ; 0; 0Þkn g; Fe M ð2Þ ¼ fð0; 0; 0; a4 ; a5 Þkn g; e M ; (ii) ½ Fe M ð1Þ; Fe M ð2Þ Fe M ð2Þ. Based on Fe M , we consider an isospectral then (i) Fe M ¼ Fe M ð1Þ Fe M ð2Þ, Fe M ð1Þ ffi G problem u1 ð19Þ /x ¼ ½U ; /; kt ¼ 0; U ¼ kI M I M ; u2 I M ; I M ; u3 I M ; u4 I M ; 4 P where ui = (ui1, ui2, . . . , uiM)T, i = 1, 2, 3, 4. Let V ¼ nm¼0 ðam ; bm ; cm ; d m ; fm Þkm . Solving the equation similar to (10) gives amx ¼ u2 cm þ bm ; bmx ¼ 2bmþ1
u1 bm 2u2 am ; 2
cmx ¼ 2cmþ1 2am þ d mx ¼ d mþ1 fmx ¼ fmþ1 þ
ð20:1Þ
u1 cm ; 2
ð20:3Þ
u1 d m þ u2 fm u3 am u4 bm ; 4
ð20:4Þ
u1 fm d m u3 cm þ u4 am ; 4
b0 ¼ c0 ¼ d 0 ¼ f0 ¼ ð0; 0; . . . ; 0Þ; b1 ¼ a u1 ;
ð20:2Þ
c1 ¼ a u2 ;
a1 ¼ 0;
ð20:5Þ
a0 ¼ a ¼ ða1 ; a2 ; . . . ; am Þ ¼ const: 6¼ 0; d 1 ¼ a u3 ;
f 1 ¼ a u4 :
ð20:6Þ ð20:7Þ
ðnÞ
Taking Dn ¼ ðcnþ1 ; 0; 0; 0; 0Þ, V ðnÞ ¼ V þ þ Dn , the zero curvature equation similar to (13) admits the following: 0 1 0 anþ1 1 0 10 anþ1 1 u1 4o 0 0 0 2 2 Bu C B C Bc C B 0 o 0 0 C B 2C B nþ1 C B CB cnþ1 C ut ¼ B C ¼ B C ¼ JB C: CB @ u3 A @ fnþ1 A @ 0 u3 0 I M A@ fnþ1 A 0 u4 I M u4 t 0 d nþ1 d nþ1
ð21Þ
From (20), a recurrence operator is presented 0 1 0 0 14 o1 u1 o 12 o 14 o1 u2 o u42 B C o B C 2I M u41 0 0 2 C: L¼B u1 B u3 o 4 IM C 2u4 @ A 2u4 o þ 2u3 u2 u4 u2 o u41 Thus, Eq. (21) can be written as 0 1 0 1 u1 0 Bu C B au C 2 C B 2C B ut ¼ B C ¼ JLn B C: @ u3 A @ a u4 A u4 t a u3
ð22Þ
When M = 1, the system (22) is the integrable coupling of the system (14). When M > 1, the system (22) is the multicomponent integrable coupling of the system (14). Remarks: The loop algebra presented here can be used to other known integrable hierarchies of soliton equation for eM generating the multi-component systems. But there exist two open problems. How do we improve the loop algebra G to work out the Hamiltonian structures, infinite conserved laws of the multi-component hierarchies? It is another open problem how to construct multi-component integrable coupling of nonisospectral problem and corresponding evolution equation hierarchy. These problems are worthwhile studying in future.
Tiecheng Xia et al. / Chaos, Solitons and Fractals 23 (2005) 1163–1167
1167
Acknowledgments One of the authors Tiecheng Xia would like to express his sincere thanks to Profs. Y.F. Zhang, F.K. Guo and M. Wadati for theirs enthusiastic guidance and help. This work was supported by the National Nature Science Foundation of China (10371070), the Special Funds for Major Specialties of Shanghai Educational Committee and Educational Committee of Liaoning Province.
References [1] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. [2] Newell AC. Soliton in mathematics and physics. Philadelphia: SIAM; 1985. [3] Wadati M. J Phys Soc Jpn 1972;32:1681. [4] Wadati M. J Phys Soc Jpn 1973;34:1289–96. [5] Wadati M, Sanuki H, Konno K. Prog Theor Phys 1975;53:419–36. [6] Tu GZ. J Math Phys 1989;30:330–7. [7] Hu XB. J Phys A: Math Gen 1994;27:2497–502. [8] Hu XB. J Phys A: Math Gen 1997;30:619–27. [9] Fan EG. Physica A 2001;301:105–10. [10] Fan EG. J Math Phys 2000;41:7769–78. [11] Tsuchida T, Wadati M. Phys Lett A 1999;257:53–64. [12] Tsuchida T, Wadati M. J Phys Soc Jpn 1999;68:2241–5. [13] Guo FK, Zhang YF. J Math Phys 2003;44:5793–803. [14] Zhang YF. A generalized multi-component Glachette–Johnson (GJ) hierarchy and its integrable coupling system. Chaos, Solitons & Fractals 2004;44:305–10. [15] Tu GZ, Meng DZ. Acta Math Appl Sin 1989;5:89–99. [16] Zeng YB. Chin Sci Bull 1992;37:769–73. [17] Guo FK. Acta Math Appl Sin 2000;23:181–8 [in Chinese]. [18] Guo FK. J Sys Sci Math Scis 2002;22:36–43 [in Chinese].