Multi-component Levi Hierarchy and Its Multi-component Integrable Coupling System

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1 Commun. Theor. Phys. (Beijing, China) 44 (2005) pp c Inernaional Academic Publishers Vol. 44, No. 6, December 5, 2005 uli-componen Levi Hierarchy and Is uli-componen Inegrable Coupling Sysem XIA Tie-Cheng,, YOU Fu-Cai,, and ZHAO Wen-Ying 2 Deparmen of ahemaics, Shanghai Universiy, Shanghai , China 2 Deparmen of ahemaics, Bohai Universiy, Jinzhou 2000, China (Received January 24, 2005; Revised April 8, 2005) Absrac A simple 3-dimensional loop algebra X is produced, whose commuaion operaion defined by us is as simple and sraighforward as ha in he loop algebra Ã. I follows ha a general scheme for generaing mulicomponen inegrable hierarchy is proposed. By aking advanage of X, a new isospecral problem is esablished, and hen by making use of he Tu scheme he well-known muli-componen Levi hierarchy is obained. Finally, an expanding loop algebra F of he loop algebra X is presened, based on he F, he muli-componen inegrable coupling sysem of he muli-componen Levi hierarchy is worked ou. The mehod in his paper can be applied o oher nonlinear evoluion equaion hierarchies. PACS numbers: Jr, Ik Key words: loop algebra, muli-componen Levi hierarchy, muli-componen inegrable coupling sysem Inroducion Searching for new inegrable hierarchies of solion equaions and heir inegrable coupling sysem is an imporan and ineresing opic in solion heory. Various efficien echniques have been proposed o obain inegrable hierarchies of solion equaions such as he AKNS sysem, he KdV sysem, he KN sysem, he Schrödinger sysem, he coupled Burgers sysem, and so on. [ 42] Tu proposed an efficien mehod for generaing inegrable Hamilonian hierarchies of solion equaions in Ref. [3]. Recenly, Guo and Zhang furher developed Tu s scheme, and some inegrable hierarchies wih muli-poenial funcions have been worked ou. [4 45] So far inegrable sysems wih muli-componen poenial funcions have been a subjec of considerable ineres for many years and many efficien approaches have been developed in Refs. [7] [8], [33], and [40]. a and Zhou in Ref. [33] have made use of he generalized Tu s scheme o arrive a he mulicomponen AKNS hierarchy and oher ineresing resuls.. Wadai used generalized Lax marix o exend nonlinear Schrödinger equaion o muli-componen Schrödinger equaion. In Ref. [34], a sysemaic approach was proposed for generaing muli-componen inegrable hierarchies. However, here are some clues o deal wih when we deduce inegrable hierarchies. A simple and efficien mehod of generaing muli-componen inegrable hierarchies was proposed in Refs. [9] and [20]. Anoher new simple mehod was presened in Ref. [40]. Consrucing a simple loop algebra and designing isospecral problem becomes a key sep in his mehod. In erms of he idea in Ref. [40], we would like o inroduce a new isospecral problem by using a new loop algebra X in his paper. By aking advanage of he Tu s scheme, he muli-componen Levi hierarchy is given. In addiion, an expanding loop algebra F of he loop algebra X is presened, which is devoed o deducing he inegrable couplings of he mulicomponen Levi hierarchy. The mehod in his paper can be applied o oher nonlinear evoluion equaion hierarchies. 2 A New Type of Lie Algebra Firsly, we define a few noaions. Definiion Le α (α, α 2,..., α ) T, β (β, β 2,..., β ) T be wo column vecors, and define heir vecor produc α β by α β β α (α β, α 2 β 2,..., α β ) T. () By inroducing he diagonal marix α diag(α, α 2,..., α ), we have α β αβ. (2) Definiion 2 Le α (α, α 2,..., α ) T, A (0,..., 0, a i, 0,..., 0) N, we define heir produc by where α A A α (0,..., 0, α a i, 0,..., 0), a i (a i, a i2,..., a i ) T, The projec suppored by Naional Naural Science Foundaion of China under Gran No , he Liuhui Cener for Applied ahemaics, Nankai Universiy, and Tianjin Universiy and he Educaional Commiee of Liaoning Province of China under Gan No. 2004C057 xiac@yahoo.com.cn fcyou2008@yahoo.com.cn

2 No. 6 uli-componen Levi Hierarchy and Is uli-componen Inegrable Coupling Sysem 99 a ij R or C are real or complex numbers and are unrelaed o he independen variables x,. Definiion 3 Le x (x, 0, 0), x 2 (0, x 2 2, 0), x 3 (0, 0, x 3 3) be linear independen real or complex 3 marices and are independen of x and. Le X denoe a linear space generaed by {x, x 2, x 3 }, i.e. X {w a w a a x + a 2 x 2 + a 3 x 3 }, (3) a i (a i, a i2,..., a i ) T, (i, 2, 3) sand for real or complex vecors, x i i (x i, x i2,..., x i ) T, x ij R or C. Define a commuaion operaion [w a, w b ] in X by [w a, w b ] w a w b (a b 2 a 2 b ) x x 2 + (a b 3 a 3 b ) x x 3 + (a 2 b 3 a 3 b 2 ) x 2 x 3, (4) x x 2 x 2 x 2x 2, x 2 x 3 x 3 x 2 x, x x 3 x 3 x 2x 3, x i x i 0, i, 2, 3. (5) I is easy o verify ha for w a, w b, w c X, [ [w a, w b ], w c ] + [ [w b, w c ], w a ] + [ [w c, w a ], w b ] 0 (6) holds. Therefore, equaion (3) along wih Eqs. (4) and (5) becomes a Lie algebra. Specially, aking x (I, 0, 0), x 2 (0, I, 0), x 3 (0, 0, I ), (7) where I (,,..., ) T, }{{} equaion (4) becomes [w a, w b ] (a 2 b 3 a 3 b 2, 2(a b 2 a 2 b ), 2(a 3 b a b 3 )), which is jus he formula (4) in Ref. [9]. Taking x (I, 0, 0), x 2 2 (0, I, 0), x 3 2 (0, 0, I ), (8) equaion (4) becomes [w a, w b ] (a 2 b 3 a 3 b 2, a b 2 a 2 b, a 3 b a b 3 ), which is jus he formula (4) in Ref. [20]. Therefore, he Lie algebra (3) wih Eqs. (4) and (5) is of he generalized form of ha presened in Refs. [9] and [20]. Thus, we can easily derive all he resuls in Refs. [9] and [20] from Lie algebra (3) wih Eqs. (4), (7), and (8) by Tu scheme. Definiion 4 Se x i (n) x i λ n x i λ n, i, 2, 3, (9a) [x (m), x 2 (n)] [x 2 (n), x (m)] 2x 2 (m + n), [x 2 (m), x 3 (n)] [x 3 (n), x 2 (m)] x (m + n), [x (m), x 3 (n)] [x 3 (n), x (m)] 2x 3 (m + n), degx i (n) n, i, 2, 3. (9b) Then, X wih Eq. (9) consiss of a loop algebra X. Le us consider he following isospecral problems: φ x [U, φ], whose compaibiliy leads o φ [V, φ], U, V, φ X, (0) φ x [U, φ] + [U, [V, φ]] φ x [V x, φ] + [V, [U, φ]], [U, φ] [V x, φ] + [U, [V, φ]] + [V, [φ, U]] 0. () By employing Eq. (6), equaion () is expressed as [U, φ] [V x, φ] + [ [U, V ], φ] 0. (2) Due o φ being arbirary, equaion (2) holds if he following condiion does, U V x + [U, V ] 0, (3) which is jus he zero curvaure equaion. This indicaes ha he solion equaions derived from he Lax pair (0) are inegrable in Lax sense. 3 uli-componen Levi Hierarchy Consider an isospecral problem, φ x [U, φ], λ 0, U 2 I x () 2 (q r) x (0) + q x 2 (0) + r x 3 (0), (4) where Se q (q, q 2,..., q ) T, r (r, r 2,..., r ) T. V (a(0, i) x ( i) + b(0, i) x 2 ( i) + c(0, i) x 3 ( i)), i0

3 992 XIA Tie-Cheng, YOU Fu-Cai, and ZHAO Wen-Ying Vol. 44 where a(0, i) (a (0) i, a(0) i2,..., a(0) i )T, b(0, i) (b (0) i, b(0) i2,..., b(0) i )T, c(0, i) (c (0) i, c(0) i2,..., c(0) i )T. Solving he adjoin equaion leads o he recurrence relaions Denoing a x (0, i) q c(0, i) r b(0, i), b x (0, i) b(0, i + ) + (q r) b(0, i) 2q a(0, i), c x (0, i) c(0, i + ) (q r) c(0, i) + 2r a(0, i), V x [U, V ], (5) b(0, 0) c(0, 0) 0 (0, 0,..., 0) T, a(0, 0) I (,,..., ) T, }{{}}{{} b(0, ) 2q, c(0, ) 2r, a(0, ) 0, (6) V (n) + V (n) equaion (5) can be wrien as n (a(0, i) x (n i) + b(0, i) x 2 (n i) + c(0, i) x 3 (n i)), i0 λ n V V (n) +, V (n) +x + [U, V (n) + ] V (n) x [U, V (n) ]. (7) I is easy o verify ha he erms on he lef-hand side of Eq. (7) are of degree no less han zero, while he erms on he righ-hand side in Eq. (7) are of degree no bigger han zero. Thus, we have Take an arbirary modified erm for V (n) + as Noice V (n) V (n) + + n, i is easy o compue V (n) +x + [U, V (n) + ] b(0, n + ) x 2 (0) c(0, n + ) x 3 (0). n 2 (c(0, n) b(0, n) + 2a(0, n)) x (0). V (n) x + [U, V (n) ] 2 (c(0, n) b(0, n) + 2a(0, n)) x x (0) + (b(0, n + ) + q (c(0, n) Thus, he zero curvaure equaion gives rise o u b(0, n) + 2a(0, n)) x 2 (0) (c(0, n + ) + r c(0, n) b(0, n) + 2a(0, n))x 3 (0). ( ) q r U V (n) x + [U, V (n) ] 0, (8) ( ) b(0, n + ) q (c(0, n) b(0, n) + 2a(0, n)). c(0, n + ) + r (c(0, n) b(0, n) + 2a(0, n) By seing b(0, n) a(0, n), c(0, n) + a(0, n), we ge ( ) ( ) ( ) ( ) q 0 u J, (9) r 0 where J is Hamilonian operaor. From Eq. (6), he following recurrence relaion exiss, ( ) ( r + q r + r q q + q r ( ) ( c(0, n ) r + q r + r L q q + q r ( ) b(0, L n ). ) ( ) c(0, n ) ) ( )

4 No. 6 uli-componen Levi Hierarchy and Is uli-componen Inegrable Coupling Sysem 993 Therefore, he sysem (9) can be wrien as ( ) ( ) q u r J L 2r n. (20) 2q The sysem (20) has he following paricular cases. Case When n, he sysem becomes muli-componen linear equaions: { q 2q x, (2) r 2r x. Case 2 When n 2, he sysem becomes muli-componen nonlinear equaions, { q 2q xx + 4 qq x 4( qr) x, When q r, he sysem (22) becomes r 2r xx 4 rr x + 4( qr) x. (22) r 2r xx + 4 rr x, (23) which is he muli-componen Burgers equaion. When, he sysem (20) is he Levi hierarchy in Ref. [39]. When >, he sysem (20) is he mulicomponen Levi hierarchy. 4 uli-componen Inegrable Couplings Sysem In order o find coupling sysem of some known hierarchy derived from he loop Ã, Guo and Zhang once se forh a simple mehod in Ref. [34]. The main idea is o consruc an expanding loop algebra G of he loop algebra Ã, which has wo subalgebras G, G2, saisfying he following condiions: (i) G G G 2, G Ã ; (ii) [ G, G 2 ] G 2 ; where he symbols and sand for direc sum and isomorphic relaion, respecively. In erms of main idea in Ref. [34], le F denoe a linear space expanded by linear independen vecor {x, x 2, x 3, x 4, x 5 } (which are independen of x, ), define a commuaive operaion among x i, i, 2,..., 5 as [x, x 2 ] [x 2, x ] 2x 2, [x 2, x 3 ] [x 3, x 2 ] x, [x, x 3 ] [x 3, x ] 2x 3, [x, x 4 ] [x 4, x ] x 4, [x, x 5 ] [x 5, x ] x 5, [x 2, x 5 ] [x 5, x 2 ] x 4, [x 3, x 4 ] [x 4, x 3 ] x 5, [x i, x i ] 0, i, 2,..., 5. (24) Then F becomes a Lie algebra. In paricular, aking x (I, 0, 0, 0, 0), x 2 (0, I, 0, 0, 0), x 3 (0, 0, I, 0, 0), x 4 (0, 0, 0, I, 0), x 5 (0, 0, 0, 0, I ), we find w a a x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 (a, a 2, a 3, a 4, a 5 ), w a F. (25) The corresponding loop algebra is given as F {(x (n), x 2 (n), x 3 (n), x 4 (n), x 5 (n))}, (26) x i (n) x i λ n x i λ n, i, 2,..., 5, [x (m), x 2 (n)] [x 2 (n), x (m)] 2x 2 (m + n), [x 2 (m), x 3 (n)] [x 3 (n), x 2 (m)] x (m + n), [x (m), x 3 (n)] [x 3 (n), x (m)] 2x 3 (m + n), [x (m), x 4 (n)] [x 4 (n), x (m)] x 4 (m + n), [x (m), x 5 (n)] [x 5 (n), x (m)] x 5 (m + n), [x 2 (m), x 5 (n)] [x 5 (n), x 2 (m)] x 4 (m + n), [x 3 (m), x 4 (n)] [x 4 (n), x 3 (m)] x 5 (m + n), [x i, x i ] 0, i, 2,..., 5, deg x i (n) n, i, 2,..., 5. Le F () and F (2) be wo subalgebras of he loop algebra F, which are expressed as, respecively, F () {(x (n), x 2 (n), x 3 (n), 0, 0)}, F(2) {(0, 0, 0, x 4 (n), x 5 (n))}, (27) hen we find (iii) F F () F (2), F () X ;

5 994 XIA Tie-Cheng, YOU Fu-Cai, and ZHAO Wen-Ying Vol. 44 (iv) [ F (), F (2) ] F (2). Thanks o he loop algebra F and he relaions (iii) and (iv), consider he following isospecral problem: where φ x [U, φ], λ 0, U 2 I x () + 2 (u u 2 ) x (0) + u x 2 (0) + u 2 x 3 (0) + u 3 x 4 (0) + u 4 x 5 (0), (28) u (u (), u(2) 2,..., u() ) T, u 2 (u (), u(2) 2,..., u() ) T,... Denoing V (a(0, i) x ( i) + b(0, i) x 2 ( i) + c(0, i) x 3 ( i) + d(0, i) x 4 (i) + f(0, i) x 5 ( i)), i0 where a(0, i) (a (0) i, a(0) i2,..., a(0) i )T, b(0, i) (b (0) similar o Eq. (5) produces a x (0, i) u c(0, i) u 2 b(0, i), i, b(0) i2,..., b(0) i )T, c(0, i) (c (0) b x (0, i) b(0, i + ) + (u u 2 ) b(0, i) 2u a(0, i), c x (0, i) c(0, i + ) (u u 2 ) c(0, i) + 2u 2 a(0, i), i, c(0) i2 d x (0, i) 2 d(0, i + ) + 2 (u u 2 ) d(0, i) + u f(0, i) u 3 a(0, i) u 4 b(0, i), f x (0, i) 2 f(0, i + ) 2 (u u 2 ) f(0, i) + u 2 d(0, i) u 3 c(0, i) + u 4 a(0, i),,..., c(0) i )T, Solving he equaion b(0, 0) c(0, 0) d(0, 0) f(0, 0) 0 (0, 0,..., 0) T, a(0, 0) I (,,..., ) T, }{{}}{{} b(0, ) 2u, c(0, ) 2u 2, a(0, ) 0, d(0, ) 2u 3, f(0, ) 2u 4,... (29) Similarly o Eq. (7), we ge V (n) x + [U, V (n) ] b(0, n + ) x 2 (0) c(0, n + ) x 3 (0) + 2 d(0, n + ) x4 (0) 2 f(0, n + ) x5 (0). Taking an arbirary modified erm for V (n) + as by seing V (n) V (n) + + n, i is easy o compue n 2 (c(0, n) b(0, n) + 2a(0, n)) x (0), V (n) x + [U, V (n) ] 2 (c(0, n) b(0, n) + 2a(0, n)) x x (0) + (b(0, n + ) + q (c(0, n) b(0, n) + 2a(0, n)) x 2 (0) (c(0, n + ) + u 2 (c(0, n) b(0, n) + 2a(0, n)) x 3 (0) + 2 (d(0, n + ) + u 3 (c(0, n) b(0, n) + 2a(0, n)) x 4 (0) 2 (f(0, n + ) + u 4 (c(0, n) b(0, n) + 2a(0, n)) x 5 (0). Thus, he zero curvaure equaion (8) gives rise o u b(0, n + ) u (c(0, n) b(0, n) + 2a(0, n)) u 2 c(0, n + ) + u 2 (c(0, n) b(0, n) + 2a(0, n)) u u 3 2 (d(0, n + ) + u 3 (c(0, n) b(0, n) + 2a(0, n))). (30) u 4 2 (f(0, n + ) + u 4 (c(0, n) b(0, n) + 2a(0, n)))

6 No. 6 uli-componen Levi Hierarchy and Is uli-componen Inegrable Coupling Sysem 995 By aking b(0, n) a(0, n), c(0, n) + a(0, n), equaion (30) becomes u 2 u 3 2 u u 4 f(0, n + ) J 2 u 4 0 d(0, n + ) ũ3 2ũ ũ4 2ũ4 2 In erms of Eq. (29), a recurrence operaor L is given as f(0, n + ) d(0, n + ) c(0, n ) L L f(0, n + ) d(0, n + ) J f(0, n + ) d(0, n + ) f(0, n + ) d(0, n + ). (3) u 2 + u u 2 + u u u + u u u 3 2u 2 u 3 2u 4 u 2u 2 u 3 2u 4 u u u 2 2u 2 2u u 4 2u 3 u 2u 4 2u u 4 2u 3 u 2 2u 2 + u u 2 c(0, n ) ũ 2 + ũ ũ 2 + ũ ũ ũ + ũ ũ ũ 3 2ũ 2ũ 3 2ũ 4 ũ 2ũ 2ũ 3 2ũ 4 ũ ũ ũ 2 2ũ 2 2ũ ũ 4 2ũ 3 ũ 2ũ 4 2ũ ũ 4 2ũ 3 ũ 2 2ũ 2 + ũ ũ 2 c(0, n ). c(0, n ) As a resul, he sysem (3) can be wrien as u 2u 2 u 2 n 2u u J L u 3 2ũ 4 (u u 2 ) + 4u 4x. (32) u 4 2ũ 3 (u 2 u 3 ) 4u 3x According o he definiion of inegrable couplings, [35,36] we conclude ha he sysem (32) is a muli-componen inegrable couplings of he muli-componen Levi hierarchy. Obviously, aking u q, u 2 r, u 3 u 4 0, he sysem (32) reduces o he sysem (20). Therefore, we regard Eq. (32) as an expanding inegrable model of he sysem (20). The sysem (32) has he following paricular cases: Case When n, u q, and u 2 r, he sysem becomes muli-componen linear equaions, q 2q x, r 2r x, u 3 2u 3x, u 4 2u 4x, (33) which is a coupling sysem of he sysem (2). Case 2 When n 2, u q, and u 2 r, he sysem becomes muli-componen nonlinear equaions, q 2q xx + 4 qq x 4( qr) x, r 2r xx 4 rr x + 4( qr) x, which is a coupling sysem of he sysem (22). When q r, he sysem (34) becomes u 3 4u 3xx + ũ 3 u x 2ũ 4 u x + 4ũ u 3x ũ 3 u 2x 4ũ u 4x 4ũ 2 u 3x, u 4 4u 4xx ũ 4 u x + 4ũ u 4x ũ 4 u 2x 4ũ 2 u 4x + 4ũ 2 u 3x + 2ũ 3 u 2x, (34) r 2r xx + 4 rr x,

7 996 XIA Tie-Cheng, YOU Fu-Cai, and ZHAO Wen-Ying Vol. 44 u 3 4u 3xx + ũ 3 u x 2ũ 4 ux + 4ũ u 3x ũ 3 u 2x 4ũ u 4x 4ũ 2 u 3x, u 4 4u 4xx ũ 4 u x + 4ũ u4x ũ 4 u 2x 4ũ 2 u 4x + 4ũ 2 u 3x + 2ũ 3 u 2x, (35) which is a coupling sysem of he sysem (23). When, he sysem (32) is he inegrable couplings of he sandard Levi hierarchy. Remark The loop algebra presened in his paper can be used o oher known inegrable hierarchies of solion equaions for generaing he muli-componen sysems. Bu here exis wo open problems: one is how o improve our mehod and anoher is how o make he obained muli-componen inegrable sysems be Liouville inegrable. Furher, how do we look for Hamilonian srucures and conserved laws of such muli-componen inegrable hierarchies? These problems are worh while sudying in he fuure. Acknowledgmens One of he auhors (Tie-Cheng Xia) would like o express his sincere hanks o Profs. Y.F. Zhang and F.K. Guo for heirs valuable discussions. References [].J. Ablowiz and P.A. Clarkson, Solions, Nonlinear Evoluion Equaions and Inverse Scaering, Cambridge Universiy Press, Cambridge (99). [2] A.C. Newell, Solion in ahemaics and Physics, SIA, Philadelphia (985). [3] G.Z. Tu, J. ah. Phys. 30 (989) 330. [4] E.G. Fan, J. Phys. A: ah. Gen. 34 (200) 53. [5] T.C. Xia, X.H. Chen, and D.Y. Chen, Chaos, Solions and Fracals 22 (2004) 939. [6] T.C. Xia, X.H. Chen, D.Y. Chen, and Y.F. Zhang, Commun. Theor. Phys. (Beijing, China) 42 (2004) 80. [7] E.G. Fan, Physica A 30 (200) 05. [8] E.G. Fan, J. ah. Phys. 4 (2000) [9] W.X. a and R.G. Zhou, J. ah. Phys. 40 (999) 449. [0] Y.F. Zhang, Chaos, Solions and Fracals 8 (2003) 855. [] Y.F. Zhang and Q.Y. Yan, Chaos, Solions and Fracals 2 (2004) 43. [2] T.C. Xia, F.J. Yu, and D.Y. Chen, Physica A 343 (2004) 238. [3] C.W. Cao, Sci. China. Ser. A 33 (990) 528. [4] Y.B. Zeng, Phys. Le. A 60 (99) 54. [5] Y.B. Zeng and J. Hiearina, J. Phys. A ah. Gen. 29 (996) 524. [6] T.C. Xia, X.H. Chen, and D.Y. Chen, Chaos, Solions and Fracals 23 (2005) 033. [7] T. Tsuchida and. Wadai, Phys. Le. A 53 (999) 257. [8] T. Tsuchida and. Wadai, J. Phys. Soc. Jpn. 69 (999) 224. [9] F.K. Guo and Y.F. Zhang, J. ah. Phys. 44 (2003) [20] Y.F. Zhang, Chaos, Solions and Fracals 2 (2004) 305. [2] T.C. Xia, F.J. Yu, and D.Y. Chen, Commun. Theor. Phys. (Beijing, China) 42 (2004) 494. [22] G.Z. Tu and D.Z. eng, Aca ah. Appl. Sin. 5 (989) 89. [23] Y.F. Zhang, Q.Y. Yan, and H.Q. Zhang, Aca Phys. Sin. 52 (2003) 5 (in Chinese). [24] F.K. Guo and Y.F. Zhang, Aca Phys. Sin. 5 (2002) 95 (in Chinese). [25] Y.F. Zhang, Aca Phys. Sin. 53 (2003) 290 (in Chinese). [26] Y.F. Zhang, Chin. Phys. 2 (2003) 94. [27] D.J. Zhang, J. Phys. A ah. Gen. 35 (2002) [28] T.C. Xia, F.J. Yu, and D.Y. Chen, Chaos, Solions and Fracals 23 (2005) 63. [29] T. Tsuchida and. Wadai, J. Phys. Soc. Jpn. 65 (996) 353. [30] T. Tsuchida and. Wadai, J. Phys. Soc. Jpn. 67 (998) 75. [3] T.C. Xia, F.C. You, and D.Y. Chen, Chaos, Solions and Fracals 23 (2005) 9. [32] Y.S. Li, e al., Phys. Le. A 275 (2000) 60. [33] W.X. a and R. Zhou, Chin. Ann. ah. B 23 (2002) 373. [34] F.K. Guo and Y.F. Zhang, J. ah. Phys. 44 (2003) [35] B. Fusseiner, Coupling of Compleely Inegrable Sysems: he Perurbaion Bundle, In Applicaions of Analyic and Geomeic ehods o Nonlinear Differenial Equaions, ed. P.A. Clarksdon, Kluwer, Dordrech (993) p. 25. [36] W.X. a and B. Fuchsseiner, Chaos, Solions and Fracals 7 (996) 227. [37] X.X. Xu, Phys. Le. A 326 (2004) 99. [38] X.X. Xu and Y.F. Zhang, Commun. Theor. Phys. (Beijing, China) 4 (2004) 32. [39] Y.F. Zhang and Q.Y. Yan, Chaos, Solions and Fracals 6 (2003) 263. [40] H.W. Tam and Y.F. Zhang, Chaos, Solions and Fracals 23 (2005) 535. [4] F.K. Guo, J. Sys. Sci. and ah. Sci. 22 (2002) 36. [42] F.K. Guo, Aca ah. Appl. Sin. 23 (2000) 8. [43] Y.F. Zhang, Phys. Le. A 37 (2003) 280. [44] F.K. Guo and Y.F. Zhang, Aca. ah. Sin. 27 (2004) 349. [45] F.K. Guo, Y.F. Zhang, and Q.Y. Yang, In. J. Theor. Phys. 43 (2004) 39.