Research Article (2+1)-Dimensional mkdv Hierarchy and Chirp Effect of Rossby Solitary Waves

Transcription

1 Advances in Mathematica Physics Voume 15, Artice ID 1658, 7 pages Research Artice (+1)-Dimensiona mkdv Hierarchy and Chirp Effect of Rossby Soitary Waves Chunei Wang, 1 Yong Zhang, Baoshu Yin, 3,4 and Xiaoen Zhang 1 Coege of Mathematics and Systems Science, Beihang University, Beijing 183, China Coege of Mathematics and Systems Science, Shandong University of Science and Technoogy, Qingdao 6659, China 3 InstituteofOceanoogy,ChineseAcademyofSciences,Qingdao6671,China 4 Key Laboratory of Ocean Circuation and Wave, Chinese Academy of Sciences, Qingdao 6671, China Correspondence shoud be addressed to Baoshu Yin; baoshuyin@16.com Received 3 January 15; Revised 17 Apri 15; Accepted 17 Apri 15 Academic Editor: Boris G. Konopechenko Copyright 15 Chunei Wang et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. By constructing a kind of generaized Lie agebra, based on generaized Tu scheme, a new ( +1)-dimensiona mkdv hierarchy is derived which popuarizes the resuts of (1 +1)-dimensiona integrabe system. Furthermore, the ( +1)-dimensiona mkdv equation can be appied to describe the propagation of the Rossby soitary waves in the pane of ocean and atmosphere, which is different from the (1 +1)-dimensiona mkdv equation. By virtue of Riccati equation, some soutions of ( +1)-dimensiona mkdv equation are obtained. With the hep of soitary wave soutions, simiar to the fiber soiton communication, the chirp effect of Rossby soitary waves is discussed and some concusions are given. 1. Introduction In soiton theory, it is an important task to find new integrabe hierarchies and their couping systems. With the deveopment of soiton theory, peope began to concern about the (+ 1)-dimensiona hierarchies. Seeking the Lax pair is a current way to get the ( + 1)-dimensiona hierarchies. Different approaches to generate the integrabe systems have been proposed [1 11.Based on the generaized Tu scheme,many (1 + 1)-dimensiona and the corresponding Hamitonian structures were obtained, such as the Dirac system [1, 13 and the NLS-mKdV system [14. Tu Guizhang put forward a scheme for generating( + 1)-dimensiona hierarchies by using a residue operator with an associative agebra A[ξ which incudes a pseudodifferentia operators N i= a iξ i, where the operator ξ is defined by ξf = fξ + ( y f), f A. Recenty, Zhang gave some ( + 1)-dimensiona integrabe hierarchies [15, 16,butitisdifficuttosovethese equations by using the usua ways which are mentioned in [15, 16. Meanwhie, the appications of these equations were not mentioned. In this paper, we wi use some tricks to dea with the coefficients of the spectra operator and make some coefficients become unreated to the variabe y. Furthermore, we obtain some soutions of the ( + 1)- dimensiona mkdv equation based on the cassica Riccati equation. In particuar, by empoying these soutions, simiar to fiber soiton communication, we wi study the chirp effect of Rossby soitary waves. Firsty, in order to get the ( + 1)-dimensiona mkdv hierarchy, we recommend the foowing formuas: ξ n f= [ n i i ( i f) ξ n i, fξ n = ( 1) i [ n i i ξn i ( i f), and introduce a residue operator (1) R:A[ξ A, R( a i ξ i )=a 1. () Secondy, we introduce the Rossby soitary waves, which are crucia for the dynamics in ocean and atmosphere. Yang et a. [17 gotthe(1 + 1)-dimensiona mkdv equation

2 Advances in Mathematica Physics and used it to describe the Rossby soitary waves in the atmosphere and ocean. The resuts showed that the ampitude of Rossby soitary waves which propagate in a ine satisfies the (1 + 1)-dimensiona mkdv equation. But actuay as we a know that the Rossby soitary waves spread in a pane because oftheterrainoftheseaandintheoceanandtheatmosphere, so we think the ( + 1)-dimensiona mkdv equation is more suitabe for describing the Rossby soitary waves. Finay, we refer to the chirp effect which is caused by the excursion of the center wave and impacted by the dispersion and the noninear effect. As we a know, there exists frequency moduation effect in the ight source, which is caed chirp effect. Because the different parts of the puse can produce different frequencies, the fiber soiton must be impacted by the chirp effect during the transmission. Simiar to the fiber soiton communication, we think the chirp effect occurs aso in the process of propagation of a kinds of waves which happen in the atmosphere and ocean. In [18, Song et a. used the noninearschrödinger equation to discuss the chirp effect of interna soitary waves in ocean by means of the chirp effect in fiber soitary communication. Then in this paper, we use the same way to describe the chirp effect of therossbysoitarywavesbasedonthe( + 1)-dimensiona mkdv equation. Theaimofthispaperistogetthe( + 1)-dimensiona mkdv equation and describe the chirp effect of the Rossby soitary waves with the hep of initia soitary wave soution of the ( + 1)-dimensiona mkdv equation. The organization of the paper is as foows: in Section, we first formuate a specia M to get a generaized Lie agebra which is different from the cassica Lie agebra. The ( + 1)-dimensiona mkdv hierarchy and the corresponding Hamitonian structures [19 5 are presented according to the generaized Tu scheme and on the basis of the Riccati equation, the soitary wave soutions, periodica wave soutions, and Rationa function wave soutions of the (+1)-dimensiona mkdv equation are obtained and paced in Section 3. Section 4 is used to discuss the chirp effect of the Rossby soitary waves by choosing the initia soitary wave soution. Finay, some concusions are paced in Section 5.. A Generaized Lie Agebra and ( + 1)-Dimensiona mkdv Hierarchy Consider the foowing set of Lie agebras: e 1 =( 1 1 ), e =( 1 ), (3) and the commutator is defined as [e i,e j =e i Me j e j Me i, i,j=1,,3. (5) It is easy to compute that [e 1,e =e, [e 1,e 3 =e e 3, [e,e 3 = e 1, which is different from the Lie agebra s(). A oop agebra of the extended Lie agebra is presented as e i (n) =e i λ n, [e i(m),e j(n) =[e i,e j λ m+n, Consider an isospectra probem φ x =UMφ, φ t =MVφ, i, j = 1,, 3. λ t =, U=(λ+u 1 +ξ)e 1 +u e +u 3 e 3, V= (Ae 1 ( n) +Be ( n) +Ce 3 ( n)). m According to the generaized Tu scheme, we sove the stationary zero curvature equation which gives rise to the fact that A x =u 3 B u C, C x = λc u 1 C ξc Cξ+u 3 A, (6) (7) (8) V x = [U, V, (9) B x = u A + λc + ξc + Cξ + u 1 C u 3 A+Bλ +u 1 B + Bξ + ξb. (1) Let A= A m λ m, B= B m λ m,andc= C m λ m ; (1) is equatothefoowingformua: A m,x =u 3 B m u C m, A m,y =, Let e 3 =( 1 1 ). M=( 1 ), (4) 1 1 B m,x = u A m +C m+1 +C m,y +ξ C m +u 1 C m u 3 A m +B m+1 +u 1 B m +B m,y +ξ B m, C m,x = C m+1 u 1 C m C m,y ξ C m +u 3 A m, (11)

3 Advances in Mathematica Physics 3 andherethesymboξ means the conjugate of an eement, where ξ A[ξ.SetA =αξ 1, B =C =;wecangetthe other unknown numbers from (11): A 1 =, B 1 =αu ξ 1, C 1 =αu 3 ξ 1, B =α (u +u 3 ) x ξ 1 α u,y ξ 1 αu 1 u ξ 1 αu, C =α u 3,y u 3,x ξ 1 αu 3 αu 1 u 3 ξ 1, A =α x ((u 3u ) x +u 3 u,y u u 3,y +u 3 u 3,x )ξ 1. (1) where the recurrence operator L is given from (11): L Q W = u 3 u 1 x y ξ, [ (u [ +u 3 ) x u 1 + x y ξ where Q= [(u 3 +u ) y +(u +u 3 )ξ +(u 3 u 1 +u u 1 )+u x, W= [u 3 ( x y ξ ) u 1 u 3. (19) () For an arbitrary number n, indicating V (n) + = n [A n e 1 (n m) +B n e (n m) +C n e 3 (n m) m =λ n V V (n), we can cacuate that (13) When n =, a simpe reduction of (17) gets the foowing generaized ( + 1)-dimensiona equation: u 1,t =α((u 3 u ) x +u 3 u,y u u 3,y +u 3 u 3,x ), u,t =α( (u +u 3 ) xx u,xy (u 1 u ) x u 1 u,x +u 1 u,y u 1 u 3,y +u 1 u +u 1 u 3), (1) V (n) +,x +[U,V(n) + =V(n),x [U,V(n), (14) and a direct cacuate can get the foowing formua: V (n) +,x +[U,V(n) + = (B n+1 +C n+1 )e () +C n+1 e 3 (). (15) Taking V (n) =V (n) + a ne 1 (), then the zero curvature equation admits that u t = [ [ u 1 u u 3 U t V (n) x + [U, V (n) + = (16) R(A n,x ) = [ R(B n,x u 1 (B n +C n )) t [ R(C n,x +u 1 C n ) R(A n ) = [ +u 1 [ R (C n ), [ u 1 [ R (B n +C n ) (17) u 3,t =α( u 3,xy u 3,xx u 1 u 3). (u 1 u 3 ) x +u 1 (u 3,y u 3,x ) When taking n=3, u =u 3,andu 1 =,weobtainthe(+1)- dimensiona mkdv hierarchy: u,t =αu,xxx +6u u,x αu,xxy αu,xyy, () and if α = 1,then() can be transated into the cassica ( + 1)-dimensiona mkdv equation: u,t =u,xxx +6u u,x u,xxy u,xyy. (3) In what foows, we wi study the Hamitonian structures of the hierarchy (17). In order to seek the Hamitonian structure, we introduce the inear function by a, b =(a 1 b 1 a 3 b a b 3 a 3 b 3 ), (4) for a a=a 1 e 1 () + a e () + a 3 e 3 (), b=b 1 e 1 () + b e () + b 3 e 3 (). The isospectra operator can be written as and then R(A n ) [ R (C n ) =L [ R(A n 1 ) [ R (C n 1 ), (18) [ R (B n +C n ) [ R (B n 1 +C n 1 ) U=[ λ+u 1 +ξ+u 3 u 3 u λ+u 1 +ξ u 3, V=[ A+C C B A C. (5)

4 4 Advances in Mathematica Physics Therefore, we can get U =[ 1 u 1 1 =e 1 (), U =[ u 1 =e (), U =[ 1 u 3 1 =e 3 (), U λ =[1 1 =e 1 (). It is easy to verify that (6) satisfies the variationa identity (6) δ δu R V, U λ =λ r λ λr R V, U u, (7) As to (3),firsty,et u = γ (x, y, t) P (ξ (x, y, t)), (31) where P(ξ) satisfies Riccati equation (3).Substitute(31) into (3);wecangetthefoowingformua: 6 i=1 R i F i (P, P,P,...) =R 1 P +R P P +R 3 P 3 +R 4 P+R 5 P +R 6 P, where R 1 =γξ 3 x γξ x ξ y γξ y ξ x, R =6γ 3 ξ x, R 3 =6γ γ x, R 4 =γ xxx γ xyy γ xxy γ t, (3) where V, U λ =tr (V, U λ ). (8) R 5 =3γ x ξ x +3γξ xξ xx γ x ξ x ξ y γξ xx ξ y γ y ξ x γξ x ξ xy γ y ξ x ξ y γξ yy ξ x γ x ξ y (33) Then the ( + 1)-dimensiona hierarchy (17) can be written as the foowing Hamitonian form: u tn = [ [ u 1 u u 3 R(A n ) =J [ R( C n ) =J δh n δu. (9) tn [ R( B n C n ) 3. The Soutions of (+1)-Dimensiona mkdv Equation In order to get the soutions of (3),weintroducethecassica Riccati equation [6 9 P (ξ) =a+cp (ξ), (3) which has the foowing specia soutions: γξ xy ξ xy, R 6 =3γ xx ξ x +3γ x ξ xx +γξ xxx γ xx ξ y γ xy ξ x γ x ξ xy γ y ξ xx γξ xxy γ yy ξ x γ xy ξ y γ y ξ xy γ x ξ yy γξ xxy γξ t. Secondy, we try to get a Riccati equation; et 6 i=1 R i F i (P, P,P,...)=R 1 P +R P P +R 3 P 3 +R 4 P +R 5 P +R 6 P=R 1 ξξ (P a cp ) +cpr 1 ξ (P a cp ) +cp R 1 (P a cp )+R 4 ξ (P a cp ) (34) (1) when a=1, c= 1, P(ξ) = tanh ξ,cothξ; () when a=1, c=1, P(ξ) = tan ξ; (3) when a= 1, c= 1, P(ξ) = cot ξ; (4) when a=1, c= 4, P(ξ) = tanh(ξ)/(1 + tanh (ξ)); (5) when a=1, c=4, P(ξ) = tan ξ/(1 tan ξ); (6) when a= 1, c= 4, P(ξ) = cot ξ/(1 cot ξ); (7) when a=1/, c=1/, P(ξ) = tan ξ±sec ξ, cscξ cot ξ; (8) when a= 1/, c= 1/, P(ξ) = sec ξ tan ξ,cotξ± csc ξ; (9) when a=1/, c= 1/, P(ξ) = tanh(ξ)/(1+sech(ξ)), tanh(ξ) ± i sech(ξ); (1) when a=, c =, P(ξ) = 1/(cξ + c ). +cr 4 P(P a cp ) =(R 1 ξξ +cpr 1 ξ +cp R 1 +R 4 ξ +cr 4 P) (P a cp ). Thirdy, compare the coefficient of (3) and (34); we can get the foowing formua: ξ x =ξ y, 6γc ξ 3 x =6γ3 ξ x, acγξ 3 x = 3γ xxξ x 3γ x ξ xx γξ xxx γξ t, ac (3γ x ξ x +3γξ xξ xx )= γ xxx γ t, c (3γ x ξ x +3γξ xξ xx )=6γ γ x. (35)

5 Advances in Mathematica Physics 5 Atast,wecangetaspeciasoutionξ=b(x+y) acb 3 t of (35),whereb is a rea number. Then we obtain thesoutions of(3) according to different conditions. Condition 1. Soitary wave soutions are as foows: (1) when a=1, c= 1: u 1 = btanh (b(x+y) b 3 t), u = bcoth (b(x+y) b 3 t), u 3 = bsech (b(x+y)+b 3 t) ; () when a=1/, c= 1/: u 4 = 1 b tanh (b(x+y) 1 b3 t) +isech (b (x + y) 1 b3 t), u 5 = 1 b tanh (b(x+y) (1/) b3 t) 1+sech (b(x+y) (1/) b 3 t) ; (36) (37) (3) when a=1, c= 4: u 6 = 4b tanh (b(x+y) b3 t) 1+tanh (b(x+y) b 3 t). (38) Condition. Periodica wave soutions are as foows: (4) when a=1, c=1: u 7 =btan (b (x + y) b 3 t) ; (39) (5) when a= 1, c= 1: u 8 = bcot (b (x + y) + b 3 t) ; (4) (6) when a=1/, c=1/: u 9 = 1 b[tan (b(x+y) 1 b3 t) ± sec (b (x + y) 1 b3 t), u 1 = 1 b[csc (b (x + y) 1 b3 t) (41) (8) when a=1, c=4: u 13 =4b tan (b(x+y) b3 t) 1 tan (b(x+y) b 3 t). (43) Condition 3. Rationa function wave soution is as foows: (9) a=, c =: 1 u 14 = cb. (44) cb (x + y) + t 4. The Chirp Effect of Rossby Soitary Waves In this part, we begin to discuss the appication of the ( + 1)-dimensiona mkdv equation. As to the Rossby soitary waves in the atmosphere, Yang et a. got the (1 + 1)-dimensiona mkdv equation to describe the propagation of Rossby soitary waves in a ine in the atmosphere and ocean,butinfact,asweaknow,thepropagationofrossby soitarywavesinapaneismoresuitabeforrefectingthe rea condition of atmosphere and ocean. So in this paper, we use the ( + 1)-dimensiona mkdv equation for anaysis of the chirp effect of the Rossby soitary waves with the hep of chirp effect in fiber soiton communication. As to the ( + 1)-dimensiona mkdv equation () u,t =αu,xxx +6u u,x αu,xxy αu,xyy, (45) where α is the coefficient of dispersion part. On the basis of initia wave form u=u tanh ( x+y ). (46) In the transmission of Rossby wave, the noninear part and the dispersive part act as an important function. In fiber soiton communication, chirp effect is caused by the excursion of the center wave; it is impacted by the dispersion and the noninear effect. Then we discuss the chirp effect of the Rossby soitary waves. Firsty, we aone consider the dispersion function; then () canbechangedinto u t =αu xxx αu xxy αu xyy, (47) cot (b(x+y) 1 b3 t) ; (7) when a= 1/, c= 1/: u 11 = 1 b[sec (b(x+y)+1 b3 t) tan (b(x+y)+ 1 b3 t), u 1 = 1 b[cot (b(x+y)+1 b3 t) csc (b(x+y)+ 1 b3 t) ; (4) and we ony observe the time t from to Δt, whereδt is an infinitesima variabe; then we can get the approximate soution of (47): u(δt,x,y)=u tanh ( x+y ) exp ( ( αδt + αδt and then the phase of the wave is φ D = ( αδt + αδt tanh5/ ( x+y ))), (48) tanh5/ ( x+y )), (49)

6 6 Advances in Mathematica Physics from (49),wecangetthechirpcausedbythedispersion: Δ D = φd = 1 ( αδt tanh3/ ( x+y ) sech ( x+y )). (5) Secondy, we consider the function of the noninear effect; then () canbechangedintothefoowingformua: u t =6u u x, (51) ikewise, the time is sti from to Δt, whereδt is an infinitesima variabe; substituting (46) into (51), wecangetthe approximate soution of (51): u(δt,x,y)=u tanh ( x+y ) exp (6 u 5/ tanh 5/ ( x+y (5) )Δt), accordingy, the phase of the wave is φ N =6 u 5/ tanh 5/ ( x+y )Δt, (53) then the chirp caused by noninear effect is Δ N = φn (54) = 3 u 5/ tanh 3/ ( x+y ) sech ( x+y )Δt. According to (54) and (5),wecangetthewhoechirp Δ S =Δ N +Δ D =( 1 α 3 u 5/ ) tanh 3/ ( x+y ) sech ( x+y )Δt, (55) if the dispersion effect is equa to the noninear effect, then Δ S =, u =( α )/5 ; (56) 3 if the dispersion effect is more than the noninear effect, Δ D > Δ N,weobtain u <( α )/5 ; (57) 3 if the dispersion effect is ess than the noninear effect, Δ D < Δ N,weobtain u > ( α )/5. (58) 3 From (55), we know that the whoe chirp effect is reated to the environment where Rossby soitary waves spread in the atmosphere and the initia ampitude; when (56) is satisfied, therossbysoitarywavescanbespreadsteadiy.theinitia ampitude is smaer, the dispersive function is stronger, the initia ampitude is bigger, and the noninear function is stronger.theformofrossbysoitarywaveisasinfigure Figure 1: The form of Rossby soitary wave at t = 1, wave speed c=. 5. Concusions In this paper, we first obtain a ( + 1)-dimensiona mkdv equation on the basis of (1+1)-dimensiona spectra operator. Athoughthe ( + 1)-dimensiona mkdv hierarchy has been obtained in this paper, we note that it is difficut to obtain a the ( + 1)-dimensiona hierarchies by using the generaized Tu scheme and not every ( + 1)-dimensiona hierarchy has its own Hamitonian structure. In the fina anaysis, we expain the chirp effect of the Rossby soitary waves on the basis of the ( + 1)-dimensiona mkdv equation. We discuss the reation between the noninear function and the dispersion and get the concusion that the whoe chirp effect is impacted by the atmosphere and ocean state of the Rossby soitary waves transmission and the initia ampitude. We can baance the whoe chirp effect according to the noninear function and dispersion in order to et the Rossby soitary waves spread pacidy. Last but definitey not east, we shoud try to find some more ( + 1)-dimensiona hierarchies which have important meanings in physics and other aspects. As we a know that there are financia soitons in economics, we beieve that we can appy some new ( + 1)- dimensiona hierarchies into the economics and describe the economica phenomenon. In addition, we can aso try to use other ways such as Darboux transformations and symmetry transformationsto get more ( + 1)-dimensiona hierarchies; these probems are worthy of discussing. Confict of Interests The authors decare that there is no confict of interests regarding the pubication of this paper. Acknowedgments This work was supported by Nationa Natura Science Foundation of China (no ), Specia Funds for Theoretica Physics of the Nationa Natura Science Foundation of China (no ), Foundation for Innovative Research 5 5 1

7 Advances in Mathematica Physics 7 Groups of the Nationa Natura Science Foundation of China (Grant no ), Science and Technoogy Pan Project of Qingdao (no jch), SDUST Research Fund (no. 1KYTD15), and Graduate Innovation Foundation from Shandong University of Science and Technoogy (no. YC1437). References [1 F. Magri, A simpe mode of the integrabe Hamitonian equation, Journa of Mathematica Physics, vo.19,no.5,pp , [ G.Z.Tu,R.I.Andrushkiw,andX.C.Huang, Atraceidentity and its appication to integrabe systems of 1+dimensions, Journa of Mathematica Physics, vo.3,no.7,pp , [3 G. Z. Tu, The trace identity, a powerfu too for constructing the Hamitonian structure of integrabe systems, Journa of Mathematica Physics,vo.3,no.,pp ,1989. [4 W. X. Ma, A new hierarchy of Liouvie integrabe generaized Hamitonian equation and its reductions, Chinese Annas of Mathematics Series A,vo.13,pp ,199. [5 W. X. Ma, An approach for constructing nonisospectra hierarchies of evoution equations, Journa of Physics A. Mathematica and Genera, vo. 5, no. 1, pp. L719 L76, 199. [6 X. B. Hu, A powerfu approach to generate new integrabe systems, Journa of Physics A: Mathematica and Genera, vo. 7, no. 7, pp , [7 H.H.DongandX.Z.Wang, LieagebrasandLiesuperagebra for the integrabe coupings of NLS MKdV hierarchy, Communications in Noninear Science and Numerica Simuation,vo.14, no. 1, pp , 9. [8 C. W. Cao, Commutative representations of spectrum-preserving equations, Chinese Science Buetin, vo. 34, no. 1, pp , [9 C. W. Cao, Noninearization of the Lax system for AKNS hierarchy, Science in China, Series A: Mathematics, Physics, Astronomy,vo.33,no.5,pp ,199. [1 Z. J. Qiao, Lax representations of the Levi hierarchy, Chinese Science Buetin,vo.17,pp ,199. [11 Z. J. Qiao, Commutator representations for the D-AKNS hierarchy of evoution equations, Mathematica Appicata,vo.4,no. 4, pp. 64 7, [1 M. R. Li and Z. J. Qiao, A proof for the theorem of eigenvaue expansion of the Dirac eigenvaue probem, Acta of Liaoning University,vo.,pp.9 18,1993. [13 Z. J. Qiao, Invoutive soutions and commutator representations of the dirac hierarchy, Chinese Quartery Journa of Mathematics,vo.1,pp.1 4,1997. [14 H.-W. Yang, H.-H. Dong, and B.-S. Yin, Noninear integrabe coupings of a noninear Schrödinger-modified Korteweg de Vries hierarchy with sef-consistent sources, Chinese Physics B, vo.1,no.1,articeid14,1. [15 Y.F.ZhangandW.J.Rui, Ongenerating(+1)-dimensiona hierarchies of evoution equations, Communications in Noninear Science and Numerica Simuation,vo.19,no.1,pp , 14. [16 Y. F. Zhang, J. Gao, and G. M. Wang, Two (+1)-dimensiona hierarchies of evoution equations and their Hamitonian structures, Appied Mathematics and Computation, vo. 43, pp , 14. [17 H.-W. Yang, B.-S. Yin, H.-H. Dong, and Z.-D. Ma, Generation of soitary rossby waves by unstabe topography, Communications in Theoretica Physics,vo.57,no.3,pp ,1. [18S.Y.Song,J.Wang,J.B.Wang,S.S.Song,andJ.M.Meng, Simuate the inter Rossby Soitonian waves by the noninear Schrödinger equation, Acta Physica Sinica, vo. 59, pp , 1. [19 H.-Y. Wei and T.-C. Xia, A new generaized fractiona Dirac soiton hierarchy and its fractiona Hamitonian structure, Chinese Physics B, vo.1,no.11,articeid113,pp.8 33, 1. [ S. X. Tao, H. Wang, and H. Shi, Binary noninearization of the super cassica-boussinesq hierarchy, Chinese Physics B,vo., no.7,articeid71,11. [1 Y.-F. Zhang, T. C. Xia, and H. Wang, The muti-component Tu hierarchy of soiton equations and its muti-component integrabe coupings system, Chinese Physics B, vo. 14, no., pp. 47 5, 5. [ H. W. Yang and H. H. Dong, Muti-component Harry-Dym hierarchy and its integrabe coupings as we as their Hamitonian structures, Chinese Physics B, vo. 18, no. 3, pp , 9. [3 T.-C.Xia, Twonewintegrabecoupingsofthesoitonhierarchies with sef-consistent sources, Chinese Physics B, vo. 19, no. 1, Artice ID 133, 1. [4 Y.-F. Zhang and Y. C. Hon, Some evoution hierarchies derived from sef-dua Yang-Mis equations, Communications in Theoretica Physics,vo.56,no.5,pp ,11. [5 X. B. Hu, An approach to generate superextensions of integrabe systems, Journa of Physics A: Mathematica and Genera, vo.3,no.,pp ,1997. [6X.R.Wang,Y.Fang,andH.H.Dong, Component-trace identity for Hamitonian structure of the integrabe coupings of the Giachetti Johnson (GJ) hierarchy and couping integrabe coupings, Communications in Noninear Science and Numerica Simuation, vo. 16, no. 7, pp , 11. [7 X. B. Hu and P. A. Carkson, Rationa soutions of a differentiadifference KDV Equation, the Toda equation and the discrete KDV Equation, Journa of Physics A. Mathematica and Genera,vo.8,no.17,pp ,1995. [8 H. C. Hu and Q. P. Liu, New Darboux transformation for Hirota Satsuma couped KdV system, Chaos, Soitons & Fractas,vo.17,no.5,pp.91 98,3. [9 Y. Chen and E.-G. Fan, Compexiton soutions of the (+1)- dimensiona dispersive ong wave equation, Chinese Physics, vo.16,no.1,pp.6 1,7.

8 Advances in Operations Research Advances in Decision Sciences Journa of Appied Mathematics Agebra Journa of Probabiity and Statistics The Scientific Word Journa Internationa Journa of Differentia Equations Submit your manuscripts at Internationa Journa of Advances in Combinatorics Mathematica Physics Journa of Compex Anaysis Internationa Journa of Mathematics and Mathematica Sciences Mathematica Probems in Engineering Journa of Mathematics Discrete Mathematics Journa of Discrete Dynamics in Nature and Society Journa of Function Spaces Abstract and Appied Anaysis Internationa Journa of Journa of Stochastic Anaysis Optimization