Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics 11 01 1 8 Journal of Applied Nonlinear Dynamics Journal homepage: www.lhscientificpublishing.com/journals/jand.html Lie Algebraic Approach to Nonlinear Integrable Couplings of Evolution Type Yufeng Zhang 1, Wen-Xiu Ma 1 College of Science, China University of Mining and Technology, Xuzhou 111, P. R. China Department of Mathematics and Statistics, University of South Florida, Tampa, FL 0-5700, USA Submission Info Communicated by Albert C.J. Luo Received 8 Nov 011 Accepted 1 Dec 011 Available online 15 Apr 01 Keywords Lie algebra Nonlinear integrable coupling AKNS hierarchy BK hierarchy KN hierarchy Abstract Based on two higher-dimensional extensions of Lie algebras, three kinds of specific Lie algebras are introduced. Upon constructing proper loop algebras, six isospectral matrix spectral problems are presented and they yield nonlinear integrable couplings of the Ablowitz- Kaup-Newell-Segur hierarchy, the Broer-Kaup hierarchy and the Kaup-Newell hierarchy. Especially, the reduced cases of the resulting integrable couplings give nonlinear integrable couplings of the nonlinear Schrödinger equation and the classical Boussinesq equation. Two linear functionals are introduced on two loop algebras of dimension and Hamiltonian structures of the obtained nonlinear integrable couplings are worked out by employing the associated variational identity. The proposed approach can also be used to generate nonlinear integrable couplings for other integrable hierarchies. 01 L&H Scientific Publishing, LLC. All rights reserved. 1 Introduction Integrable couplings are coupled systems of integrable equations, which contain given integrable equations as their sub-systems [1 ]. The general definition on integrable couplings is as follows: for a given integrable system of evolution type u t = Ku, an integrable coupling is a new bigger triangular integrable system { ut = Ku, 1 v t = Su,v, where the vector-valued function S should satisfy the non-triviality condition S/ [u] 0, [u] denoting a vector consisting of all derivatives of u with respect to the space variable. In the paper [4], a kind of matrix Lie algebras was introduced and the associated integrable coupling of the TD hierarchy was obtained. A general procedure for generating integrable couplings was proposed in [], based on semi-direct sums of Lie algebras. The basic idea is as follows. Corresponding author. Email address: mathzhang@1.com ISSN 14 457, eissn 14 47/$-see front materials 01 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/JAND.011.1.001

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 Let G be a matrix Lie algebra. Assume that a continuous integrable system of evolution type u t = Ku = Ku,u x,u xx,... is linked with G. That is, there exists a pair of Lax matrices U and V in G so that the matrix spectral problem and the associated matrix spectral problem φ x = Uφ = Uu,λφ φ t = V φ = Vu,u x,..., n 0 u x n ;λ 4 0 generate the integrable system through the isospectral λ t = 0 compatibility condition U t V x +[U,V] = 0. 5 To construct an integrable coupling of Eq., we enlarge the Lie algebra G by using semi-direct sums of Lie algebras as follows Ḡ = G G c, where [G,G c ] = {[A,B] A G,B G c }, and G and G c satisfy that [G,G c ] G c. 7 Therefore, G c is an ideal Lie sub-algebra of Ḡ. Take a pair of new Lax matrices in the semi-direct sum Ḡ : Ū = U +U c, V = V +V c,u c,v c G c. 8 Then the compatibility condition of the Lax pair Ū and V, i.e., the enlarged zero curvature equation Ū t V x +[Ū, V] = 0 is equivalent to { Ut V x +[U,V] = 0, 9 U c,t V c,x +[U,V c ]+[U c,v]+[u c,v c ] = 0. The first equation 9 presents Eq.. The whole system 9 provides an integrable coupling of Eq.. In particular, we can choose the enlarged spectral matrices Ū and V as follows []: U U a1... U aν V V a1... V aν. Ū = 0 U.......... Ua1,. V = 0 V.......... Va1, 10 0... 0 U 0... 0 V Thus, the coupling system 9 becomes that u t V x +[U,V] = 0, U ai,t V ai,t + l+k=i,k,l 0 [U ak,v al ] = 0, 11

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 where U a0 = U,V a0 = 0. Besides the form of Lax pairs in 10, another enlarged form of the spectral matrices [5] U Ua Ū = 1 0 0 corresponds to the semi-direct sum of Lie algebras { A 0 G G c,g = 0 0 }, G c = { } 0 B. 0 0 More generally, the enlargement of Lax spectral matrices []: U Ua1 Ū = 0 U a 1 can be taken from the following semi-direct sum of Lie algebras { } { } A 0 0 B1 G G c,g =, G 0 0 c =, 0 B where A,B 1 and B have the same sizes as U,U a1 and U a, respectively. Even more general matrix forms than 1 and 1 can be taken as follows [] U U a1 U a U U a1 U a Ū = 0 U U a,ū = 0 U U a. 14 0 0 0 0 0 U a4 All this presents a few simple categories of Lax pairs which yield integrable couplings under the framework of semi-direct sums of Lie algebras. Based on this idea, some integrable couplings of integrable systems were obtained [ 9], from which we see that there are much richer mathematical structures behind integrable couplings than scalar integrable equations. Tu s trace identities have been broadly generalized to semi-direct sums of Lie algebras recently [10,11]. The generalizations form a general identity called the variational identity, which provides a powerful tool for generating Hamiltonian structures of integrable couplings [11]. More recently, the variational identity has been extensively studied and extended to the case of super Lie algebras, and the supertrace identity has been established to deduce super-hamiltonian structures of a super-akns soliton hierarchy and a super-dirac soliton hierarchy [1]. Usually, integrable equations can have different integrable couplings. For example, a known equation or system by Eq. 1 may have two integrable couplings [1]: Ku u ū 1,t = K 1 ū = Su, v Ku ū,t = K ū = Tu,v, ū 1 =, ū = v u w, 15. 1 Putting 15 and 1 together forms a bigger system Ku u ū,t = K û = Su, v, û = v. 17 Tu,w w

4 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 If 17 is still integrable, then we call it a bi-integrable coupling of Eq. 1 [1]. Two kinds of matrix Lie algebras consisting of square matrices of block forms were constructed to obtain different bi-integrable couplings of the standard nonlinear schrödinger equation. Following this idea, two kinds of higher-dimensional Lie algebras were presented in [14,15] to produce bi-integrable couplings of the KdV hierarchy and the BPT hierarchy. However, the integrable couplings obtained this way are all linear with respect to the second variable v in the system v t = Su,v, that is, the whole system is linear with respect to the variable v and its derivatives with respect to the space variable. One of the reasons may be the fact that the Lie algebra G c is nilpotent. Recently based on a kind of new special non-semisimple Lie algebras, two feasible schemes for constructing nonlinear continuous and discrete integrable couplings were proposed in [1,17]. Variational identities [9,18] over the corresponding loop algebras were used to furnish Hamiltonian structures for the resulting nonlinear integrable couplings. In the paper, enlightened by the idea adopted in [1,17], we want to establish three kinds of specific Lie algebras and make use of them to generate nonlinear continuous integrable couplings of evolution type for given integrable evolution equations. We take the AKNS hierarchy, the BK hierarchy and the KN hierarchy as examples to illustrate the suggested approach. Nonlinear integrable couplings of the nonlinear Schrödiner equation and the classical Boussinesq equation are presented as reductions of the computed examples. Moreover, two linear functionals are introduced on the corresponding loop algebras and Hamiltonian structures of the resulting nonlinear integrable couplings are furnished by employing the associated variational identity. Three Lie Algebras: G, H and Q Let us consider a vector space [14,15]: Denote by L = {a = a 1,a,a T,a i R}. A 1 KL = {A = A,A 1,A,A L }. 18 A For A = A 1,A,A T,B = B 1,B,B T KL, define an operation [A,B] = [A 1,B 1 ] [A 1,B ]+[A,B 1 ] [A 1,B ]+[A,B 1 ]+[A,B ]. 19 It can be verified that KL is a Lie algebra equipped with 19. In the vector space L, two different commutative operations are given by a b a b [a,b] = a 1 b a b 1 0 a b 1 a 1 b and a b a b [a,b] = a 1 b a b 1. 1 a 1 b a b 1

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 5 It can be verified that L is a Lie algebra if equipped with 0 or 1, where a = a 1,a,a T,b = b 1,b,b T L. Now we extend the vector space L into a higher-dimensional one as follows: L 9 = {a = a 1,...,a 9 T,a i R}. For a = a 1,...,a 9 T,b = b 1,...,b 9 T L 9, note a = A 1,A,A T,b = B 1,B,B T, where A 1 = a 1,a,a T,A = a 4,a 5,a T,A = a 7,a 8,a 9 T, B 1 = b 1,b,b T,B = b 4,b 5,b T,B = b 7,b 8,b 9 T. According to 0 and 1 combining with 19, we define the following two operations in L 9 : [A 1,B 1 ] 1 [a,b] = [A 1,B ] 1 +[A,B 1 ] 1, [A 1,B ] 1 +[A,B 1 ] 1 +[A,B ] 1 where a b a b [A 1,B 1 ] 1 = a 1 b a b 1, a b 1 a 1 b a b a b + a 5 b a b 5 [A 1,B ] 1 +[A,B 1 ] 1 = a 1 b 5 a 5 b 1 + a 4 b a b 4, a b 1 a 1 b + a b 4 a 4 b a b 9 a 9 b + a 8 b a b 8 + a 8 b 9 a 9 b 8 [A 1,B ] 1 +[A,B 1 ] 1 +[A,B ] 1 = a 1 b 8 a 8 b 1 + a 7 b a b 7 + a 7 b 8 a 8 b 7, a 9 b 1 a 1 b 9 + a b 7 a 7 b + a 9 b 7 a 7 b 9 [A 1,B 1 ] [a,b] = [A 1,B ] +[A,B 1 ], 4 [A 1,B ] +[A,B 1 ] +[A,B ] where a b a b [A 1,B 1 ] = a 1 b a b 1, a 1 b a b 1 a b 5 a 5 b + a b a b [A 1,B ] +[A,B 1 ] = a 1 b a b 1 + a 4 b a b 4, a 1 b 5 a 5 b 1 + a 4 b a b 4 a b 8 a 8 b + a 9 b a b 9 + a 9 b 8 a 8 b 9 [A 1,B ] +[A,B 1 ] +[A,B ] = a 1 b 9 a 9 b 1 + a 7 b a b 7 + a 7 b 9 a 9 b 7, a 1 b 8 a 8 b 1 + a 7 b a b 7 + a 7 b 8 a 8 b 7

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 { 1, i = j, Noting e i = e i1,...,e i9 T,e i j =, in terms of, we have a commutative relation as follows: 0, i j [e 1,e ] = e,[e 1,e ] = e,[e,e ] = e 1,[e 1,e 4 ] = 0,[e 1,e 5 ] = e 5,[e 1,e ] = e, [e 1,e 7 ] = 0,[e 1,e 8 ] = e 8,[e 1,e 9 ] = e 9,[e,e 4 ] = e 5,[e,e 5 ] = 0,[e,e ] = e 4, [e,e 7 ] = e 8,[e,e 8 ] = 0,[e,e 9 ] = e 7,[e,e 4 ] = e,[e,e 5 ] = e 4,[e,e ] = 0, [e,e 7 ] = e 9,[e,e 8 ] = e 7,[e,e 9 ] = [e 4,e 5 ] = [e 4,e ] = [e 4,e 7 ] = [e 4,e 8 ] = [e 4,e 9 ] = [e 5,e ] = [e 5,e 7 ] = [e 5,e 8 ] = [e 5,e 9 ] = 0,[e,e 7 ] = [e,e 8 ] = [e,e 9 ] = 0, [e 7,e 8 ] = e 8,[e 7,e 9 ] = e 9,[e 8,e 9 ] = e 7. Let us make a linear transformation then we have a new Lie algebra where the commutative operations are the following: f 1 = 1 e 7, f = 1 e 8 + e 9, f = 1 e 8 e 9, G = span{e 1,e,e, f 1, f, f }, 5 [e 1,e ] = e,[e 1,e ] = e,[e,e ] = e 1,[e 1, f 1 ] = 0,[e 1, f ] = f,[e 1, f ] = f, [e, f 1 ] = f f,[e, f ] = f 1,[e, f ] = f 1,[e, f 1 ] = f f,[e, f ] = [e, f ] = f 1, [ f 1, f ] = f,[ f 1, f ] = f,[ f, f ] = f 1. Denote and then we have G 1 = span{e 1,e,e },G = span{ f 1, f, f }, G = G 1 G,[G 1,G ] G, which satisfy the sufficient condition for generating integrable couplings. We remark that here G 1 and G are all simple Lie algebras. If we make another linear transformation h 1 = i f 1,h = i f + f,h = i f f, where then we obtain that [ f 1, f ] = f,[ f 1, f ] = f,[ f, f ] = f 1,i = 1, [h 1,h ] = h h,[h 1,h ] = h + h,[h,h ] = h 1,[e 1,h 1 ] = 0, [e 1,h ] = i h + h,[e 1,h ] = i h + h,[e,h 1 ] = i i 1 [e,h ] = h 1,[e,h ] = 1+ i h 1,[e,h 1 ] = i+ h + i [e,h ] = 1 i h 1. h + + i h, h,[e,h ] = 1+ i h 1,

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 7 Denoting by H = span{e 1,e,e,h 1,h,h }, then H is also a Lie algebra equipped with the above commutator. Similarly, according to 4, a basis of the Lie algebra R 9, { 1, i = j, e i = e i1,...,e i9 T,e i j = 1 i, j 9, 0, i j, possesses the following commutative relations [e 1,e ] = e,[e 1,e ] = e,[e,e ] = e 1,[e 1,e 4 ] = 0,[e 1,e 5 ] = e,[e 1,e ] = e 5,[e 1,e 7 ] = 0, [e 1,e 8 ] = e 9,[e 1,e 9 ] = e 8,[e,e 4 ] = e,[e,e 5 ] = 0,[e,e ] = e 4,[e,e 7 ] = e 9, [e,e 8 ] = 0,[e,e 9 ] = e 7,[e,e 4 ] = e 5,[e,e 5 ] = e 4,[e,e ] = 0,[e,e 7 ] = e 8, [e,e 8 ] = e 7,[e,e 9 ] = [e 4,e 5 ] = [e 4,e ] = [e 4,e 7 ] = [e 4,e 8 ] = [e 4,e 9 ] = [e 5,e ] = [e 5,e 7 ], [e 5,e 8 ] = [e 5,e 9 ] = [e,e 7 ] = [e,e 8 ] = [e,e 9 ] = 0, [e 7,e 8 ] = e 9,[e 7,e 9 ] = e 8,[e 8,e 9 ] = e 7. If make a linear transformation p 1 = e 7, p = e 8 + e 9, p = e 8 e 9, then we have [p 1, p ] = p,[p 1, p ] = p,[p, p ] = 4p 1,[e 1, p 1 ] = 0,[e 1, p ] = p,[e 1, p ] = p, [e, p 1 ] = 1 p p,[e, p ] = p 1,[e, p ] = p 1,[e, p 1 ] = 1 p + p,[e, p ] = p 1, [e, p ] = p 1. Setting Q = span{e 1,e,e, p 1, p, p }, 7 then Q forms a Lie algebra equipped with the above commutative operations. Nonlinear Integrable Couplings In the section, we shall use the loop algebras of the Lie algebras G, H and Q to construct nonlinear integrable couplings for the AKNS hierarchy, the BK hierarchy and the KN hierarchy, respectively. Specially, we shall obtain nonlinear integrable couplings of the standard nonlinear Schrödinger equation and the classical Boussinesq equation..1 Two Nonlinear Integrable Couplings of the AKNS Hierarchy Introduce a loop algebra of the Lie algebra G: G = span{e 1 n,e n,e n, f 1 n, f n, f n}, where e i n = e i λ n, f i n = f i λ n, i = 1,,, n Z.

8 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 The corresponding commutative operations read that [e i m,e j n] = [e i,e j ]λ m+n, [ f i m, f j n] = [ f i, f j ]λ m+n, 1 i, j, m,n Z. By using the loop algebra G, we introduce a Lax pair of zero curvature equations U = e 1 1+qe 0+re 0+u 1 f 0+u f 0, V = m 0 i=1 V im e i m+ m 0 j=4 V jm f j m. A compatibility condition of the Lax pair 8 gives rise to a recursion relation V 1,mx = qv m rv m, V,mx = V,m+1 qv 1m, V,mx = V,m+1 + rv 1m, V 4,mx = q r+u V 5m q+r+u 1 V m +u 1 + u V m +u u 1 V m, V 5,mx = V,m+1 +r qv 4m u V 1m +V 4m, V,mx = V 5,m+1 q+rv 4m u 1 V 1m +V 4m. 8 9 Note that V n + = λ n V + = n m=0 i=1 V im e i n m+ n m=0 j=4 V jm f j n m = λ n V V n. According to the Tu scheme [19,0], we need to compute the left-hand side of the following equation A direct calculation gives that V n +x +[U,V n + ] = V n x [U,V n ]. V n +x +[U,V n + ] = V,n+1 e 0+V,n+1 e 0 V 5,n+1 f 0 V,n+1 f 0. Let V n = V n +. Then the zero curvature equation admits a Lax integrable hierarchy u tn = q r F u 1 u U t V n x +[U,V n ] = 0 t n V,n+1 = V,n+1 V,n+1 V 5,n+1 = L V n V n V n V 5n, 0 where L is a recurrence operator, i.e., q 1 r q 1 q 0 0 r L = 1 r + r 1 q 0 0 1 q r+u 1 u 1 u 1, q r+u 1 u 1 + u l 1 l s l l 4 l 5 l

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 9 with l 1 = 1 q r+u 1 q+r+u 1,l = + 1 q r+u 1 q r+u, l = 1 q+r u 1 1 u u 1 u 1 1 r,l 4 = u 1 1 q+ 1 q+r u 1 1 u 1 + u, It is easy to check that l 5 = 1 q+r u 1 1 q+r+u 1,l = 1 q+r u 1 1 q r+u. V,n+1 V,n+1 V,n+1 V 5,n+1 = L V n V n V n V 5n When set u 1 = u = 0, the hierarchy 0 reduces to the well-known AKNS hierarchy. Setting V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then from 9 we have. V 4,0 = 0,V,1 = αq,v,1 = αr,v,1 = αu,v 5,1 = αu 1,V 1,1 = V 4,1 = 0,V, = α q x, V, = α r x,v 1, = α qr,v, = α 4 q xx α q r,v, = α 4 r xx α qr,v 1, = α 4 qr x q x r, V, = α u 1x,V 5, = α u x,v 4, = α q ru α q+ru 1 α 4 u 1 u, V 5, = α 4 u 1xx + α 4 q+r+u 1[q ru q+ru 1 ] α 8 q+r+u 1u 1 u α qru 1, V, = α 4 u xx + α 4 q r+u [q ru q+ru 1 ] α 8 q r+u u 1 u α qru,..., When n =, the hierarchy 0 reduces to a nonlinear integrable coupling of the standard Schrödinger equation q t = α q xx αq r, r t = α r xx + αqr, u 1,t = α u xx + α q r+u [q ru q+ru 1 ] α 4 q r+u u 1 u αqru, u,t = α u 1xx + α q+r+u 1[q ru q+ru 1 ] α 4 q+r+u 1u 1 u αqru 1. 1 When taking u 1 = u = 0, 0 reduces to the nonlinear Schrödinger equation. When taking n =,,..., the hierarchy 0 all reduces to nonlinear integrable coupling equations. Hence, 0 presents a hierarchy of nonlinear integrable coupling for the AKNS hierarchy. In what follows, we employ the Lie algebra H to construct a second hierarchy of nonlinear integrable couplings for the AKNS hierarchy. A loop algebra of the Lie algebra H is chosen as H = span{e 1 n,e n,e n,h 1 n,h n,h n},

10 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 where e i n = e i λ n,h i n = h i λ n, i = 1,,, n Z. The corresponding commutative operations read [e i m,e j n] = [e i,e j ]λ m+n, [h i m,h j n] = [h i,h j ]λ m+n, 1 i, j, m,n Z. By utilizing H, a Lax pair is introduced as follows U = e 1 1+qe 0+re 0+s 1 h 0+s h 0, V = V im e i m+ V jm h j m. m 0 i=1 j=4 The stationary zero curvature equation [U,V] = V x is equivalent to the following V 1,mx = qv m rv m, V,mx = V,m+1 qv 1m, V,mx = V,m+1 + rv 1m, V 4,mx = 1+ i s + 1 i s 1 V m + i 1 s 1+ s 1 V m + i 1 q+ 1+ i r+s V 5m + V 5,mx = i V 5,m+1 4i V,m+1 + 1 i r 1+ i q s 1 V m, i q i+ r+s 1 s V 4m 4i + s is 1 V 1m, V,mx = 4i V 5,m+1 i i V,m+1 + q+ i r+s 1 s V 4m i + s 4i s 1 V 1m. Setting V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then we obtain from

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 11 V 4,0 = 0,V 5,1 = αs 1,V,1 = αs,v 4,1 = 0,V,1 = αq,v,1 = αr,v, = α q x,v, = α r x, V 1, = α qr,v 5, = α i s x s 1x,V, = α i s x s 1x, i+1 V 4, = αqs + 1 i αqs 1 + 1 i αrs + 1+ i αrs 1 + α i s 1 + s s 1s, V 5, = α 4 s 1xx + iα i + 1+ i s 1 r 1 q+ i r+ s 1 ] i s 1 + s s 1s α qrs 1, [ i+1 qs + 1 i qs 1 + V, = α 4 s xx αi i q i+ r [ i+1 s s + 1 i s 1 q 1 i + s + 1+ ] i i s 1 r s 1 + s s 1s α qrs. 1 i s Setting V n = n m=0 i=1 V im e i n m+ j=4 V jm h j n m, a direct calculation gives rise to V x n +[U,V n ] = V,n+1 e 0+V,n+1 e 0+ i V,n+1 V 5,n+1 h 0 + i V,n+1 V 5,n+1 h 0. The equation leads to u tn = q r s 1 s U tn V n x +[U,V n ] = 0 t n V,n+1 V,n+1 i = V 5,n+1 V,n+1. 4 i V 5,n+1 V,n+1 When setting s 1 = s = 0, 4 reduces to the AKNS hierarchy, and so it gives a hierarchy of nonlinear integrable couplings for the AKNS hierarchy. Specially, if we take n =, the hierarchy 4 reduces to

1 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 another nonlinear integrable coupling of the nonlinear Schrödinger equation: q t = α q xx αq r, r t = α r xx + αqr, s 1,t = iα s xx s 1xx + ıα s s 1 qr+ α i [ i+1 s + 1 i s 1 i 1 q+ s + 1+ i r s,t = iα s xx s 1xx + iα s s 1 qr α i [ i+1 i 1 i i s + 1 s 1 q+ s + 1+ s 1 q+ i+ r s 1 s i s 1 + s s 1s ] q+ i r+ s 1 s r i s 1 + s s 1s ]., 5. Three Nonlinear Integrable Couplings of the BK Hierarchy In this sub-section, we first make use of the Lie algebra G to establish two different Lax pairs to generate the corresponding different nonlinear integrable couplings of the BK hierarchy. As reduced cases, various nonlinear integrable couplings of the classical Boussinesq equation are obtained. Employing the Lie algebra H, we introduce a Lax pair whose compatibility condition gives rise to the third nonlinear integrable couplings of the BK hierarchy. Applying the loop algebra G of the Lie algebra G introduces a Lax pair as follows: U = e 1 1+ v e 10+e 0 we 0+u 1 f 0+u f 0, V = m 0 j=1 V jm e j m+ m 0 k=4 V km f k m. Equation V x = [U,V] gives that V 1,mx = V m + wv m, V,mx = V,m+1 + vv m V 1m, V,mx = V,m+1 vv m wv 1m, V 4,mx = 1+w+u V 5m + 1+w u 1 V m +u 1 u 1 V m +u 1 + u V m, V 5,mx = V,m+1 + vv m + 1+w u V 4m u V 1m, V,mx = V 5,m+1 + vv 5m + 1+w u 1 V 4m u 1 V 1m. 7 Noting V n + = lambda n V + = n m=0 j=1 V jm e j n m+ k=4 V km f k n m = λ n V V n,

then Eq.V x = [U,V] can be written as A direct computation reads Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 1 V n +x +[U,V n + ] = V n x [U,V n ]. V n +x +[U,V n + ] = V,n+1 e 0 V,n+1 e 0+V 5,n+1 f 0+V,n+1 f 0. Setting V n = V n + + 1, 1 = V,n+1 e 1 0, we have V n x +[U,V n ] = V,n+1x e 1 0+ wv,n+1 V,n+1 e 0 u tn = + u V,n+1 + V,n+1 f 0+ u 1 V,n+1 + V 5,n+1 f 0. Thus, the compatibility of the Lax pair U and V n gives rise to v V,n+1x w u 1 u t n = wv,n+1 V,n+1 u V,n+1 V,n+1 u 1 V,n+1 V 5,n+1 = V,n+1x V 1,n+1x u V,n+1 V,n+1 u 1 V,n+1 V 5,n+1. 8 Letting u 1 = u = 0, 8 reduces to the BK hierarchy. Setting V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then one infers from 7 that V 1,1 = 0,V,1 = α,v,1 = αw,v 1, = α w,v, = α v,v,1 = αw,v 1, = α 4 w xx + α wv, V, = α w x + wv,v, = α 4 v x α 4 v α w,v, = α 4 w x + wv x + α 4 vw x + wv+ α w, V,1 = 0,V, = α u 1x vu,v 5, = α u x vu 1, V 4, = α u u 1 + wu + 1 u 1 u 1 + wu 1, V, = α 4 u xx vu 1 x + α 4 vu 1x vu α 4 1+w+u u u 1 + wu 1 + wu 1 u 1 1 u α u w, V 5, = α 4 u 1xx vu x + α 4 vu x vu 1 + α 4 1+w u 1 u u 1 + wu 1 + wu + 1 u 1 u 1 α u 1w,... Letting n =, then 8 reduces to the following nonlinear integrable equations v t = α v xx αvv x αw x, w t = α w xx αwv x, u 1,t = α u v x α u v αu w+ α u xx vu 1 x α vu 1x vu + α 1+w+u [u u 1 + wu 1 + u 1 u1 + u ] + αu w, 9 u,t = α u 1v x α u 1v αu 1 w α u 1xx vu x α 1+w u 1[u u 1 + wu 1 + u + 1 u u 1 ]+αu 1w.

14 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 Setting u 1 = u = 0,α = 1,t = t, 9 reduces to the BK system v t = 1 v xx + vv x + w x, w t = vw+ 1 w x, x 40 which was obtained in [1]. Two basic Darboux transformations and some new solutions were obtained in []. Li and Zhang [] obtained the third Darboux transformation of the BK system 40 and produced some multi-soliton solutions. 9 presents a nonlinear integrable coupling for the BK system 40. The Lax integrable hierarchy 8 gives a hierarchy of nonlinear integrable couplings for the BK hierarchy. In what follows, we deduce a nonlinear integrable coupling for the classical Boussinesq equation. By making a linear transformation [,]: v = u,w = ξ + 1+ v x, 41 the BK system 40 is transformed to the following classical Boussinesq equation u t + uu x + ξ x = 0, ξ t +1+ξu x + 1 4 u xxx = 0, 4 where ξ is the elevation of the water wave, u is the surface velocity of water along x-direction. Substituting 41 into 9 yields a nonlinear integrable coupling of 4: u t + uu x + ξ x = 0, ξ t + 1+ξu x + 1 4 u xxx = 0, u 1t = 1 u u x 1 u u + u 1+ξ + v x 1 [ +ξ + v x + u u u 1 + 1+ξ + v x u t = 1 u 1u x + 1 u 1u + u 1 1+ξ + v x + 1 ξ + v [ x u 1 u u 1 +u 1 + u 1 u xx +uu x x 1 u 1x + uu + 1 u 1xx +uu x u 1 + u 1 u 1 + u ] u 1+ξ + v x, 1+ξ + v x + 1 ] u u 1 u 1 1+ξ + v x, In the following, we deduce the second nonlinear coupling of the BK hierarchy. We still use the loop algebra G to introduce a Lax pair: 4 U = e 1 1+ v e 10+e 0 we 0+s 1 f 1 0+s f 0, V = V im e i m+ V jm f j m. m 0 i=1 j=4 44

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 15 Similar to the above discussion, a recurrence relation reads as follows V 1,mx = V m + wv m, V,mx = V,m+1 + vv m V 1m, V,mx = V,m+1 vv m wv 1m, V 4,mx = w 1V m +1+w+s V 5m + s V m +V m, V 5,mx = V,m+1 +v+s 1 V m 1+w+s V 4m s V 1m + s 1 V m V m, V,mx = V 5,m+1 +v+s 1 V 5m + 1+wV 4m + s 1 V m +V m. 45 Taking some initial values such as V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then from 45 one infers that V 1,1 = V 4,0 = 0,V,1 = α,v,1 = αw,v 1, = α w,v, = α v,v,1 = αw, V 1, = α 4 w xx + α wv,v, = α w x + wv,v, = α 4 v x α 4 v α w, V, = α 4 w x + wv x + α 4 vw x + wv+ α w,v,1 = αs,v 5,0 = 0,V 4,0 = 0, V, = α s v+s 1 s + s 1 + s 1 w,v 5, = α s x s 1 + s 1 w, V 4, = α s + s w+ 1 s, V, = α 4 s xx s 1x +s 1 w x α 4 v+s 1s v+s 1 s + s 1 + s 1 w α 4 1+w+s s + s w+ 1 s α s w α s 1v+s 1 w x + s 1 wv, V 5, = α 4 s v+s 1 s + s 1 + s 1 w x + α 4 v+s 1s x s 1 + s 1 w+ α 4 w 1 s + s w+ 1 s Setting we obtain that Noting α 4 s 1v s 1 w x s 1 wv,... V n n + = m=0 i=1 V im e i n m+ j=4 V jm f j n m V n +x +[U,V n + ] = V,n+1 e 0 V,n+1 e 0+V 5,n+1 f 0+V,n+1 f 0. V n = V n + +, = V,n+1 e 1 0+ V,n+1 s V,n+1 f 1 0, 1+w+s then we have V x n +[U,V n V,n+1 s V,n+1 ] = V,n+1x e 1 0 V 1,n+1x e 0 f 1 0+V 5,n+1 1+w+s x + w 1 V,n+1 s V,n+1 f 0. 1+w+s,

1 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 The compatibility condition of the Lax pair U and V n leads to v u tn = w s 1 s t n V,n+1x V 1,n+1x = V,n+1 s V,n+1. 4 1+w+s x V 5,n+1 + 1 w V,n+1 s V,n+1 1+w+s When s 1 = s = 0, 4 reduces to the BK hierarchy. When taking n =,α = 1,t = t, 4 reduces to a nonlinear integrable coupling of the BK system 40: v t = 1 v xx + vv x + w x, w t = s 1t = vw+ 1 w x, x [ 1 1 1+w+s s xx s 1x +s 1 w x + 1 v+s 1s v+s 1 s + s 1 + s 1 w + 1 1+w+s s + s w+ 1 s + s 1 v+s 1 w x + s 1 wv+ 1 s v x s v ], x s t = 1 s v+s 1 s + s 1 + s 1 w x + 1 v+s 1s x s 1 + s 1 w+ 1 s w 1 + s w+ 1 s 1 [ s 1 w 1v s 1 w x s 1 wv+ s xx s 1x +s 1 w x +v+s 1 s v+s 1 s + s 1 + s 1 w 1+w+s + 1+w+s s + s w+ 1 ] s + s 1 v+s 1 w x + s 1 wv+s v x s v. 47 Obviously, 47 is different from 9. They are all nonlinear integrable couplings of the BK system. When making the linear transformation, it is easy to transform the hierarchy 47 into a hierarchy of nonlinear integrable couplings for the classical Boussinesq equation. Here we omit it due to complicatedness. Next, we construct the third nonlinear integrable coupling of the BK hierarchy by using a loop algebra H of the Lie algebra H. Setting U = e 1 1+ v e 10+e 0 we 0+w 1 h 0+w h 0, V = V im e i m+ V jm h j m, m 0 i=1 j=4 48 Eq. [U,V] = V x leads to

V 1,mx = V m + wv m, Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 17 V,mx = V,m+1 + vv m V 1m, V,mx = V,m+1 vv m wv 1m, i 1 V 4,mx = 1+ i w+w V 5m + 1 i w+w 1 V m + 1+ i 1 i w 1 + 1+ i w V m, V 5,mx = i V 5,m+1 + 4i V,m+1 + i vv 5m i i vv m + i + w 1 + 4i V 1m, + i+ w+w 1 + w V 4m V,mx = 4i V 5,m+1 + i V,m+1 + i vv 5m i + i vv m + + i w+w 1 + w V 4m + 4i w 1 + i w V 1m. Setting V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then from 49 we have V 4,0 = 0,V 5,1 = αw 1,V,1 = αw,v 4,1 = 0,V, = iα w 1x iα w x + α V 5, = iα w 1x iα w x + α 5 vw 1 vw, i V 4, = αw 1 + i+ αw 1+ i αww 1 + V in 1 i, j are the same with the previous those in 45. Defining we obtain that V n n + = m=0 i=1 V im e i n m+ j=4 vw 1 5 vw, 49 i 1 i i i αww αw 1 w + αw 1w, V jm h j n m = λ n V V n, V n +x +[U,V n + ] = V,n+1 e 0 V,n+1 e 0+ i V 5,n+1 + V,n+1 h 0 If setting V n = V n + +V,n+1 e 1 0, we have + i V 5,n+1 +V,n+1 h 0. V x n +[U,V n ] = V,n+1x e 1 0 wv,n+1 + V,n+1 e 0+ i [ V 5,n+1 + V,n+1 +w w 1 V,n+1 ]h 0+ i [ V 5,n+1 +V,n+1 +w w 1 V,n+1 ]h 0.

18 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 Hence, we get a nonlinear integrable coupling of the BK hierarchy from the zero curvature equation: u tn = v w w 1 w t n V,n+1x V 1,n+1x = i [ V 5,n+1 + V,n+1 +w w 1 V,n+1 ]. 50 i [ V 5,n+1 +V,n+1 +w w 1 V,n+1 ] When setting w 1 = w = 0, 50 reduces to the BK hierarchy. When taking n =,α = 1,t = t, we get a nonlinear integrable coupling of the BK system 40: v t = 1 v xx + vv x + w x, w t = vw+ 1 w x, x w 1t = i [ 1 4 w 1xx + 1 w xx + i 5 vw 1 vw 5i + 1 i vw 1 + i vw x 1 vw 1 5 vw i i + +w w 1 14 v x + 14 v + 1 w, w t = i [ 1 w 1xx + 1 4 w xx + 1 4 v w +w w 1 i 1 x 4 w 1x + 1 w x i vw + i+ w+w 1 +w V 4, α= 1 1 ww 1 w ] i + i 14 v x+ 14 v + 1 w. vw 1 5 vw x + i x vw 1x i 4 vw x + 1 v w 1 + i w+w 1 +w V 4, α= 1 + 1 ww w 1 ] 51. A Nonlinear Integrable Coupling of the KN Hierarchy In the section, we shall use a loop algebra of the Lie algebra Q to introduce a Lax pair, from which a nonlinear integrable coupling of the KN hierarchy is obtained. Set where Q = span{e 1 n,e n,e n, p 1 n, p n, p n}, e 1 n = e 1 λ n,e n = e λ n+1,e n = e λ n+1, p 1 n = p 1 λ n, p n = p λ n+1, p n = p λ n+1,n Z.

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 19 It is easy to verify that the following commutative relations hold: [e 1 m,e n] = e m+n,[e 1 m,e n] = e m+n,[e m,e n] = e 1 m+n+1, [p 1 m, p n] = p m+n,[p 1 m, p n] = p m+n,[p m, p n] = 4p 1 m+n, [e 1 m, p 1 n] = 0,[e 1 m, p n] = p m+n,[e 1 m, p n] = p m+n, [e m, p 1 n] = 1 p m+n p m+n,[e m, p n] = p 1 m+n+1, [e m, p n] = p 1 m+n+1,[e m, p 1 n] = 1 p m+n+ p m+n, [e m, p n] = p 1 m+n+1,[e m, p n] = p 1 m+n+1,m,n Z. By employing the loop algebra Q, we consider a Lax pair U = e 1 1+qe 0+re 0+u 1 p 0+u p 0, V = V im e i m+ m 0 i=1 j=4 V jm p j m The stationary zero curvature equation of the compatibility condition of 5 leads to the following V 1,mx = qv,m+1 + rv,m+1 = qv,mx + rv,mx, V,mx = V,m+1 rv 1m, V,mx = V,m+1 qv 1m,. 5 Setting V 4,mx = q+r+4u 1 V,m+1 + q+r 4u V 5,m+1 +u 1 u V,m+1 u 1 + u V,m+1 = q+r+4u 1 V,mx + q+r 4u V 5,mx +u 1 u V,mx u 1 + u V,mx, 1 V 5,mx = V 5,m+1 V,mx = V,m+1 + then we obtain that q+ 1 r+u 1 V 4m u 1 V 1m, 1 q 1 r+u V 4m + u V 1m. V n n + = m=0 i=1 V im e i m+ j=4 V jm p j mλ n = λ n V V n V n +x +[U,V n + ] = rv,n+1 + qv,n+1 e 1 0 V,n+1 e 0 V,n+1 e 0 V 5,n+1 p 0+V,n+1 p 0 +[q r+4u V 5,n+1 q+r+4u 1 V,n+1 +u u 1 V,n+1 +u 1 + u V,n+1 ]p 1 0 = V 1,nx e 1 0 V,n+1 e 0 V,n+1 e 0 V 5,n+1 p 0+V,n+1 p 0 V 4,nx p 1 0. Choosing a modified term of V n + as that V n = V n + V 1,n e 1 0 V 4,n p 1 0, a direct calculation yields that V x n +[U,V n ] = V,n+1 + rv 1n e 0+ V,n+1 + qv 1n e 0+ +u 1 V 4n p 0+ V,n+1 + 1 q+ 1 r u = V,nx e 0 V,nx e 0 V 5,nx p 0 V,nx p 0., 5 V 5,n+1 + 1 qv 4n + 1 rv 4n + u 1 V 1n V 4n u V 1n p 0

0 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 Therefore, the zero curvature equation admits a hierarchy of equations of evolution type U t V n x +[U,V n ] = 0 q r u tn = u 1 = V,nx V,nx V 5,nx. 54 u t n V,nx A recurrence operator of 54 is q 1 q + q 1 r 0 0 r L = 1 q r 1 r 0 0 A 1 A A A 4, B 1 B B B 4 which satisfies that V,n+1 V,n+1 V 5,n+1 = L V,n V,n V 5,n, V,n+1 V,n where A 1 = u 1 q 1 q+r+4u 1 1 u 1 + u, A = u 1 1 r + 1 q+r+4u 1 1 u 1 u, A = 1 q+r+4u 1 1 q+r 4u, A 4 = 1 q+r+4u 1 1 q+r+4u 1, B 1 = u 1 q 1 q r+4u 1 u 1 + u, B = u 1 r + 1 q r+4u 1 u 1 u, B = 1 q r+4u 1 q+r 4u, B 4 = 1 q r+4u 1 q+r+4u 1..

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 1 Letting V 1,0 = α,v,0 = V,0 = V 5,0 = V,0 = 0, then we have from 5 V,1 = αr,v,1 = αq,v 1,0 = 0,V 5,1 = αu 1,V,1 = αu,v 4,0 = 0,V 1,1 = α r q, V 4,1 = α qu qu 1 + ru 1 ru 4u 1 u,v, = αr x + α qr q, V, = αq x + α rr q, V 5, = αu 1x + α q+r+4u 1 qu qu 1 + ru 1 ru 4u 1 u + 1 u 1r q, V, = αu x + α q r+4u qu qu 1 + ru 1 ru 4u 1 u + α u r q,... When taking n =, the hierarchy 54 reduces to a nonlinear evolution equation q t = αr xx + α qr q x, r t = αq xx + 1 r rq x, u 1,t = αu 1xx + α [q+r+4u 1ru 1 u qu 1 + u 4u 1 u ] x + α u 1r q x, 55 u,t = αu xx + α q r+4u qu 1 + u +ru 1 u 4u 1 u x + α u r q x. Setting u 1 = u = 0, 55 reduces to a nonlinear coupled KN equation q t = αr xx + α qr q x, r t = αq xx + 1 r rq x. 5 Obviously, 55 is a nonlinear integrable coupling of 5. Thus, 54 is a hierarchy of nonlinear integrable couplings of the KN hierarchy. 4 Vector representations of the Lie algebras G and Q as well as some Hamiltonian structures of nonlinear integrable couplings We find that it is difficult to express the Lie algebras G and Q as square matric representations. Therefore, we consider their vector representations so that we can deduce Hamiltonian structures of the obtained nonlinear integrable couplings by using the variational identities [11,18]. For a, b G, we can express them as and we have a = i=1 a i e i + j=4 a j f j,b = i=1 b i e i + j=4 b j f j G, [a,b] = a b a b e 1 +a 1 b a b 1 e +a b 1 a 1 b e +a b 5 a 5 b + a 5 b a b 5 + a b a b + a b a b + a b 5 a 5 b f 1 +a 1 b a b 1 + a 4 b a b 4 + a b 4 a 4 b + a 4 b a b 4 f +a 1 b 5 a 5 b 1 + a 4 b 5 a 5 b 4 + a 4 b a b 4 + a 4 b a b 4 f. 57

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 If noting a = a 1,...,a T,b = b 1,...,b T, then we can write 57 as [A1,B 1 ] 1 [a,b] =, 58 [a,b] 1 where a b 5 a 5 b + a 5 b a b 5 + a b a b + a b a b + a b 5 a 5 b [a,b] 1 = a 1 b a b 1 + a 4 b a b 4 + a b 4 a 4 b + a 4 b a b 4. a 1 b 5 a 5 b 1 + a 4 b 5 a 5 b 4 + a 4 b a b 4 + a 4 b a b 4 It can be verified that the vector space R = {a = a 1,...,a T } is a Lie algebra if equipped with 58. Hence, the Lie algebra G is isomorphic to the Lie algebra R. In order to apply the variational identity to deduce Hamiltonian structures of nonlinear integrable couplings, we need to get a constant symmetric matrix F which satisfies the matrix equation where Rb comes from rewriting 58 as the following form RbF = RbF T,F T = F, 59 [a,b] = a T R 1 b. 0 It is easy to see that 0 b b 0 b b 5 b b 1 0 b 5 b b 4 b 4 b 0 b 1 b 5 b b 4 b 4 R 1 b =. 0 0 0 0 b b + b b 5 + b + b 0 0 0 b + b b 0 b 1 b 4 0 0 0 b + b + b 5 b 1 b 4 0 Solving 59 yields Similarly, for define that a = i=1 a i e i + 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 F 1 =. 1 0 0 1 0 0 0 1 1 0 1 0 j=4 0 1 1 0 0 1 a j p j,b = i=1 b i e i + j=4 b j p j Q, [A1,B 1 ] [a,b] = = a T R b, [a,b]

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 where a b a b + a b a b + a 5 b a b 5 + 4a 5 b 4a b 5 + a b 5 a 5 b [a,b] = a 1 b 5 a 5 b 1 + 1 a 4b 1 a b 4 + 1 a 4b 1 a b 4 + a 4 b 5 a 5 b 4. a b 1 a 1 b + 1 a b 4 1 a 4b + 1 a 4b 1 a b 4 + a b 4 a 4 b Solving Eq.59 for F gives 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 F =. 1 0 0 0 0 0 1 1 0 0 4 0 1 1 0 4 0 In what follows, we construct Hamiltonian structures of the nonlinear integrable couplings of 0 and 54 by using the variational identity [11,18]. At first, we make use of F 1 and F to define two linear functionals. For a = a 1,...,a T,b = b 1,...,b T R, using F 1 we define Using F, we define that Rewrite the Lax pair 8 as follows {a,b} = a 1 b 1 + a 4 b 1 + a 1 b 4 + a b + a b + a 5 b +a b 5 + a 5 b + a b 5 + a b + a b a b a b +a 4 b 4 + a 5 b 5 a b. 4 {a,b} = a 1 b 1 + a 4 b 1 + a 1 b 4 + a b + a 5 b + a b 5 + a b + a b a b a 5 b a b 5 + a b + a b + a 4 b 4 + 4a b 5 + 4a 5 b. 5 where V 1 = V 1m λ m,... Using 4, we obtain m 0 { } } V, U q { V, U u U = λ,q,r,0,u 1,u T,V = V 1,...,V T, { = V +V 5 V, V, U r } { = V +V V, V, U λ { = V +V 5 +V, } = V 1 + V 4. Substituting the above into the variational identity gives rise to ˆ δ x V 1 + V 4 dx = λ γ δ λ λ γ V, U u 1 } = V +V +V 5, V +V 5 V V +V 5 +V V +V +V 5. V +V V

4 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 Comparing the coefficients of λ n on both sides in yields ˆ δ x V 1,n+ + V 4,n+ dx = n 1+γ δu It is easy to verify that γ = 0. Thus, we have where V,n+1 +V 5,n+1 V,n+1 V,n+1 +V 5,n+1 +V,n+1 V,n+1 +V,n+1 +V 5,n+1 = δ δu V,n+1 +V,n+1 V,n+1 H n+1 = ˆ x Therefore, 0 can be written as a Hamiltonian form q r u tn = u 1 u t n 0 0 = 0 0 V,n+1 +V 5,n+1 V,n+1 V,n+1 +V 5,n+1 +V,n+1 V,n+1 +V,n+1 +V 5,n+1. V,n+1 +V,n+1 V,n+1 xv1,n+ + V 4,n+ dx n+1 V 1,n+ + V 4,n+ dx. n+1 =: δh n+1 δu, V,n+1 +V 5,n+1 V,n+1 V,n+1 +V 5,n+1 +V,n+1 V,n+1 +V,n+1 +V 5,n+1 = J δh n+1 δu, 7 V,n+1 +V,n+1 V,n+1 where J is obviously Hamiltonian. From 9, we can obtain V 1,4 and V 4,4. Thus, we can obtain the Hamiltonian structure of the nonlinear integrable coupling 1 of the nonlinear Schrödinger equation if substituting V 1,4,V 4,4 into 7. In the following, we deduce the Hamiltonian structure of 54. Rewrite the Lax pair 5 as { U = λ,qλ,rλ,0,u 1 λ,u λ T, V = V 1,V λ,v λ,v 4,V 5 λ,v λ T, 8 where V 1 = V 1m λ m,... Using the linear functional 5 along with 8, we have m 0 { V, U } = V +V 5 +V λ,{v, U { q r } = V V 5 +V λ, V, U } = V V + 4V λ, u 1 { V, U } = V +V + 4V 5 λ, u { V, U } = [V 1 +q+u 1 + u V + r u 1 + u V + V 4 + r+4u + qv 5 +q+r+4u 1 V ]λ. λ

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 5 Substituting them into the variational identity gives ˆ δ x {V 1 +q+u 1 +u V + r u 1 +u V +V 4 δu + r+4u +qv 5 +q+r+4u 1 V }λdx V +V 5 +V λ = λ γ λ λ γ V V 5 +V λ V V + 4V λ. 9 V + v + 4V 5 λ Comparing the coefficients of λ n+1 on both sides of 9 yields ˆ δ x V 1n +q+u 1 + u V n dx = r u 1 + u V n + V 4n δu V n +V 5n +V n V + r+4u + qv 5n +q+r+4u 1 V n dx = n++γ n V 5n +V n V n V n + 4V n. V n +V n + 4V 5n Utilizing the initial values in 5, we have γ =. Hence, 54 can be written as the Hamiltonian form where H n = 1 4 n q r u tn = u 1 u t n 0 V n +V 5n +V n 0 V = n V 5n +V n 0 V n V n + 4V n = J δh n δu, 70 V n +V n + 4V 5n 0 ˆ x V 1n +q+u 1 u V n + r u 1 + u V n + V 4n +q r+ 4u V 5n +q+r+ 4u 1 V n dx, and J is a Hamiltonian operator. As for Hamiltonian structures of the other obtained nonlinear integrable couplings, we can make a similar analysis but we omit here. 5 Discussions We have constructed three Lie algebras of semi-direct sum form: Ḡ = G G c, where G and G c are all simple Lie sub-algebras and satisfy that [G,G] G,[G c,g c ] G c,[g,g c ] G c. 71

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 We require that the commutator of G is different from that of G c to generate non-trivial nonlinear integrable couplings. Various nonlinear integrable couplings of the AKNS hierarchy and the BK hierarchy were obtained from different loop algebras. We also remark that the problem of how to apply the approach in [1-15] to nonlinear bi-integrable couplings of the AKNS hierarchy and the BK hierarchy deserves future investigation. In order to generate more interesting nonlinear integrable couplings, it should be good to study different isospectral matrix spectral problems. Ablowitz et al. [4] proposed a simple expression of the self-dual Yang-Mills equation by the following isospectral problem { σ + ξ τ ψ = A σ + ξ A τ ψ, 7 τ ξ σ ψ = A τ ξ A σ ψ, where σ, σ,τ, τ are all null coordinates, and A = A µ dµ = A σ dσ +A σ d σ +A τ dτ +A τ d τ. The compatibility condition of 7 leads to A στ A τσ + A σ A τ A τ A σ = 0, A τ τ A σσ + A σ σ A ττ +[A τ,a τ ]+[A σ,a σ ] = 0, A σ τ A τ σ +[A σ,a τ ] = 0. From this, some reduced cases can be presented as follows: 1 Assume that A X, X = σ,τ, σ, τ, are functions of x = σ and t = τ only. Set A σ = 0,A τ = P,A σ = Q,A τ = R. Then we have { Pt +[P,R] = 0, 7 Q t R x +[Q,R] = 0. Let x = σ + σ,y = τ,t = τ,a σ = 0,A σ = P,A τ = R,A τ = Q. Then we obtain P y R x +[P,R] = 0, R t + P x P y +[R,Q] = 0, Q x = 0. 74 75 If A σ is a function of x and t only, x = τ + τ,t = σ,a τ = 0,P = A σ,q = A τ,r = A σ, then we get [4] P x = 0, P t Q x +[P,R] = 0, R x Q t + R,Q] = 0. 4 If A X, X = σ,τ, σ, τ, are functions of x = σ and t = τ only, taking A σ = U + B y,a τ = V +C y,ψ = Ge ξy, leads to [4] { x G = U + B y G, 77 t G = V +C y G. 7 This gives rise to { t U x V +[U,V] C y U + B y V = 0, [B,V] = [C,U]. 78

Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 7 We can generate new nonlinear integrable couplings of both 1+1-dimension and +1-dimension, by using our general construction procedure and combining the above generalized zero curvature equations. It is also interesting for us to see if the resulting integrable couplings possess the linear superposition principle for exponential waves, or more generally, to see if there are linear subspaces of their solution spaces see [5] for details. These will be one of our future research topics. Acknowledgements: The work was supported in part by the State Administration of Foreign Experts Affairs of China, the National Natural Science Foundation of China Nos.17147 and 11071159, Chunhui Plan of the Ministry of Education of China, the Natural Science Foundation of Shanghai, Shanghai Leading Academic Discipline Project No. J50101 and the fundamental Research Funds of the Central University 010LKSX08 and the Natural Science Foundation of Liaoning Province009171. References [1] Ma, W.X., and Fuchssteiner B. 199, Integrable theory of the perturbation equations, Chaos, Solitons and Fractals, 7, 17 150. [] Ma W.X.000, Integrable couplings of soliton equations by perturbation I. A general theory and application to the KdV hierarchy, Meth. Appl. Anal., 7, 1 5. [] Ma,W.X., Xu, X.X. and Zhang,Y.F. 00, Semi-direct sums of Lie algebras and comtinuous integrable couplings, Phys. Lett. A, 51, 15 10. [4] Zhang,Y.F. and Zhang, H.Q. 00, A direct method for integrable couplings of TD hierarchy, J. Math. Phys., 41, 4 47. [5] Ma, W.X.00, Enlarging spectral problems to construct integrable couplings, Phys Lett A, 1, 7 7. [] Zhang, Y.F. and Guo, F.K.00, Matrix Lie algebras and integrable couplings, Commun. Theor. Phys., 45, 81 818. [7] Fan, E.G. and Zhang, Y.F.00, Vector loop algebra and its applications to integrable system, Chaos, Solitons and Fractals, 8, 9 971. [8] Zhang, Y.F. and Liu, J. 008, Induced Lie algebras of a six-dimensional matrix Lie algebra,commun. Theor. Phys., 50, 89 94. [9] Ma, W.X.009, Variational identities and applications to Hamiltonian structures of soliton equations,nonl. Anal., 71, e171 e17. [10] Guo,F.K. and Zhang, Y.F.005, The quadratic-form identity for constructing the Hamiltonian structure of integrable system, J. Phys. A, 8, 857 854. [11] Ma, W.X. and Chen,M. 00, Hamiltonian and qusi-hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A, 9, 10787 10801. [1] Ma,W.X., He, J.S.and Qin,Z.Y. 008, A supertrace identity and its applications to superintegrable systems, J. Math. Phys, 49., 0511. [1] Ma, W.X. and Gao, L. 009, Coupling integrable couplings, Mod. Phys. Lett. B, 15, 1847-180. [14] Zhang,Y.F. and Tam, W.H. 010, Four Lie algebras associated with R and their applications, J. Math. Phys., 51, 04510. [15] Zhang, Y.F. and Fan,E.G. 010, Coupling integrable couplings and bi-hamiltonian structure associated with Boiti-Pempinelli-Tu hierarchy, J. Math. Phys., 51, 0850. [1] Ma,W.X. and Zhu, Z.N. 010, Constructing nonlinear discrete integrable Hamiltonian couplings,comput. Math. Appl., 0, 01 08. [17] Ma,W.X. 011, Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput., 17, 78 744. [18] Ma,W.X.011, Variational identities and Hamiltonian structures, in:nonlinear and Modern Mathematical Physics, pp.1-7, edited byma,w.x., Hu, X.B. and Liu, Q.P., AIP Conference. Proceedings, Vol.11American Institute of Physics, Melville, NY.

8 Y.F. Zhang, W.-X. Ma / Journal of Applied Nonlinear Dynamics 11 01 1 8 [19] Tu, G.Z.1989, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 0, 0 8. [0] Ma, W.X.199, A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reductions, Chin. J. Contemp. Math., 1, 79 89. [1] Ma, W.X., Xu,X.X. and Zhang, Y.F.00, Semidirect sums of Lie algebras and discrete integrable couplings, J. Math. Phys., 47, 05501. [] Li,Y.S., Ma, W.X. and Zhang, J.E.000, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A, 75, 0. [] Li Y.S.,and Zhang, J.E. 001, Darboux transformations of classical Boussinesq system and its multi-soliton solutions, Phys. Lett. A, 84, 5 58. [4] Ablowitz, M.J., Chakravarty, S.and Halburd, R.G. 00, Integrable systems and reductions of the self-dual myang-mills equations, J. Math. Phys., 44, 147 15. [5] Ma, W.X. and Fan, E.G.011, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl., 1, 950 959.