Applicable Analysis Vol. 89, No. 4, April 2010, 457–472
Component-trace identities for Hamiltonian structures Wen-Xiu Maab* and Yi Zhanga a
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R. China; b Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Communicated by Willy Hereman
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
(Received 2 June 2009; final version received 6 August 2009) We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called componenttrace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy. Keywords: Hamiltonian structures; integrable couplings; zero-curvature equations; matrix Lie algebras AMS Subject Classifications: 37K05; 35Q53; 37K10; 35Q58
1. Introduction One of the important problems in soliton theory is to classify integrable equations and identify their integrable structures. From the point of view of zero-curvature equations, there are two classes of integrable equations [1]: the first one associated with semisimple Lie algebras and the second one associated with non-semisimple Lie algebras. It is known that the theory of semisimple Lie algebras provides a solid foundation to study Hamiltonian structures of the integrable equations of the first class [2]. Variational identities on semi-direct sums of Lie algebras [3–5] provide direct approaches to Hamiltonian structures of the integrable equations of the second class [6,7]. The integrable equations of the second class are called integrable couplings [8,9] and they possess a very rich variety of mathematical structures [8–15]. The study of integrable couplings advances the complete classification of integrable equations, based on Lie algebras. In what follows, we will discuss Hamiltonian structures of integrable couplings [4,5], in an effort to supplement the theory of multi-component Hamiltonian equations (see, e.g. [16–18]).
*Corresponding author. Email:
[email protected] ISSN 0003–6811 print/ISSN 1563–504X online ß 2010 Taylor & Francis DOI: 10.1080/00036810903277143 http://www.informaworld.com
458
W.X. Ma and Y. Zhang
Let us now introduce our basic notations and terminology. We start with an evolution equation ut ¼ KðuÞ ¼ Kðx, t, u, ux , uxx , . . .Þ,
ð1:1Þ
where u ¼ (u1, . . . , uq)T is a column vector of dependent variables. We will write vectors in column form throughout this article. A zero-curvature representation [19] of an evolution equation (1.1) is
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
Ut Vx þ ½U, V ¼ 0,
ð1:2Þ
where the Lax pair, U and V, belongs to a given matrix loop algebra g. Let us recall that the Gateaux derivative of a function or an operator P ¼ P(x, t, u) in a direction is given by @ ð1:3Þ P0 ðuÞ½ ¼ Pðu þ "Þ: @" "¼0 R A functional P ¼ P dx is defined as Z dQ j Q-arbitrary function , ð1:4Þ P ¼ P dx ¼ P þ dx where P ¼ P(x, t, u) is a function. An inner product on a space of vector functions is defined by Z ð, Þ ¼ T dx, ð1:5Þ T where and are R two vector functions and denotes the transpose of . A functional P ¼ P dx (or a function P) is called a conserved functional (or a (1.1). conserved density) of an evolution equation (1.1), if dP dt ¼ 0 when u solves R P P T The variational derivative P ¼ ð , . . . , Þ of a functional P ¼ P dx is u u1 uq defined by P @ , ¼ Pðu þ "Þ, ð1:6Þ u @" "¼0 R where is an arbitrary vector field. If P ¼ P dx is local, then we have
P X @P ¼ ð@Þ j ð j Þ , ui @ui j0
@¼
@ , @x
1 i q,
ð1:7Þ
j
where uði j Þ ¼ @@xuji . The adjoint operator Jy of a linear operator J ¼ J(x, t, u) that maps vector fields to vector fields is determined by ð, Jy Þ ¼ ð, JÞ,
ð1:8Þ
where and are arbitrary vector fields. If Jy ¼ J, then J is called skew-symmetric. A linear operator J ¼ J(x, t, u) mapping vector fields to vector fields is called a Hamiltonian operator, if it is skew-symmetric and satisfies the Jacobi identity: ð, J 0 ðuÞ½JÞ þ cycleð, , Þ ¼ 0,
ð1:9Þ
Applicable Analysis
459
where , , are arbitrary vector fields. Associated with a Hamiltonian operator J, the Poisson bracket P Q ,J ð1:10Þ fP, Qg ¼ fP, QgJ ¼ u u defines a Lie bracket, and thus, we have the Jacobi identity fP, fQ, Rgg þ cycleðP, Q, RÞ ¼ 0,
ð1:11Þ
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
where P, Q, R are arbitrary functionals. Any vector field S ¼ S(x, t, u) is a symmetry of an evolution equation (1.1), if it satisfies @S ¼ K 0 ðuÞ½S S 0 ðuÞ½K : @t
ð1:12Þ
If S does not depend explicitly on time t, the above condition (1.12) is reduced to a commutativity condition between K and S, namely ½K, S :¼ K 0 ðuÞ½S S 0 ðuÞ½K ¼ 0:
ð1:13Þ
This is a Lie product on the space of vector fields. If S satisfies the adjoint equation of (1.12): @S ¼ ðK 0 Þy ðuÞ½S S 0 ðuÞ½K , @t
ð1:14Þ
then S is called an adjoint symmetry of an evolution equation (1.1). A linear operator ¼ (x, t, u) mapping vector fields to vector fields is called a recursion operator of an evolution Equation (1.1) [20], if it satisfies: @ þ 0 ðuÞ½K K 0 ðuÞ½ þ K 0 ðuÞ½ ¼ 0 @t
ð1:15Þ
for any vector filed . A recursion operator maps symmetries to symmetries of the same equation. A linear operator ¼ (x, t, u) is called a hereditary operator [21], if it satisfies 0 ðuÞ½ 0 ðuÞ½ 0 ðuÞ½ þ 0 ðuÞ½ ¼ 0
ð1:16Þ
for any vector fields and . Obviously, if a time-independent recursion operator ¼ (x, u) of an evolution equation (1.1) is hereditary, then is also a recursion operator of any evolution equation ut ¼ mK with m 1. If there is a Hamiltonian operator J ¼ J(x, t, u) and a functional H ¼ H(x, t, u) such that an evolution equation (1.1) can be written as Z HðuÞ ð1:17Þ , HðuÞ ¼ HðuÞdx, ut ¼ KðuÞ ¼ JðuÞ u then we say that the evolution equation (1.1) possesses a Hamiltonian structure (1.17), and the functional H and the function H is called the Hamiltonian functional and the Hamiltonian function associated with the Hamiltonian structure, respectively. Computer algebra systems were used to search for symmetries and
460
W.X. Ma and Y. Zhang
conserved densities [22,23]. A Hamiltonian structure (1.17) tells us the following relations: conserved functional I $ adjoint symmetry
I I $ symmetry J , u u
ð1:18Þ
and J u is a Lie homomorphism: J
I 1 I 2 ,J fI 1 , I 2 gJ ¼ J , u u u
ð1:19Þ
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
where I 1 and I 2 are arbitrary functionals. A Hamiltonian pair J and M means that any linear combination of J and M is Hamiltonian, and a bi-Hamiltonian equation [24] is given by ut ¼ KðuÞ ¼ JðuÞ
H1 ðuÞ H2 ðuÞ ¼ MðuÞ , u u
ð1:20Þ
where J and M constitute a Hamiltonian pair. Hamiltonian structures link two Lie algebras of conserved functionals and symmetries together closely and play an important role in exploring Liouville integrability. This article is organized as follows. In Section 2, component-trace identities will be introduced on a particular class of non-semisimple matrix Lie algebras, which can be applied to Hamiltonian structures of integrable couplings. In Section 3, a new approach to Hamiltonian structures of the perturbation equations will be presented by using component-trace identities. In Section 4, an application will be made for the KdV hierarchy, to illustrate the resulting general theory. In Section 5, a few concluding remarks and an open question on Hamiltonian structures of integrable couplings will be discussed.
2. Component-trace identities Let us take a matrix Lie algebra g consisting 2 A0 A1 6 6 A0 A1 6 6 .. A¼6 . 6 6 4
of the following matrices: 3 AN .. 7 . 7 7 .. 7 .. , . 7 . 7 7 A0 A1 5
0
ð2:1Þ
A0
where Ai, 0 i N, are square matrices of the same order. The lower triangular case is completely similar. For convenience, we rewrite an element of the Lie algebra g as a vector of matrices: A ¼ (A0, A1, . . . , AN). Then, for A ¼ ðA0 , A1 , . . . , AN Þ,
B ¼ ðB0 , B1 , . . . , BN Þ 2 g,
ð2:2Þ
the matrix product AB reads AB ¼ ðC0 , C1 , . . . , CN Þ,
Ck ¼
X iþj¼k
Ai Bj ,
0 k N,
ð2:3Þ
461
Applicable Analysis and the matrix commutator is given by X
½A, B ¼ AB BA ¼ . . . ,
! ½Ai , Bj , . . . :
ð2:4Þ
iþj¼k
These Lie algebras are semi-direct sums of two classes of Lie algebras: ]
g ¼ fðA0 , 0 , 0ÞjA0 -arbitraryg fð0, A1 , . . . , AN ÞjAi -arbitraryg,
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
and thus, they are non-semisimple. Let i, 0 i N, be given constants. We introduce a bilinear form on the Lie algebra g as follows: hA, Bi ¼
N X
k trðCk Þ ¼
k¼0
N X
!
X
k tr
Ai Bj ,
A, B 2 g,
ð2:5Þ
iþj¼k
k¼0
where A and B are given by (2.2) and C ¼ AB ¼ (C0, . . . , CN) is the matrix product defined by (2.3). This is a linear combination of the traces of the components of AB. It is obvious that this bilinear form is non-degenerate on g iff N 6¼ 0. Such a special case yields a non-degenerate bilinear form generated only by the last component trace: !
X
hA, Bi ¼ tr
Ai Bj :
ð2:6Þ
iþj¼N
If we replace N with a smaller integer in (2.6), the resulting bilinear form is degenerate. Moreover, the bilinear form (2.5) is symmetric hA, Bi ¼ hB, Ai,
A, B 2 g,
ð2:7Þ
since tr(ab) ¼ tr(ba); and it is invariant under the matrix product, since we have hA, BC i ¼
¼
N X
k tr
k¼0
iþj¼k
N X
X
k tr
¼
¼
N X
! Ai ðBC Þj
Ai
k tr
Ai Br Cs
iþrþs¼k
N X
X
X
k¼0
jþs¼k iþr¼j
N X
X
k¼0
k tr
jþs¼k
Br Cs !
X
k¼0
k tr
!
X rþs¼j
iþj¼k
k¼0
¼
X
!
!
Ai Br Cs
ðABÞj Cs
! ¼ hAB, C i,
A, B, C 2 g:
ð2:8Þ
462
W.X. Ma and Y. Zhang
Thus, by the symmetric property (2.7), the bilinear form (2.5) is invariant under the Lie product: hA, ½B, C i ¼ hA, BC CBi ¼ hA, BC i hA, CBi ¼ hAB, C i hAC, Bi ¼ hAB, C i hB, AC i ¼ hAB, C i hBA, C i
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
¼ hAB BA, C i ¼ h½A, B, C i,
A, B 2 g:
ð2:9Þ
This way, for a given spectral matrix U ¼ U(u, ) ¼ (U0, U1, . . . , UN) 2 g, the variational identity [4]
Z @U @ @U V, dx ¼ V, , u @ @ @u where V ¼ V(u, ) ¼ (V0, V1, . . . , VN) 2 g is a solution to Vx ¼ ½U, V ,
ð2:10Þ
and is defined by ¼
1 d ln jhV, V ij, 2 d
ð2:11Þ
yields the following result. THEOREM 2.1 Let g be a matrix Lie algebra consisting of block matrices defined by (2.1). For a given spectral matrix U ¼ U(u, ) ¼ (U0, U1, . . . , UN) 2 g, we have the variational identity ! ! Z N N X X X X @Uj @Uj @ k tr Vi k tr Vi dx ¼ , ð2:12Þ u k¼0 @ k¼0 @ @u iþj¼k iþj¼k where V ¼ V(u, ) ¼ (V0, V1, . . . , VN) 2 g satisfies the stationary zero-curvature equation (2.10), all k’s are arbitrary constants with N 6¼ 0 and is the constant determined by (2.11). We call this variational identity (2.12) the component-trace identity, since it only @U involves the traces of the components of the matrix products V @U @ and V @u . For N ¼ 1, we consider two special cases, namely, a bi-trace and a tri-trace identity, as follows: Z @U1 @U0 @U1 @U0 @ tr V0 tr V0 þ tr V1 dx ¼ þ tr V1 , ð2:13Þ u @ @ @ @u @u and Z @U0 @U1 @U0 tr V0 þ tr V0 þ tr V1 dx @ @ @ @ @U0 @U1 @U0 þ tr V0 þ tr V1 : ¼ tr V0 @ @u @u @u
u
ð2:14Þ
463
Applicable Analysis
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
The second one has been used to construct Hamiltonian structures of multicomponent soliton equations in [25]. In particular, the above bi-trace and tri-trace identities will help generate Hamiltonian structures of the first-order perturbation equations. For a general integer N, we can have a special variational identity ! ! Z X X @Uj @Uj @ tr tr Vi Vi dx ¼ , ð2:15Þ u @ @ @u iþj¼N iþj¼N associated with the bilinear form (2.6). This identity only involves the lastcomponent trace, and thus, we call it the last-component-trace identity. We will show that this identity (2.15) can be used to furnish Hamiltonian structures for the perturbation equations of any order. When N ¼ 2, the identity (2.15) gives another tri-trace identity Z @U2 @U1 @U0 tr V0 þ tr V1 þ tr V2 dx u @ @ @ @ @U2 @U1 @U0 þ tr V1 þ tr V2 : ð2:16Þ ¼ tr V0 @ @u @u @u This will be a tool for generating Hamiltonian structures of the second-order perturbation equations.
3. Hamiltonian structures of the perturbation equations For a given square matrix A ¼ A(u, ), an be taken as 2 A A1 6 6 A 6 6 A ¼ A^ N ¼ 6 6 6 4
enlarged square matrix A ¼ AˆN 2 g can 3 AN .. 7 . 7 A1 7 .. 7 .. .. , . 7 . . 7 7 A A1 5
0
ð3:1Þ
A
where the components Ai, 1 i N, are defined by Ai ¼
1 @i Aðu^ N Þ, i! @"i "¼0
u^ N ¼ u þ
N X
"i i ,
1 i N:
ð3:2Þ
i¼1
Let us assume that a zero-curvature equation Ut Vx ¼ ½U, V ¼ 0
ð3:3Þ
yields an integrable Hamiltonian equation ut ¼ KðuÞ ¼ JðuÞ
HðuÞ , u
Z HðuÞ ¼
HðuÞdx:
ð3:4Þ
464
W.X. Ma and Y. Zhang
Thus, the enlarged zero-curvature equation by perturbation U^ N,t V^ N,x þ ½U^ N , V^ N ¼ 0,
ð3:5Þ
where UˆN and V^ N are defined in (3.1) and (3.2), gives rise to the perturbation equation of N-th order: T 1 @ 1 @N T T ð3:6Þ ^ ^ K ð u Þ, . . . , K ð u Þ , ^ N,t ¼ K^ N ð^ N Þ ¼ KT ðuÞ, N N 1! @" "¼0 N! @"N "¼0 where the column vector ^ N of dependent variables is ð3:7Þ
Now based on the generating function of Hamiltonian functions for the original equation @U , ð3:8Þ Hg ðuÞ ¼ tr V @ we can compute the generating function of Hamiltonian functions for the perturbation equations of N-th order as follows: 1 @N H^ g,N ð^ N Þ ¼ Hg ðu^ N Þ N!@"N "¼0 1 @N @ ^ ^ Uð u ¼ tr Vð u Þ Þ N N N! @"N "¼0 @ ! 1 X i @i @ j @ Uðu^ N Þ ¼ tr Vðu^ N Þ j N! iþj¼N N @"i "¼0 @" "¼0 @ ! X 1 @i @ 1 @ j ¼ tr Vðu^ N Þ Uðu^ N Þ i! @"i "¼0 @ j! @"j "¼0 iþj¼N ! X @Uj ¼ tr Vi ð3:9Þ @ iþj¼N
i
.
..
.
the first step of which is guaranteed by a theory dealing with the perturbation equations [8]. This implies that the last-component-trace identity (2.15) provides the generating function of Hamiltonian functions for the perturbation equations. ^ ^ N Þ We further introduce the enlarged Hamiltonian operator J^ ¼ Jð 3 2 0 J 6 J J1 7 7 6 7 6 . 6 .. 7 ð3:10Þ J ¼ J^N ¼ 6 7, 6 .. 7 7 6 4 J J1 . 5 J J1 JN ..
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
^ N ¼ ðuT , T1 , . . . , TN ÞT :
where Ji ¼ i!1 @"@ i j"¼0 Jðu^ N Þ, 1 i N: Then, a Hamiltonian structure of the perturbation equation (3.6) is given by Z @ H^ N ð^ N Þdx, ^ N,t ¼ K^ N ð^ N Þ ¼ J^N ð3:11Þ @^ N
465
Applicable Analysis where the Hamiltonian function
1 @N ð3:12Þ H^ N ð^ N Þ ¼ Hðu^ N Þ N! @"N "¼0 is generated from the generating function H^ g,N ð^ N Þ defined by (3.9) like the Hamiltonian function H(u) is generated from the generating function Hg(u) defined by (3.8). Therefore, the Hamiltonian structures of the perturbation equations are just a consequence of the last-component-trace identity (2.15). The whole process provides a new approach to Hamiltonian structures of the perturbation equations by using component-trace identities.
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
4. Application to the KdV hierarchy 4.1. The KdV hierarchy Let us start with the well-known spectral problem of the KdV hierarchy: 1 0 1 , ¼ x ¼ U ¼ Uðu, Þ , U ¼ : u 0 2
ð4:1Þ
Setting
a V¼ c
X 1 b ¼ Vi i , a i¼0
ai Vi ¼ ci
bi , ai
i 0,
ð4:2Þ
the stationary zero-curvature equation Vx ¼ [U, V] becomes ax ¼ c ð uÞb,
bx ¼ 2a,
cx ¼ 2ð uÞa:
ð4:3Þ
This is equivalent to 1 c ¼ bxx þ ð uÞb, 2
1 a ¼ bx , 2
1 1 bxxx ux b þ ð uÞbx ¼ 0: 4 2
It follows then that 8 1 1 > > > biþ1 ¼ Lbi , L ¼ @2 þ u @1 ux , b0,x ¼ 0, > > 4 2 > < 1 ci ¼ bi,xx þ biþ1 ubi , > 2 > > > > 1 > : ai ¼ bi,x , 2
i 0, i 0,
ð4:4Þ
i 0:
As in [26], we select the initial data b0 ¼ 0, b1 ¼ 1, which yields c0 ¼ 1, c1 ¼ 12 u, a0 ¼ a1 ¼ 0, and suppose that biju¼0 ¼ ciju¼0 ¼ aiju¼0 ¼ 0, i 2 (i.e. setting the constant of integration equal to zero). Thus, we can have 1 b2 ¼ u, 2
1 3 b3 ¼ uxx þ u2 , 8 8
b4 ¼
1 5 5 5 uxxxx þ u2x þ uuxx þ u3 : 32 32 16 16
ð4:5Þ
The compatibility conditions of Lax pairs x ¼ U ,
tm ¼ V ½m ,
V ½m ¼ ðmþ1 V Þþ þ Dm ,
ð4:6Þ
466
W.X. Ma and Y. Zhang
where (P)þ denotes a selection of the polynomial part of P and 0 0 Dm ¼ , m 0, bmþ1 0
ð4:7Þ
give rise to the KdV hierarchy (see, e.g. [26]): utm ¼ Km ¼ 2bmþ2,x ¼ 2@Lbmþ1 ¼ ¼ 2@Lm b2 ,
m 0:
ð4:8Þ
Its first nonlinear equation is the KdV equation 3 1 ut1 ¼ uux þ uxxx : 2 4
ð4:9Þ
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
Since we have @U @U tr V ¼ b, tr V ¼ b, @ @u the trace identity [27] or the last-component-trace identity (2.15) with N ¼ 0 gives Z @ b dx ¼ ðbÞ: ð4:10Þ u @ This identity with ¼ 12 leads to bi ¼
2biþ1 , u 2i 1
i 0:
ð4:11Þ
Therefore, the KdV hierarchy has the bi-Hamiltonian structure [24]: Hm Hm1 ¼M , u u
m 1,
ð4:12Þ
1 1 J ¼ @, M ¼ JL ¼ @3 þ u@ þ ux , 4 2
ð4:13Þ
u tm ¼ J with the Hamiltonian pair
and the Hamiltonian functionals Z Hm ¼
4bmþ3 dx, 2m þ 3
m 0:
ð4:14Þ
The corresponding recursion operator reads 1 1 ¼ Ly ¼ MJ1 ¼ @2 þ u þ ux @1 , 4 2
ð4:15Þ
which is hereditary. We emphasize that the approach to Hamiltonian structures by the trace identity produces Hamiltonian functions explicitly, and thus, we can compute conserved densities readily by computer algebra systems, such as Maple, Mathematica and MuPAD.
467
Applicable Analysis 4.2. The perturbation equations
We will focus on the perturbation equations of the first order, but a similar analysis can be made for the perturbation equations of higher order. To generate the perturbation equations of the first order [8,28] we take u, Þ ¼ Uð
U0
U1
0
U0
,
U0 ¼ U,
U1 ¼
0
0
v 0
,
u u ¼ , v
ð4:16Þ
and solve the enlarged stationary zero-curvature equation by assuming that
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
u, Þ ¼ Vð
V0
V1
0
V0
,
V0 ¼ V,
V1 ¼
e
f
g
e
¼
X ei i0
gi
fi ei
i ,
ð4:17Þ
where U and V are defined by (4.1) and (4.2). The enlarged stationary zero-curvature V becomes equation V x ¼ ½U, ex ¼ g ð uÞ f þ vb,
fx ¼ 2e,
gx ¼ 2ð uÞe 2va,
ð4:18Þ
together with (4.3). Equivalently, this system leads to 8 > >
1 1 1 > : fxxx ux f þ ð uÞ fx ¼ ðvbÞx þ 1 vbx : 4 2 2 2 So, we have a recursive formula for determining f: 1 1 fiþ1 ¼ Lfi þ vbi þ @1 vbi,x , 2 2
ð4:19Þ
i 0,
where L is defined by (4.4). If we choose the initial data: f0 ¼ 0 and f1 ¼ 0, then we have 8 1 1 3 > < f2 ¼ v, f3 ¼ vxx þ uv, 2 8 4 5 5 5 15 > :f ¼ 1 v ux vx þ uxx v þ uvxx þ u2 v: 4 xxxx þ 32 16 16 16 16
ð4:20Þ
Now, the Lax operator V ½m is selected to be " ½m
V
¼
V ½m
V1½m
0
V ½m
# ,
V1½m ¼ ðmþ1 V1 Þþ þ
0 fmþ1
0 , 0
ð4:21Þ
V ½m ¼ 0 yield the and then the enlarged zero-curvature equations U tm V x½m þ ½U, hierarchy of the first-order perturbation equations: utm ¼ 2bmþ2,x ,
vtm ¼ 2fmþ2,x ,
m 0:
ð4:22Þ
468
W.X. Ma and Y. Zhang
Its first nonlinear equation is the first-order perturbation equation of the KdV equation [10,29]: 8 3 1 > < dut1 ¼ uux þ uxxx , 2 4 3 > : dv ¼ ðuvÞ þ 1 v : t1 xxx x 2 4 It is straightforward to compute that ! X @Uj e @U0 tr Vi ¼ tr V1 ¼ tr @ @ g iþj¼1
ð4:23Þ
f e
0 1
0 0
¼f
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
and
tr
X iþj¼1
Vi
@Uj @u
!
2 3 3 2 e f 0 0 @U0 6 tr V1 @u 7 6 tr g e 1 0 7 f 6 7 6 7 7 ¼ 6 ¼6 : 7 ¼ 4 5 a b 0 0 @U1 5 4 b tr V0 tr @v c a 1 0
Now the last-component-trace identity (2.15) gives rise to Z f @ f dx ¼ : u @ b
ð4:24Þ
Checking a term of a special power of determines that ¼ 12. Next, an application of the last-component-trace identity (2.15) yields fi 2fiþ1 , i 0: ð4:25Þ ¼ u 2i 1 bi Therefore, the hierarchy (4.22) of the perturbation equations has the bi-Hamiltonian structure: u tm ¼ K m ¼
2bmþ2,x 2fmþ2,x
¼ J
m H Hm1 , ¼M u u
m 0,
ð4:26Þ
with the Hamiltonian pair J ¼
0
@
@
0
,
" ¼ M
1 3 4@
0 1 3 4@
þ u@ þ 12 ux
þ u@ þ 12 ux v@ þ 12 vx
# ,
ð4:27Þ
and the Hamiltonian functionals m¼ H
Z
4fmþ3 dx, 2m þ 3
m 0:
ð4:28Þ
The above computation on the Hamiltonian structures of the perturbation equations of the KdV hierarchy is given for the first time by using component-trace identities.
469
Applicable Analysis
It follows from the bi-Hamiltonian theory [24,30] that the bi-Hamiltonian structure (4.26) also gives a common hereditary recursion operator for the hierarchy (4.22): " # 1 2 1 1 @ þ u þ u @ 0 x 2 J1 ¼ 4 ¼M : ð4:29Þ 1 2 1 1 v þ 12 vx @1 4 @ þ u þ 2 ux @ Furthermore, we have ½K m , K n ¼ 0,
m, H n g ¼ fH m, H n g ¼ 0, fH J M
m, n 0:
ð4:30Þ
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
In words, every perturbation equation has infinitely many commuting symmetries 1 fK n g1 n¼0 and conserved functionals fHn gn¼0 . Higher-dimensional generalizations of the Hamiltonian pair (4.27) and its recursion operator (4.29) are presented in [15].
5. Discussion and concluding remarks Component-trace identities have been presented and discussed, based on variational identities on semi-direct sums of Lie algebras. A special case of such componenttrace identities yields Hamiltonian structures of the perturbation equations. To illustrate the theory, an application has been made for the KdV hierarchy. Applications to integrable couplings of other integrable equations, such as the KdV-6 equation [31] should be interesting. Similar to the algebra in (2.1), the algebra of lower triangular square block matrices can have similar zero-curvature representations for integrable couplings and can be used to construct their Hamiltonian structures. There are also different enlarged Lax pairs for given integrable couplings. For example, all pairs " # " # V k V1 U k U1 , V¼ U¼ 0 U 0 V with different values of k present the same integrable coupling system. Moreover, direct sums of enlarged Lax pairs produce new Lax pairs for coupled systems. For instance, an enlarged spectral matrix Uˆ(uˆ) can be taken as either of the following matrices (see [32] for more details about the second example): 2 3 2 3 UðuÞ Ua,1 ðu, vÞ 0 0 0 0 UðuÞ Ua,1 ðu, vÞ 6 0 7 6 0 7 UðuÞ 0 0 UðuÞ 0 0 6 7 6 7 6 7, 6 7: 4 0 0 UðuÞ Ua,2 ðu, wÞ 5 4 0 0 UðuÞ Ua,2 ðu, wÞ 5 0
0
0
UðuÞ
0
0
0
0
Then using the Kronecker product [33,34] we can generate many different zerocurvature representations for integrable couplings. Furthermore, we can form a general coupled system by n given integrable couplings: ut ¼ KðuÞ,
v1,t ¼ S1 ðu, v1 Þ, . . . , vn,t ¼ Sn ðu, vn Þ:
470
W.X. Ma and Y. Zhang
Similar integrable properties to the original equation ut ¼ K(u) can be verified for this large coupled system. Such coupled systems of integrable couplings can also provide concrete examples of soliton equations exhibiting solution diversity (see, e.g. [35–43]). There remains, however, an open question [44]. Assume that the original equation ut ¼ K(u) is Hamiltonian. Does the above coupled system of integrable couplings possess any Hamiltonian structure? In particular, is there a Hamiltonian structure for the following coupled system: ut ¼ KðuÞ,
vt ¼ K 0 ðuÞ½v,
wt ¼ K 0 ðuÞ½w,
0
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
where K (u)[] denotes the Gateaux derivative as in (1.3)? Extensions of polynomial Virasoro algebras [45] might be helpful to answer this question. If w ¼ 0 or v ¼ 0, the coupled system reduces to the first-order perturbation equation, and thus, these reductions possess Hamiltonian structures [8,9].
Acknowledgements The work was supported in part by the Established Researcher Grant and the CAS faculty development grant of the University of South Florida and Chunhui Plan of the Ministry of Education of China.
References [1] W.X. Ma, Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal. 70 (2009), doi:10.1016/j.na.2009.02.045. [2] A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I, Birkha¨user Verlag, Basel, 1990. [3] F.K. Guo and Y.F. Zhang, The quadratic-form identity for constructing the Hamiltonian structure of integrable systems, J. Phys. A: Math. Gen. 38 (2005), pp. 8537–8548. [4] W.X. Ma and M. Chen, Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A: Math. Gen. 39 (2006), pp. 10787–10801. [5] W.X. Ma, A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A: Math. Theor. 40 (2007), pp. 15055–15069. [6] W.X. Ma, X.X. Xu, and Y.F. Zhang, Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. Lett. A 351 (2006), pp. 125–130. [7] W.X. Ma, X.X. Xu, and Y.F. Zhang, Semidirect sums of Lie algebras and discrete integrable couplings, J. Math. Phys. 47 (2006), pp. 053501–053516. [8] W.X. Ma and B. Fuchssteiner, Integrable theory of the perturbation equations, Chaos Solitons Fractals 7 (1996), pp. 1227–1250. [9] W.X. Ma, Integrable couplings of soliton equations by perturbations: I. A general theory and application to the KdV hierarchy, Methods Appl. Anal. 7 (2000), pp. 21–55. [10] W.X. Ma and B. Fuchssteiner, The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A 213 (1996), pp. 49–55. [11] F.G. Guo and Y.F. Zhang, A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling, J. Math. Phys. 44 (2003), pp. 5793–5803. [12] T.C. Xia, X.H. Chen, and D.Y. Chen, A new Lax integrable hierarchy N Hamiltonian structure and its integrable couplings system, Chaos Solitons Fractals 23 (2005), pp. 451–458. [13] Z. Li, Y.J. Zhang, and H.H. Dong, Integrable couplings of the TC hierarchy and its Hamiltonian structure, Modern Phys. Lett. B 21 (2007), pp. 595–602.
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
Applicable Analysis
471
[14] F.J. Yu and H.Q. Zhang, Hamiltonian structure of the integrable couplings for the multicomponent Dirac hierarchy, Appl. Math. Comput. 197 (2008), pp. 828–835. [15] W.X. Ma, A bi-Hamiltonian formulation for triangular systems by perturbations, J. Math. Phys. 43 (2002), pp. 1408–1421. [16] M. Boiti, P.J. Caudrey, and F. Pempinelli, A hierarchy of Hamiltonian evolution equations associated with a generalized Schro¨dinger spectral problem, Nuovo Cimento B 83 (1984), pp. 71–87. [17] W.X. Ma and M. Pavlov, Extending Hamiltonian operators to get bi-Hamiltonian coupled KdV systems, Phys. Lett. A 246 (1998), pp. 511–522. [18] W.X. Ma, A Hamiltonian structure associated with a matrix spectral problem of arbitraryorder, Phys. Lett. A 367 (2007), pp. 473–477. [19] A. Das, Integrable Models, World Scientific, Teaneck, NJ, 1989. [20] P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), pp. 1212–1215. [21] B. Fuchssteiner, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. 3 (1979), pp. 849–862. [22] W. Hereman, Review of symbolic software for Lie symmetry analysis: Algorithms and software for symbolic analysis of nonlinear systems, Math. Comput. Model. 25 (1997), pp. 115–132. [23] U¨. Go¨ktas and W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbolic Comput. 24 (1997), pp. 591–621. [24] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), pp. 1156–1162. [25] F.K. Guo and Y.F. Zhang, Two unified formulae, Phys. Lett. A 366 (2007), pp. 403–410. [26] W.X. Ma, New finite-dimensional integrable systems by symmetry constraint of the KdV equations, J. Phys. Soc. Japan 64 (1995), pp. 1085–1091. [27] G.Z. Tu, On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. Phys. A: Math. Gen. 22 (1989), pp. 2375–2392. [28] W.X. Ma and X.G. Geng, New completely integrable Neumann systems related to the perturbation KdV hierarchy, Phys. Lett. B 475 (2000), pp. 56–62. [29] K.M. Tamizhmani and M. Lakshmanan, Complete integrability of the Korteweg-de Vries equation under perturbation around its solution: Lie Ba¨cklund symmetry approach, J. Phys. A: Math. Gen. 16 (1983), pp. 3773–3782. [30] B. Fuchssteiner and A.S. Fokas, Symplectic structures, their Ba¨cklund transformations and hereditary symmetries, Phys. D 4 82 (1981), pp. 47–66. [31] A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich, and R. Turhan, A new integrable generalization of the Korteweg-de Vries equation, J. Math. Phys. 49 (2008), pp. 073516–073526. [32] W.X. Ma, Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. Lett. A 316 (2003), pp. 72–76. [33] W.X. Ma and F.K. Guo, Lax representations and zero-curvature representations by the Kronecker product, Int. J. Theor. Phys. 36 (1997), pp. 697–704. [34] F.J. Yu and L. Li, A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product, Phys. Lett. A 372 (2008), pp. 3548–3554. [35] W.X. Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A 301 (2002), pp. 35–44. [36] W.X. Ma, Diversity of exact solutions to a restricted Boiti-Leon-Pempinelli dispersive long-wave system, Phys. Lett. A 319 (2003), pp. 325–333. [37] H.C. Hu, B. Tong, and S.Y. Lou, Nonsingular positon and complexiton solutions for the coupled KdV system, Phys. Lett. A 351 (2006), pp. 403–412. [38] H.Y. Wang, X.B. Hu, and Gegenhasi, 2D Toda lattice equation with self-consistent sources: Casoratian type solutions, bilinear Ba¨cklund transformation and Lax pair,
472
[39] [40] [41]
[42] [43]
Downloaded By: [University of South Florida] At: 22:03 21 June 2010
[44] [45]
W.X. Ma and Y. Zhang in Special Issue: Topics on Integrable Systems, S.B. Damelin and W.X. Ma, eds., pp. 133–143, 202 (2007), J. Comput. Appl. Math. Y. Zhang, Y.N. Lv, L.Y. Ye, and H.Q. Zhao, The exact solutions to the complex KdV equation, Phys. Lett. A 367 (2007), pp. 465–472. W.X. Ma, An application of the Casoratian technique to the 2D Toda lattice equation, Mod. Phys. Lett. B 22 (2008), pp. 1815–1825. Y. Zhang, J.B. Li, and Y.N. Lv, The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vries equation, Ann. Physics 323 (2008), pp. 3059–3064. W.X. Ma, R.G. Zhou, and L. Gao, Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2þ1 dimensions, Mod. Phys. Lett. A 24 (2009), pp. 1677–1688. W.X. Ma and J.H. Lee, A transformed rational function method and exact solutions to the 3þ1 dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 42 (2009) 1356–1363. W.X. Ma and L. Gao, Coupling integrable couplings, Mod. Phys. Lett. B 23 (2009), pp. 1847–1860. P. Casati and G. Ortenzi, Bihamiltonian equations on polynomial Virasoro algebras, J. Nonlinear Math. Phys. 13 (2006), pp. 352–364.
