Research Article Two Expanding Integrable Models of the Geng-Cao Hierarchy

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1 Abstract ad Applied Aalysis Volume 214, Article ID 86935, 7 pages Research Article Two Epadig Itegrable Models of the Geg-Cao Hierarchy Xiurog Guo, 1 Yufeg Zhag, 2 ad Xupig Zhag 1 1 Departmet of Basic Courses, Shadog Uiversity of Sciece ad Techology, Tai a, Shadog 27119, Chia 2 College of Sciece, Chia Uiversity of Miig ad Techology, Xuzhou , Chia Correspodece should be addressed to Xiurog Guo; sprigmig7@126.com Received 21 November 213; Accepted 13 Jauary 214; Published 27 February 214 Academic Editor: Tiecheg Xia Copyright 214 Xiurog Guo et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. As far as liear itegrable coupligs are cocered, oe has obtaied some rich ad iterestig results. I the paper, we will deduce two kids of epadig itegrable models of the Geg-Cao (GC) hierarchy by costructig differet 6-dimesioal Lie algebras. Oe epadig itegrable model (actually, it is a oliear itegrable couplig) reduces to a geeralized Burgers equatio ad further reduces to the heat equatio whose epadig oliear itegrable model is geerated. Aother oe is a epadig itegrable model which is differet from the first oe. Fially, the Hamiltoia structures of the two epadig itegrable models are obtaied by employig the variatioal idetity ad the trace idetity, respectively. 1. Itroductio Itegrable coupligs are a kid of epadig itegrable models of some kow itegrable hierarchies of equatios. Based o this theory, oe has obtaied some itegrable coupligs of the kow itegrable hierarchies [1 8]. These itegrable coupligs are all liear with respect to the coupled variables. That is, if we itroduce a evolutio equatio U t = K(u thecoupledvariablev satisfyig V t = S(u, V) is liear i V.ThereasoforthismaybegivebyspecialLiealgebras. That is, such a Lie algebra G ca be decomposed ito a sum of the two subalgebras G 1 ad G 2,whichmeets G=G 1 G 2, [G 1,G 2 ] G 2. (1) If the subalgebra G 2 is ot simple, the the itegrable couplig U t =K(u), V t =S(u, V) is liear with respect to the variable V, whichisobtaiedby itroducig La pairs through the Lie algebra G. However, it is more iterestig to seek for oliear itegrable coupligs because most of the coupled dyamics from physics, mechaics, ad so forth are oliear. Recetly, Ma ad Zhu [9] itroduced a kid of Lie algebra to deduce the (2) oliear itegrable coupligs of the oliear Schrödiger equatio ad so forth, the Lie subalgebras are simple ad are differet from the above. Based o this, Zhag [1] proposed a simple ad efficiet method for geeratig oliear itegrable coupligs ad obtaied the oliear itegrable coupligs of the Giachetti-Johso (GJ) hierarchy ad the Yag hierarchy, respectively. I additio, Zhag ad Ho [11] proposed aother Lie algebra which is differet from those i [9, 1] to deduce oliear itegrable coupligs. Wei ad Xia [12] alsoobtaiedsomeoliearitegrable coupligs of the kow itegrable hierarchies. Ithepaper,wewattostartfromaspectralproblem proposed by Geg ad Cao [13] todeduceaitegrable hierarchy (called the GC hierarchy) uder the frame of zero curvature equatios by the Tu scheme [14]ad obtai its ew Hamiltoia structure. The with the help of a 6-dimesioal Lie algebra, a oliear epadig itegrable model of the GC hierarchy is obtaied, whose Hamiltoia structure is geerated by the variatioal idetity preseted i [15]. The epadig itegrable model ca reduce to a geeralized Burgers equatio ad further reduce to the heat equatio. Aother ew 6-dimesioal Lie algebra is costructed for which the secod epadig itegrable model is produced by usig the Tu scheme whose Hamiltoia structure is derived from the trace idetity proposed by Tu [14]. We shall fid the two epadig itegrable models of the GC hierarchy are differet.

2 2 Abstract ad Applied Aalysis 2. The GC Itegrable Hierarchy ad Its Hamiltoia Structure We have kow that h=( e=( f=( 1 (3) the oe gets [h, e] = 2e, [h, f] = 2f, [e, f] =h. (4) It is well kow that spa{h, e, f} = A 1 is a Lie algebra. A loop algebra of A 1 is give by A 1 = spa {h (),e(),f()}, (5) h () =h() λ, e() =e() λ, h () =h() λ, Z. By usig the loop algebra A 1, itroduce a isospectral problem [13]: (6) λ λu U=( )= h(1) +ue(1) + Vf (). (7) V λ V=V 1 h () +V 2 e () +V 3 f (), (8) V i = V im h ( m), i = 1,2,3. (9) m The statioary equatio V =[U,V]admits that a solutio for the V is as follows: which gives rise to (V 1m ) =uv 3m VV 2m, (V 2m ) = 2V 2,m+1 2uV 1,m+1, (V 3m ) =2V 3,m+1 +2VV 1,m+1, (1) (V 1,m+1 ) = 1 2 u(v 3m) 1 2 V(V 2m). (11) V 1, =V 2, =V 3, =, V 1,1 =α; (12) from (1)ad(11)wehave deote by V () = We have (V 1m h ( m) +V 2m e ( m) +V 3m f ( m)). m= (14) V () +[U,V () ]=(2V 2,+1 +2uV 1,+1 )e(1) 2(V 3,+1 +2V 1,+1 )f(). The compatibility of the followig La pair V () = gives rise to U= h(1) +ue(1) + Vf () (15) (V 1m h ( m) +V 2m e ( m) +V 3m f ( m)) m= (16) ( u V ) t =( 2V 2,+1 2uV 1,+1 2V 3,+1 +2VV 1,+1 )=( (V 2) (V 3 ) ) =( )(V 3 V 2 )=J( V 3 V 2 (17) J=( ) (18) is a Hamiltoia operator. By the trace idetity preseted i [14], we have u =λv 3, V =λv 2, λ = 2λV 1 +uv 3. Substitutig the above results to the trace idetity yields (19) w ( 2λV 1 +uv 3 )=λ γ λ λγ ( λv 3 (2) λv 2 w = ( u, V ) T. (21) Comparig the coefficiets of λ of both sides i (2)gives w ( 2V 1,+1 +uv 3,+1 )=( +1+γ)( λv 3 λv 2 ). (22) V 2,1 = αu, V 3,1 = αv, V 1,2 = α 2 uv, V 2,2 = α 2 (u +u 2 V), V 3,2 = α 2 ( V +uv 2... (13) It is easy to see γ= 1.Thus,wehave ( λv 3 λv 2 )= u (2V 1,+1 uv 3,+1 )= H u, (23)

3 Abstract ad Applied Aalysis 3 H = 2V 1,+1 uv 3,+1 (24) are Hamiltoia coserved desities of the La itegrable hierarchy (17). Therefore, we get a Hamiltoia form of the hierarchy (17) as follows: ( u V ) =J H t u. (25) Let us cosider the reduced cases of (17). Whe =1,weget that u t1 = αu, V t1 = αv. (26) Takig =2, oe gets a geeralized Burgers equatio: u t2 = α 2 u +αuu V + α 2 u2 V, V t2 = α 2 V + α 2 u V 2 +αuvv. (27) Remark 1. The Hamiltoia structure (23) is differet from thati [14].We call (17) the GC hierarchy. 3. The First Epadig Itegrable Model of the GC Hierarchy Zhag ad Tam [16] proposed a few kids of Lie algebras to deduce oliear itegrable coupligs. I the sectio we will chooseoeofthemtoivestigatetheoliearitegrable couplig of the hierarchy (17). Cosider the followig Lie algebra: F=spa {f 1,...,f 6 }, (28) f 1 =( e 1 f e 2 =( e 2 f 1 e 3 =( e 3 2 e 3 f 4 =( e 1 e 1 f 5 =( e 2 e 2 f 6 =( e 3 e 3 e 1 =( 1 1 e 2 =( 1 e 3 =( 1 ). (29) Defie [a, b] = ab ba, a, b F. (3) It is easy to compute that [f 1,f 2 ]=2f 2, [f 1,f 3 ]= 2f 3, [f 2,f 3 ]=f 1, [f 1,f 4 ]=, [f 1,f 5 ]=2f 5, [f 1,f 6 ]= 2f 6, [f 2,f 4 ]= 2f 5, [f 2,f 5 ]=, [f 2,f 6 ]=f 4, [f 3,f 4 ]=2f 6, F 1 = spa {f 1,f 2,f 3 }, F 2 = spa {f 4,f 5,f 6 }; (32) we have F 1 =F 1 F 2, [F 1,F 2 ] F 2 ; (33) F 1 ad F 2 are all simple Lie-subalgebras of the Lie algebra F. The correspodig symmetric costat matri M appearig i the variatioal idetity is that 2η 1 2η 2 η 1 η 2 ( η 1 η 2 ). 2η 2 2η 2 (34) η 2 η 2 ( η 2 η 2 ) A loop algebra correspodig to the Lie algebra F is defied by F =spa {f 1 (),...,f 6 ()}, f i () =f i λ, [f i (m),f j ()] =[f i,f j ]λ m+, 1 i, j 6,m, Z. (35) We use the loop algebra F to itroduce a La pair: U= f(1) +uf 2 (1) + Vf 3 () +u 1 f 5 (1) +u 2 f 6 (), V= (V 1m f 1 (1 m) +V 2m f 2 (1 m) +V 3m f 3 ( m) m +V 4m f 4 (1 m) +V 5m f 5 (1 m) +V 6m f 6 ( m)). (36) The statioary equatio V =[U,V]is equivalet to (V 1m ) =uv 3m VV 2m, (V 2m ) = 2V 2,m+1 2uV 1,m+1, (V 3m ) =2V 3,m+1 +2VV 1,m+1, (V 4m ) =u 1 V 3m u 2 V 2m (V +u 2 )V 5m +(u+u 1 )V 6m, (V 5m ) = 2V 5,m+1 2u 1 V 1,m+1 2(u+u 1 )V 4,m+1, (V 6m ) =2V 6,m+1 +2u 2 V 1,m+1 +2(V +u 2 )V 4,m+1 (37) from which we have (V 1,m+1 ) = 1 2 u(v 3m) 1 2 V(V 2m), [f 3,f 5 ]= f 4, [f 3,f 6 ]=, [f 4,f 5 ]=2f 5, (V 4,m+1 )= 1 2 u 1(V 3m ) u 2(V 2m ) (38) [f 4,f 6 ]= 2f 6, [f 5,f 6 ]=f 4. (31) (u + u 2)(V 5m ) (u + u 1)(V 6m ).

4 4 Abstract ad Applied Aalysis V 1, =V 2, =V 3, =V 4, =V 5, =V 6, =, we obtai from (37) V 1,1 =α; (39) itegrable model of the geeralized Burgers equatio (27)as follows: u t2 = α 2 u +αuu V + α 2 u2 V, V t2 = α 2 V + α 2 u V 2 +αuvv, V 3,1 = αv, V 2,1 = αu, V 4,1 =, V 4,1 = αu 1, V 6,1 = αu 2, V 1,2 = α 2 uv, V 4,2 = α 2 (u 1V +uv 2 +u 1 u 2 V 5,2 = α 2 (u 1+u 1 uv +(u+u 1 )(u 1 V +uu 2 +u 1 u 2 )), V 6,2 = α 2 ( u 2+u 2 uv +(V +u 2 )(u 1 V +uu 2 +u 1 u 2 )... (4) u 1t2 u 2t2 = α 2 (u 1 +u 1 uv +u 1 (uv) +(u+u 1 ) (u 1 V +uu 2 +u 1 u 2 ) +(u+u 1 )(u 1 V +uu 2 +u 1 u 2 ) = α 2 ( u 2 +u 2 uv +u 2 (uv) +(V +u 2 ) (u 1 V +uu 2 +u 1 u 2 ) +(V +u 2 )(u 1 V +uu 2 +u 1 u 2 ) ). (45) Note V () = (V 1m f 1 (1+ m) +V 2m f 2 (1+ m) m= +V 3m f 3 ( m) +V 4m f 4 (1+ m) adirectcalculatioyields V () +[U,V () ] +V 5m f 5 (1+ m) +V 6m f 6 ( m)); = 2(V 3,+1 + VV 1,+1 )f 3 () +(2V 2,+1 +2uV 1,+1 )f 2 (1) 2((V +u 2 )V 4,+1 +u 2 )V 1,+1 +V 6,+1 f 6 () +2((u+u 1 )V 4,+1 +u 1 )V 1,+1 +V 5,+1 f 5 (1) = (V 3 ) f 3 () (V 2 ) f 2 (1) (V 6 ) f 6 () (V 5 ) f 5 (1). Therefore, zero curvature equatio (41) (42) U t V + [U, V() ] = (43) Obviously, the coupled equatios are oliear with respect to the coupled variables u 1 ad u 2. Therefore, the hierarchy (44) is a oliear epadig itegrable model of the itegrable system (17); actually, it is a oliear itegrable couplig. The oliear epadig itegrable model (45) ca be writteastwoparts,oeisjustright(27); aother oe is the latter two equatios i (45 which ca be regarded as a coupled oliear equatio with variable coefficiets u, V, ad their derivatives i the variable,thefuctiosu, V satisfy (27).Iparticular,wetakeatrivialsolutioof(27) to be u=v =;the(45) reduces to the followig equatios: u 1t2 = α 2 [u 1, +(u 2 1 u 2) ], u 2t2 = α 2 [ u 2, +(u 2 2 u 1) ]. (46) Whe we set u 2 =,theaboveequatiosreducetothewellkow heat equatio. I order to deduce Hamiltoia structure of the oliear itegrable couplig (44 we defie a liear fuctioal [11]: {a, b} =a T Mb, (47) a=(a 1,...,a 6 ) T, b=(b 1,...,b 6 ) T. It is easy to see that the Lie algebra F is isomorphic to the Lie algebra R 6 if equipped with a commutator as follows: admits that u (V 2 ) V (V ( )=( 3 ) ). (44) u 1 (V 5 ) u 2 (V 6 ) [a, b] T =(a 2 b 3 a 3 b 2,2a 1 b 2 2a 2 b 1,2a 3 b 1 2a 1 b 3,a 2 b 6 a 6 b 2 +a 5 b 3 a 3 b 5 +a 5 b 6 a 6 b 5,2a 1 b 5 2a 5 b 1 +2a 4 b 2 2a 2 b 4 +2a 4 b 5 2a 5 b 4,2a 3 b 4 2a 4 b 3 +2a 6 b 1 u 1 = u 2 =,(44) reduces to the itegrable hierarchy (17). Whe we take = 2, we get a epadig oliear 2a 1 b 6 +2a 6 b 4 2a 4 b 6 ). (48)

5 Abstract ad Applied Aalysis 5 Thus, uder the Lie algebra R 6,theLapair(36) cabe writte as U = ( λ, uλ, V,,u 1,λ,u 2 ) T, (49) V=(V 1 λ, V 2 λ, V 3,V 4 λ, V 5 λ, V 6 ) T. I terms of (48)ad(49)weobtaithat u }=(η 1V 3 +η 2 V 6 )λ, V }=(η 1V 2 +η 2 V 5 )λ, u 1 }=(η 2 V 3 +η 2 V 6 )λ, u 2 }=(η 2 V 2 +η 2 V 5 )λ, λ }= 2η 1V 1 +(η 1 u+η 2 u 1 )V 3 (5) 2η 2 V 4 +(η 2 u+η 2 u 1 )V 6. Substitutig the above results ito the variatioal idetity yields w ( 2η 1 V 1 +(η 1 u+η 2 u 1 )V 3 +η 2 (u + u 1 )V 6 )d (η 1 V 3 +η 2 V 6 )λ (51) =λ γ (η λ λγ ( 1 V 2 +η 2 V 5 )λ (η 2 V 3 +η 2 V 6 )λ (η 2 V 2 +η 2 V 5 )λ w = ( T u, V,, ). (52) u 1 u 2 Comparig the coefficiets of λ o both sides i (51)gives w ( 2η 1 V 1,+1 +(η 1 u+η 2 u 1 )V 3,+1 +2η 2 V 4,+1 +η 2 (u + u 1 )V 6,+1 )d η 1 V 3 +η 2 V 6 η =( +1+γ)( 1 V 2 +η 2 V 5 ). η 2 V 3 +η 2 V 6 η 2 V 2 +η 2 V 5 From (37)we have γ= 1.Thus,we get that (53) η 1 V 3 +η 2 V 6 η ( 1 V 2 +η 2 V 5 H )= +1 η 2 V 3 +η 2 V 6 u, (54) η 2 V 2 +η 2 V 5 H +1 = (2η 1 V 1,+1 (η 1 u+η 2 u 1 )V 3,+1 1/ +2η 2 V 4,+1 η 2 (u + u 1 )V 6,+1 )d. (55) Therefore, we obtai the Hamiltoia structure of the oliear itegrable couplig (44) as follows: w ( 2η 1 V 1,+1 +(η 1 u+η 2 u 1 )V 3,+1 u V W t =( ) u 1 u 2 2η 2 V 4,+1 +η 2 (u + u 1 )V 6,+1 )d, t = ( ( η 1 V 3 +η 2 V 6 η ( 1 V 2 +η 2 V 5 ) η 2 V 3 +η 2 V 6 η 2 V 2 +η 2 V 5 η 1 V 3 +η 2 V 6 η =J( 1 V 2 +η 2 V 5 )=J η 2 V 3 +η 2 V 6 η 2 V 2 +η 2 V 5 J is obviously Hamiltoia. η 1 ( )η 2 H +1 u, η 1 ( )η 2 ) ) 4. The Secod Epadig Itegrable Model of the GC Hierarchy (56) I this sectio we costruct a ew 6-dimesioal Lie algebra to discuss the secod itegrable couplig of the GC hierarchy. It is easy to see that h 1 =f 1, h 2 =f 2, h 3 =f 3, h j =( e (57) j 3 e j 3 ), j = 4, 5, 6. [h 1,h 2 ]=2h 2, [h 1,h 3 ]= 2h 3, [h 2,h 3 ]=h 1, [h 1,h 4 ]=, [h 1,h 5 ]=2h 5, [h 1,h 6 ]= 2h 6, [h 2,h 4 ]= 2h 5, [h 2,h 5 ]=, [h 2,h 6 ]=h 4, [h 3,h 4 ]=2h 6, [h 3,h 5 ]= h 4, [h 3,h 6 ]=, [h 4,h 5 ]=2h 2, [h 4,h 6 ]= 2h 3, [h 5,h 6 ]=h 1. (58)

6 6 Abstract ad Applied Aalysis If we set G=spa{h 1,...,h 6 }, G 1 = spa{h 1,h 2,h 3 },adg 2 = spa{h 4,h 5,h 6 },thewehavethat G=G 1 +G 2, [G 1,G 2 ] ot i G 2. (59) Hece, the itegrable coupligs of the GC hierarchy caot be geerated by the Lie algebra G as above uder the frame of the Tu scheme. I what follows, we will deduce a oliear epadig itegrable model of the GC hierarchy. U= h 1 (1) +uh 2 (1) + Vh 3 () +w 1 h 5 (1) +w 2 h 6 (), V= (V 1m h 1 (1 m) +V 2m h 2 (1 m) +V 3m h 3 ( m) m +V 4m h 4 (1 m) +V 5m h 5 (1 m) +V 6m h 6 (1 m) h i (m) = h i λ m, i = 1, 2, 3, 4, 5, 6. Solvig the statioary zero curvature equatio gives rise to (6) V = [U, V] (61) (V 1m ) =uv 3m VV 2m w 2 V 5m +w 1 V 6m, (V 2m ) = 2V 2,m+1 2uV 1,m+1 2w 1 V 4,m+1, (V 3m ) =2V 3,m+1 +2VV 1,m+1 +2w 2 V 4,m+1, (V 4m ) =uv 6m VV 5m +w 1 V 3m w 2 V 2m, (V 5m ) = 2V 5,m+1 2uV 4,m+1 2w 1 V 1,m+1, (V 6m ) =2V 6,m+1 +2VV 4,m+1 +2w 2 V 1,m+1. (62) Let V 2,1 = αu, V 3,1 = αv, V 5,1 = αw 1,adV 6,1 =αw 2 ; the oe gets from (62)that V 1,2 = α 2 uv, V 2,2 = α 2 (u +u 2 V V 3,2 = α 2 ( V +uv 2 ), V 4,1 =, V 4,2 = α 2 (uw 2 w 1 V V 5,2 = α 2 (w 1, u 2 w 2 +w 1 uv V 6,2 = α 2 (w 2, + V 2 w 1... (63) Notig V () + = m= (V 1mh 1 (1+ m)+v 2m h 2 (1+ m)+ V 3m h 3 ( m)+v 4m h 4 (1+ m)+v 5m h 5 (1+ m)+v 6m h 6 ( m)) = λ V V (),oeifersthat V() +, +[U,V() + ] = (2V 2,+1 + 2uV 1,+1 +2w 1 V 4,+1 )h 2 (1) 2(V 3,+1 +VV 1,+1 +w 2 V 4,+1 )h 3 ()+ (2V 5,+1 +2uV 4,+1 +2w 1 V 1,+1 )h 5 (1) 2(V 6,+1 + VV 4,+1 + w 2 V 1,+1 )h 6 (). V () =V () +,byemployigthezerocurvatureequatio U t V () + [U, V () ] = (64) we have u V u t = ( ) w 1 w 2 t Whe =2, α=2,(65)reducesto 2V 2,+1 2uV 1,+1 2w 1 V 4,+1 2V = ( 3,+1 +2VV 1,+1 +2w 2 V 4,+1 ). 2V 5,+1 2uV 4,+1 2w 1 V 1,+1 2V 6,+1 +2VV 4,+1 +2w 2 V 1,+1 u t2 =u +(u 2 V), V t2 = V +(uv 2 ), w 1,t2 =w 1, (u 2 w 2 ) +(w 1 uv), w 2,t2 =w 2, +(V 2 w 1 ). (65) (66) It is remarkable that (66) is liear with respect to the variables w 1, w 2 ;however,itisoliear. Equatio (6)cabewritteas By computig that λ uλ w 1 λ V λ w U=( 2 w 1 λ λ uλ w 2 V λ V 1 λ V 2 λ V 4 λ V 5 λ V V=( 3 V 4 λ V 1 λ V 5 λ V 6 V 1 λ V 4 λ ). V 2 λ V 6 V 4 λ V 3 V 1 λ λ u =( λ 1 =( w 2 1 thus, we have u =2λV 3, V, w 1 =2λV 6, λ =( w 1 λ V =( 1 1 (67) 1 u w 1 λ =( 1 w 1 1 u 1 (68) V =2λV 2, V, w 2 =2λV 5, λ = 4λV 1 +2uV 3 +2w 1 V 6. (69)

7 Abstract ad Applied Aalysis 7 Substitutig the above cosequeces ito the trace idetity proposed by Tu [14]yieldsthat 2λV 3 u ( 4λV 1 +2uV 3 +2w 1 V 6 )=λ γ 2λV λ λγ ( 2 ). (7) 2λV 6 2λV 5 Comparig the coefficiets of λ gives u ( 4V 1, +2uV 3, 1 +2w 1 V 6, 1 ) 2V 3 2V 2 =( +1+γ)( ). 2V 6 2V 5 (71) Isertig the iitial values i (62)gives γ= 1. Therefore, we obtai that 2V 3 2V ( 2 )= 2V 6 u (4V 1 2uV 3, 1 2w 1 V 6, 1 ) H u, 2V 5 (72) H = (1/)(4V 1 2uV 3, 1 2w 1 V 6, 1 ) are coserved desities of the epadig itegrable model (65). Thus, (65) cabewritteasthehamiltoiastructure u V u t =( ) w 1 w 2 t =J H u, (73) J = (1/2)( = /,isahamiltoia operator. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. [3] W.-X. Ma, X.-X. Xu, ad Y. Zhag, Semi-direct sums of Lie algebras ad cotiuous itegrable coupligs, Physics Letters A,vol.351,o.3,pp ,26. [4] Y. Zhag, A geeralized multi-compoet Glachette-Johso (GJ) hierarchy ad its itegrable couplig system, Chaos, Solitos ad Fractals,vol.21,o.2,pp.35 31,24. [5] Y.F.ZhagadH.Q.Zhag, Adirectmethodforitegrable coupligs of TD hierarchy, Joural of Mathematical Physics,vol. 43, o. 1, pp , 22. [6] T.C.XiaadF.C.You, Multi-compoetDiracequatiohierarchy ad its multi-compoet itegrable coupligs system, Chiese Physics B,vol.16,o.3,p.65,27. [7] Y.F.ZhagadF.K.Guo, MatriLiealgebrasaditegrable coupligs, Commuicatios i Theoretical Physics, vol. 46, o. 5, pp , 26. [8] E.FaadY.Zhag, Asimplemethodforgeeratigitegrable hierarchies with multi-potetial fuctios, Chaos, Solitos ad Fractals,vol.25,o.2,pp ,25. [9] W.-X. Ma ad Z.-N. Zhu, Costructig oliear discrete itegrable Hamiltoia coupligs, Computers & Mathematics with Applicatios,vol.6,o.9,pp ,21. [1] Y. F. Zhag, Lie algebras for costructig oliear itegrable coupligs, Commuicatios i Theoretical Physics, vol. 56, p. 85, 211. [11] Y. F. Zhag ad Y. C. Ho, Some evolutio hierarchies derived from self-dual Yag-Mills equatios, Commuicatios i Theoretical Physics,vol.56,p.856,211. [12] H. Y. Wei ad T.C. Xia, Noliear itegrable coupligs of super Kaup-Newell hierarchy ad its super Hamiltoia structures, Acta Physica Siica, vol. 62, Article ID 1222, 213. [13] X. G. Geg ad C. W. Cao, Quasi-periodic solutios of the 2+ 1 dimesioal modified Korteweg-de Vries equatio, Physics Letters A,vol.261,o.5-6,pp ,1999. [14] G. Z. Tu, The trace idetity, a powerful tool for costructig the Hamiltoia structure of itegrable systems, Joural of Mathematical Physics, vol. 3, o. 2, pp , [15] W.-X. Ma ad M. Che, Hamiltoia ad quasi-hamiltoia structures associated with semi-direct sums of Lie algebras, Joural of Physics A,vol.39,o.34,pp ,26. [16] Y.-F. Zhag ad H. Tam, A few ew higher-dimesioal Lie algebras ad two types of couplig itegrable coupligs of the AKNS hierarchy ad the KN hierarchy, Commuicatios i Noliear Sciece ad Numerical Simulatio, vol.16,o.1,pp , 211. Ackowledgmets This work was supported by the Natural Sciece Foudatio of Chia ( ) ad the Natural Sciece Foudatio of Shadog Provice (ZR212AQ11, ZR212AQ15). Refereces [1] W. X. Ma ad B. Fuchssteier, Itegrable theory of the perturbatio equatios, Chaos, Solitos ad Fractals,vol.7,o. 8, pp , [2] W.-X. Ma, Itegrable coupligs of solito equatios by perturbatios I. A geeral theory ad applicatio to the KdV hierarchy, Methods ad Applicatios of Aalysis,vol.7,o.1,pp , 2.

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