Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations

Applied Mathematics and Computation 217 (2010) 1497–1508

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations Hong-Xiang Yang a,*, Jun Du b,c, Xi-Xiang Xu d, Jin-Ping Cui a a

College of Information Science and Technology, Taishan College, Taian 271021, China College of Information Science and Engineering, Shandong University, Jinan 250100, China School of Communication, Shandong Normal University, Jinan 250014, China d College of Science, Shandong University of Science and Technology, Qingdao 266510, China b c

a r t i c l e

i n f o

Keywords: Isospectral eigenvalue problem Integrable coupling Generalized trace identity Super-trace identity Super-Hamiltonian structure

a b s t r a c t By considering an isospectral eigenvalue problem, a hierarchy of soliton equations are derived. Two types of extensions are presented by enlarging the associated spectral problem. With the aid of generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures for the integrable extensions are established. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction During the past two decades, there has been a great progress of what is nowadays called the theory of soliton [1–10]. The representation of a nonlinear system as the compatibility condition of linear equations is an idea that is central to our understanding of the word ‘‘integrability”. An effective approach, i.e., the trace identity, that produces Hamiltonian structures of infinite-dimensional integrable systems has been established [4], by using of which many infinite-dimensional Liouville integrable Hamiltonian systems are constructed [4–10]. A hierarchy of evolution equations

ltm ¼ K m ðlÞ; m P 0

ð1:1Þ

is said to be Lax integrable, if it can be written as a compatibility condition [4]

Uðl; kÞt  V m ðl; kÞx þ ½Uðl; kÞ; V m ðl; kÞ ¼ 0;

ð1:2Þ

of a suitable eigenvalue problem

ux ¼ Uðl; kÞu; and a sequence of associated auxiliary eigenvalue problems

utm ¼ V m ðl; kÞu; m P 0; where l ¼ ðl1 ; l2 ; . . . ; ll ÞT , li ¼ li ðx; tÞ; i ¼ 1; 2; . . . ; l are the potentials, k is spectral parameter with kt ¼ 0. Usually, for the system (1.1), the Hamiltonian structure can be constructed using Tu scheme, if there is a Hamiltonian operator J and a set of e m ; m ¼ 1; 2; . . ., such that Eq. (1.1) may be represented as the Hamiltonian form conserved functionals H * Corresponding author. E-mail address: [email protected] (H.-X. Yang). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.030

1498

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

ltm ¼ K m ðlÞ ¼ J

em dH ; dl

m P 0;

where the functionals

em ¼ H @¼

Z

d ¼ dl

Hm dx;

d ; dx



d

d

;

dl1 dl2

;...;

d dll

T

em X em dH dH ¼ ð@Þm m ; dli dli mP0

;

lmi ¼ @ m li ; i ¼ 1; 2; . . . ; l:

Wherein, choosing a suitable isospectral problem is well known to be of paramount importance in constructing integrable Hamiltonian systems of evolution. Given a spectral problem [8]

ux ¼ U u; U ¼



k þ u

kv

1

ku

 ;

ð1:3Þ

u ¼ ðu1 ; u2 ÞT . By means of constructing a proper time evolution

utm ¼ V m u; V m ¼

m  X ai

kbi

ci

ai

i¼0



kmi þ



cmþ1

0

0

cmþ1

 ;

and using the zero-curvature equation (1.2), we obtain a hierarchy of bi-Hamiltonian soliton equations as follows:

ltm ¼ K m ðlÞ ¼



amx  cmþ1x

 ¼J

2amx

em e m1 H H ¼M ; dl dl

m > 1;

ð1:4Þ

e m are given by where the Hamiltonian operators J; M and the Hamiltonian functional H

J¼ em ¼ H

1 @ 2

@

@ Z

0

!

;

M¼

1 2 @ 2

2amþ1  v cmþ1 dx; m

0

 12 @ 2  @u

 u@

v @  @ v

!

; ð1:5Þ

m P 1:

The first non-trivial system of hierarchy (1.4) reads

lt 2 ¼

  u

v

t2

¼

 12 uxx  14 v xx  12 ðuv Þx  2uux 1 2

v xx  2ðuv Þx  32 vv x

! :

ð1:6Þ

By taking use of two kinds of different constraints, the soliton system (1.4) (briefly called Geng hierarchy) is non-linearized into complete integrable Bargmnn and Neumann system, respectively [8]. This paper is devoted to discussing two types of extensions of the hierarchy (1.4). On the one hand, a new system from largeness point of view of both potentials and dimensions is given. On the other hand, a super-integrable system, with the basis of super-Lie algebras Bð0; 1Þ, is established. Then, the generalized trace identity and super-trace identity are used to construct the Hamiltonian and super-Hamiltonian structures for the extended systems, respectively. This letter is organized as follows. In Section 2, a 4  4 matrix spectral problem based on (1.3) is introduced, and, a hierarchy of Lax integrable equations are derived by zero-curvature representation. The Hamiltonian structure of the enlarged system is constructed by using of generalized trace identity through a non-degenerate symmetric bilinear form. In Section 3, a super-Hamiltonian system is deduced on the basis of super-Lie algebras Bð0; 1Þ. With the help of super-trace identity on super-Lie algebras with non-degenerate Killing forms, the super-Hamiltonian structure is presented. Finally, in Section 4, there will be some concluding remarks. 2. An integrable coupling system and Hamiltonian structure of hierarchy (1.4) With the development of the soliton theory, integrable coupling [11] have become a new and very important topic in the study of integrable systems. The concept of integrable couplings and related theories were brought forward in recent years. It not only generalizes the symmetry problem, but also provides information on complete classification of integrable systems. Methods to construct the integrable couplings of the integrable systems of evolution equations have been widely reported (see, e.g., Refs. [12–20] and references therein). By enlarging associated spectral problem, the integrable coupling system of relativistic Toda type lattice is discussed [21]. In Ref. [12–15], by considering semi-direct sum Lie algebras, a technologically-practicable approach to derive integrable couplings is proposed, where the integrable couplings of some classical systems are studied. In order to look for the Hamiltonian structures of the corresponding integrable coupling systems, a generalized trace identity [16], in the case of non-semi-simple Lie algebras, is obtained, which undoes the constraint on the standard trace identity [4].

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

1499

To construct the integrable couplings of the Geng hierarchy (1.4), let us first consider the semi-direct sum of Lie algebras by enlarging the following Lie algebra:

 1; w  2; w  3 g; F ¼ spanfw     1 0 0 1 1 ¼ 2 ¼ w ; w ; 0 1 0 0

2 ¼ w



0 0 1 0

 ;

to matrix Lie algebra of 4  4 matrix

0

1

0

0

C 0 C C C; 0 C A

B B 0 1 0 B w1 ¼ B B0 0 1 @ 0

1

0

0

0 1 1 0 0 1 0 C B B 0 0 0 1 C C B w4 ¼ B C; B0 0 0 0 C A @ 0

0 0 0

0

0 1 0 0

B B0 0 B w2 ¼ B B0 0 @

1

C 0 0C C C; 0 1C A

0 0

0 0 1 0 0 0 1 C B B0 0 0 0C C B w5 ¼ B C; B0 0 0 0C A @ 0

0

0

0 0 0 0

1

C B B1 0 0 0C C B w3 ¼ B C; B0 0 0 0C A @ 0 0 1 0 1 0 0 0 0 C B B0 0 1 0C C B w6 ¼ B C: B0 0 0 0C A @ 0

0 0 0 0

0 0

0 0

F 0 ¼ spanfw1 ; w2 ; w3 g;

F c ¼ spanfw4 ; w5 ; w6 g;

It is easy to verify that

F ¼ spanfw1 ; w2 ; w3 ; w4 ; w5 ; w6 g;

construct three Lie algebras with the communication operation

½wi ; wj  ¼ wi wj  wj wi

ði; j ¼ 1; 2; . . . ; 6Þ;

and

F ¼ F0 ] Fc;

½F; F c  ¼ FF c  F c F # F c :

Let

e F ¼ fAjA 2 R½k  Fg;

e F 0 ¼ fAjA 2 R½k  F 0 g;

e F c ¼ fAjA 2 R½k  F c g;

where R½k  F means the loop algebra defined by spanfkn Ajn P 0; A 2 Fg. Obviously, e F c is an Abelian ideal of the loop Lie algebra e F , and e F 0 and e F c is closed under the multiplication of matrix. Thus, e F forms a semi-direct sum of e F 0 and e F c. In terms of e F , the spectral matrix W is of the form

 W¼

Wc W0

W0 0



2e F;

where W0 ; Wc are 2  2 matrices, 0 stands for a 2  2 zero-matrix. For Geng equation (1.4), we consider an isospectral problem

1 1 u B u C 2C  ¼B u C; B 3 A @u 4 u 0

 x ¼ Uu  ¼ u



U

Uc

0

U



; u

ð2:1Þ

where U is the same as that in (1.3), and

Uc ¼



 r ks : 0 r

The stationary zero-curvature equation

V x  ½U; V ¼ 0;

ð2:2Þ

with

0 V¼



V

Vc

0

V



a

kb

e

B c a g B ¼B @0 0 a 0

0

c

kf

1

e C C e C 2 F; kb A a

1500

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

becomes

ax ¼ v kc þ kb;

bx ¼ 2kb þ 2ub  2v a; ex ¼ v kg þ skc þ kf ;

cx ¼ 2kc  2uc  2a;

fx ¼ 2kf þ 2uf  2v e þ 2rb  2sa;

ð2:3Þ

g x ¼ 2kg  2ug  2rc  2e:

Upon setting

0 V¼

X

V i ki ¼

iP0

1

ai

kbi

ei

kfi

XB B ci B @0 iP0

ai

gi

0

ai

ei C C i Ck ; kbi A

0

ci

ai

0 and choosing the initial data

a0 ¼ 1;

b0 ¼ c0 ¼ 0;

e0 ¼ 1;

f 0 ¼ g 0 ¼ 0;

then, (2.3) is equivalent to the following recursion relations:

1 bmþ1 ¼  bmx þ ubm  v am ; m P 0; 2 1 cmþ1 ¼ cmx þ ucm þ am ; m P 0; 2 1 1 amx ¼ v cmx  bmx þ uv cm þ ubm ; m P 0; 2 2 1 fmþ1 ¼  fmx þ ufm  v em þ rbm  sam ; m P 0; 2 1 g mþ1 ¼ g mx þ ug m þ rcm þ em ; m P 0; 2 1 1 1 emx ¼ v g mx þ scmx  fmx þ rbm þ ðr v þ suÞcm þ ufm þ uv g m ; 2 2 2

ð2:4Þ

m P 0:

Assume that the constants of integration are selected to be zero. Then the recursion relations (2.4) uniquely define a series of  with respect to x. The first few are listed as follows: sets of differential polynomial functions in l

1 1 v ; b1 ¼ v ; c1 ¼ 1; e1 ¼ ðv þ sÞ; f 1 ¼ ðv þ sÞ; g 1 ¼ 1; 2 2 1 1 1 1 1 a2 ¼  v x þ uv þ v 2 ; b2 ¼ v x  uv  v 2 ; c2 ¼ u þ v ; 4 8 2 2 2 1 1 2 3 e2 ¼  ðv x þ sx Þ þ uv þ v þ us þ v r þ v s; 4 4 2 1 1 2 f2 ¼ ðv x þ sx Þ  uv  us  v s  v r  v ; 2 2 1 g 2 ¼ u þ r þ ðv þ sÞ; . . . 2

a1 ¼

Now, we set

0 ðVkm Þþ ¼

ai kmi mi B ci k B @ 0

m B X i¼0

bi kmiþ1 ai kmi

ei kmi g i kmi

0

ai kmi

1 fi kmiþ1 ei kmi C C C: bi kmiþ1 A

0

ci kmi

ai kmi

0 A direct computation gives

ðVkm Þþx  ½U; ðVkm Þþ  ¼





H11 H12 ; 0 H11

here 0 denotes zero-matrix of 2  2, and



H11 ¼

amx

2kbmþ1

2cmþ1

amx

 ;



H12 ¼

emx 2g mþ1

 2kfmþ1 : emx

To generate associated soliton equations through zero-curvature equations, we take a modification



Dm ¼

D11 0

 D12 ; D11

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

1501

in which



D11 ¼

cmþ1

0

0

cmþ1





D12 ¼

;

g mþ1

0

0

g mþ1

 ;

and define

V m ¼ ðVkm Þþ þ Dm : Let the time evolution of the eigenfunction of the spectral problem (2.1) obey the differential equations

 tm ¼ V m u  ; m P 0: u

ð2:5Þ

Then the compatibility condition of (2.1) and (2.5), i.e.,

U tm ¼ ðV m Þx  ½U; V m  gives rise to the following equations:

l tm

0 1 u Bv C B  m ðl Þ ¼ B C ¼K C @rA s

0

amx  cmþ1x

1

C B 2a mx C B ¼B C; @ emx  g mþ1x A

m P 0;

ð2:6Þ

2emx

tm

which is a kind of integrable coupling system of Geng hierarchy. The first non-trivial equation reads

0

1

 12 uxx  14 v xx  12 ðuv Þx  2uux

C 1 v  2ðuv Þx  32 vv x C 2 xx C: @  12 ðu þ v Þxx  14 ðv þ sÞxx  u2x  2ðurÞx  12 ðuv þ us þ v rÞx A B

l t2 ¼ K 2 ðl Þ ¼ B B

1 ð 2

v þ sÞxx  2ðuv Þx  12 v 2x  ð2us þ 2v r þ 3v sÞx

The first two equations are same as those in (1.6), hence, it is a kind of integrable system of Geng equation (1.6). In what follows, we are going to construct the Hamiltonian structure of the system (2.6). In order to do so, we should introduce a non-degenerate symmetric bilinear form. Let us consider the following map [16,17]:

0

X:e F ! R6 ;

A#a ¼ ða1 ; a2 ; a3 ; a4 ; a5 ; a6 ÞT ;

a1 Ba B 3 A¼B @0 0

a5

1

a2

a4

a1

a6

0

a1

a4 C C e C 2 F; a2 A

0

a3

a1

ð2:7Þ

which induces a Lie algebraic structure on R , isomorphic to the matrix loop algebra e F . The commutator ½; R6 on R6 is derived e by the commutator ½; e on F , 6

F

½a; bTR6 ¼ Xð½A; BeT Þ ¼ aT RðbÞ; F

where a; b 2 R ; A; B 2 e F ; RðbÞ is a square matrix defined actually by communication operation in e F [16]. Also, we introduce the matrix 6

0

2 0

0 2 0 0

1

B0 0 1 0 0 1C C B C B B0 1 0 0 1 0C C H¼B B 2 0 0 0 0 0 C: C B C B @0 0 1 0 0 0A 0 1 0

0

0 0

Therefor, we can define a non-degenerate symmetric bilinear form on R6

ha; bi ¼ aT Hb:

ð2:8Þ

Now, a non-degenerate bilinear form on e F may be determined by

hA; Bie ¼ hXðAÞ; XðBÞiR6 ¼ ða1 ; a2 ; a3 ; a4 ; a5 ; a6 ÞHðb1 ; b2 ; b3 ; b4 ; b5 ; b6 ÞT F

¼ 2a1 b1 þ 2a1 b4 þ a2 b3 þ a2 b6 þ a3 b2 þ a3 b5 þ 2a4 b1 þ a5 b3 þ a6 b2 ; which is symmetric and invariant associated with the Lie product, i.e.,

hA; Bie ¼ hB; Aie; hA; ½B; Cie ¼ h½A; B; Cie; F

F

F

F

A; B; C 2 e F:

ð2:9Þ

1502

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

Through a direct computation, by utilizing (2.9), we have

hV; U k ie ¼ 2a þ v c þ sc  2e þ v g; F

hV; U u ie ¼ 2a þ 2e; F

hV; U r ie ¼ 2a;

hV; U v ie ¼ kðc þ gÞ;

ð2:10Þ

F

hV; U s ie ¼ kc:

F

F

The substitution of (2.10) with

a¼ e¼

1 X m¼0 1 X

am km ;

b¼

em km ;

f ¼

m¼0

1 X

bm km ;

m¼0 1 X

fm km ;

c¼ g¼

m¼0

1 X

cm km ;

m¼0 1 X

g m km ;

m¼0

into generalized trace identity [13,14]

d  dl

Z

hV; U k iedx ¼ F

  @ kc kc hV; U l ie; F @k

ð2:11Þ

and equating the coefficients of km1 on both sides of (2.11) yield

0 d  dl

2am þ 2em

1

C B B cmþ1 þ g mþ1 C C: ð2amþ1 þ v cmþ1 þ scmþ1  2emþ1 þ v g mþ1 Þdx ¼ ðc  mÞB C B 2am A @

Z

cmþ1 To fix the constant c, we simply set m ¼ 0 and we have c ¼ 0. Therefor, we have

1 0 2am þ 2em  Z  C Bc d 2amþ1  v cmþ1  scmþ1 þ 2emþ1  v g mþ1 B mþ1 þ g mþ1 C dx ¼ B C:  A @ dl m 2am cmþ1 Then, we could construct the Hamiltonian structure of the system (2.6) by

l tm

0 1 u Bv C C  ÞB ¼ K m ðl B C @rA s

0

amx  cmþ1x

1

B 2a C dHm mx B C ¼B ; C¼J  @ emx  g mþ1x A dl

m P 1;

ð2:12Þ

2emx

tm

in which the Hamiltonian operator J and the Hamiltonian functionals Hm are given by

0

0

0

1 @ 2

@

1

C ! B B 0 0 J 0 @ 0 C C B ; J¼B C¼ B 1 @ @  1 @ @ C J J A @2 2 @ 0 @ 0  Z  2amþ1  v cmþ1  scmþ1 þ 2emþ1  v g mþ1 Hm ¼ dx; m and, J is defined by (1.5). Now, if we set

dHmþ1 dHm ¼U ;   dl dl by utilizing the recursion relation (2.4), we get

0

@ 1 u@  @2 B   B 1 1 B 2 @ u@  @2 U¼B B B 0 @ 0

@ 1 v @ þ v 1 ð@ 1 2

v @ þ v þ @ þ 2uÞ 0 0

@ 1 r@ 1 1 @ r@ 2

@ 1 u@  2@   1 @ 1 u@  @2 2

@ 1 s@ þ s 1 ð@ 1 s@ 2

þ s þ 2rÞ

@ 1 v @ þ v 1 ð@ 1 2

v @ þ v þ @ þ 2uÞ

1 C C C C: C C A

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

1503

Then, the soliton hierarchy (2.6) or Hamiltonian system (2.12) has following bi-Hamiltonian structure [4,6,16,17]

dH

dH

l tm ¼ K m ðl Þ ¼ J m ¼ M m1 ; m > 1;  dl dl where the second compatible operator reads

M ¼ JU ¼



0

M

M

Mc

 ;

in which 0 stands for zero-matrix of 2  2, M is given in (1.5), and

Mc ¼

0

@ðu  rÞ þ 12 @ 2

ðu  rÞ@  12 @ 2

@ðv  sÞ þ ðv  sÞ@

! :

According to the general theory on the Liouville integrability of zero curvature equations [2–7,14], we obtain

fHm ; Hl g :¼

Z

! dHm dHl dx ¼ 0; ;J   dl dl

m; l P 1:

Then,

d Hl ¼ dt m

! Z dHm  tl dx ¼ ;l  dl

Z

! dHm dHl dx ¼ fHm ; Hl g ¼ 0; ;J   dl dl

m; l P 1:

So we have following proposition. Proposition 1. Every equation in hierarchy (2.6) has infinitely many commuting conserved functionals fHm g1 m¼1 , which are in involution in pairs. In addition, the nonlinear soliton equations in hierarchy (2.6) are all integrable in the Liouville sense.

3. Super-Hamiltonian equations associated with hierarchy (1.4) The super-symmetric integrable systems have received much attention in past decades, especially in the explorations of the application to the super-symmetric conformal field theories and string theories[22]. These super-integrable systems are shown to have many common features, for example, the intimate relationship between super-integrability and exact solvability is illustrated by Fordy [23], Liu discussed the soliton solutions by extending Hirota’s method [24] and through constructing Darboux–Bäcklund transformation to sMKdV and sKP equations [25], etc. Furthermore, it is a common belief that they also possess Lax representations and bi-Hamiltonian structures [26,27] that define the dynamical flows on the corresponding Poisson super-manifolds. Once, the standard trace identity is generalized to construct the super-Hamiltonian structures of super-integrable equations, i.e., the super-trace identity by using of which some super-systems are discussed [28–30]. In contrast with the general soliton equations, the super-trace identity associated with commutative super-algebra A defined over R or C with the non-degenerated Killing form. Let G be a matrix loop super-algebra over A with the non-degenerate Killing form [31]. For an operator J ¼ ðJij Þqq from Aq to Aq , the corresponding bracket is defined by

H1 ; H2 g ¼

  Z X q ðiÞ ðjÞ ðH2 Þ dH2 dH1 ð1Þp p p Jij dx; dUj dUi i;j¼1

ð3:1Þ

where H1 ; H2 are two functionals, and pðiÞ ¼ pðUi Þ and pðH2 Þ are the degrees of Ui and H2 (either 0 or 1). An operator J is called a super-Hamiltonian operator if the corresponding bracket (3.1) is super-Lie algebra, i.e., it is super-skew symmetric

fH1 ; H2 g ¼ ð1Þp

ðH1 Þ ðH2 Þ p

fH1 ; H2 g;

and satisfies the super-Jacobi identity

ð1Þp

ðH1 Þ ðH2 Þ p

fH1 ; fH2 ; H3 gg þ cycleðH1 ; H2 ; H3 Þ ¼ 0;

ð3:2Þ

Hi ði ¼ 1; 2; 3Þ are pure in the Z 2 grading. An evolution equation Utm ¼ Km ðUÞ is called a super-Hamiltonian system [28], if there is a super-Hamiltonian operator J and functionals Hm such that

Utm ¼ Km ðUÞ ¼ J

dHm : dU

Consequently, the evolution equation is said to possess super-Hamiltonian structure.

1504

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

This section is devoted to introducing super-integrable equations related to Geng hierarchy. Set Bð0; 1Þ be a super-Lie algebras given by

0

g0 B g ¼ g 0 E0 þ g 1 E1 þ g 2 E2 þ g 3 E3 þ g 4 E4 ¼ @ g 2

g1 g 0

1 g3 C g 4 A;

g4

g 3

0

g i 2 A; 0 6 i 6 4;

where A is a commutative super-algebra over R or C, g 0 ; g 1 ; g 2 are even, and g 3 ; g 4 are odd.[28–31] The commutation and anti commutation relations are defined by

½a; b ¼ ab  ð1ÞpðaÞpðbÞ ba; i.e.,

½E0 ; E1  ¼ 2E1 ;

½E0 ; E2  ¼ 2E2 ;

½E0 ; E4  ¼ E4 ;

½E1 ; E3  ¼ 0;

½E2 ; E4  ¼ 0;

½E1 ; E2  ¼ E0 ;

½E1 ; E4  ¼ E3 ;

½E3 ; E3 þ ¼ 2E1 ;

½E0 ; E3  ¼ E3 ;

½E2 ; E3  ¼ E4 ;

½E3 ; E4 þ ¼ E0 ;

½E4 ; E4 þ ¼ 2E2 :

Then we have super-Lie algebra

e ¼ Bð0; e 1Þ ¼ Bð0; 1Þ  C½k; k1 ; G

or Bð0; 1Þ  R½k; k1 :

e 1Þ as follows: Now, let us consider the spectral problem associated with (1.3) on the basis of Bð0;

0 e /x ¼ U/;

e ¼B U @

k þ u

kv

1

ku

b

ka

ka

1

C b A; 0

0 1 u Bv C B C U ¼ B C; @aA

ð3:3Þ

b

where k is the spectral parameter, u; v are commuting variables, and a; b are anti-commuting variables [28–30]. The above super spectral problem reduces to (1.3) if we set a ¼ b ¼ 0. Solving the stationary zero-curvature equation

0

e x ¼ ½ U; e V e ; V

A

e ¼B V @C

kq

kB

1

rC A

A

r kq

ð3:4Þ

0

yields

Ax ¼ kðv c þ ar þ B  qbÞ; Bx ¼ 2kB þ 2uB  2v A  2kaq; C x ¼ 2kC  2uC  2A þ 2br;

ð3:5Þ

qx ¼ kq þ uq þ v r  aA  bB; rx ¼ kr  ur þ bA  kq  kaC: The substitution of the selection

A¼

1 X i¼0

Ai ki ;

B¼

1 X

Bi ki ;

C¼

i¼0

1 X i¼0

C i ki ;

q¼

1 X

qi ki ; r ¼

1 X

i¼0

ri ki

i¼0

into (3.5) leads to the initial relation

A0 ¼ 1;

B0 ¼ C 0 ¼ q0 ¼ r0 ¼ 0

and the recursion relations ðm P 0Þ,

1 Bmþ1 ¼  bmx þ uBm  v Am  aqmþ1 ; 2 1 C mþ1 ¼ C mx þ uC m þ Am  brm ; 2 qmþ1 ¼ qmx þ uqm þ v rm  aAm  bBm ; rmþ1 ¼ rmx þ urm  bAm þ qmþ1 þ aC mþ1 ; 1 1 Amx ¼ v C mx  Bmx þ armx  bqmx þ uv C m þ uBm þ uarm þ ubqm : 2 2

ð3:6Þ

Under the initial value conditions of selecting zero constants for the integration operation, the recursion relations (3.6) uniquely determine a series of sets of differential polynomial functions in U with respect to x. The first few quantities are as follows:

1505

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

A1 ¼

v 2

B1 ¼ v ;

 ab;

C 1 ¼ 1;

q1 ¼ a; r1 ¼ b;

  1 3 1 3 v x  abx þ ax b þ aax þ uv þ v 2 v þ 2u ab; 4 2 8 2 1 1 2 1 B2 ¼ v x  aax  v  uv þ v ab; C 2 ¼ u þ v  ab; 2 2 2 1 1 q2 ¼ ax  ua  v a; r2 ¼ bx þ ax  ub  v b; . . . 2 2 A2 ¼ 

Now we set

0 Ai kmi m X B m e ð V k Þþ ¼ @ C i kmi i¼0

ri k

Bi kmiþ1 Ai kmi

mi

1

qi kmiþ1 C ri kmi A:

qi kmiþ1

ð3:7Þ

0

It is not difficult to find

0 ek Þ ðV þx m

e ðV ek Þ  ¼ B  ½ U; @ þ

Amx

2kðBmþ1 þ aqmþ1 Þ

2C mþ1

Amx

rmþ1  aC mþ1  qmþ1

kqmþ1

m

kqmþ1

1

rmþ1  aC mþ1  qmþ1 C A: 0

We take a modification

0 em ¼ B D @

C mþ1

0

0

C mþ1

0

0

0

1

C 0 A; 0

and define

e km Þ þ D e m ¼ ðV e m; V þ

m P 0;

and, introduce the following auxiliary spectral problem:

e m /; /tm ¼ V

m P 0:

ð3:8Þ

The compatibility conditions of (3.3) and (3.8), i.e., the zero-curvature equations

et ¼ V e m  ½ U; e V e m U m x give rise to the following hierarchy of super-integrable equations:

Ut m

0 1 u Bv C B C ¼ Km ðUÞ ¼ B C @aA b

0

Amx  C mþ1x B 2ðBmþ1 þ v C mþ1 þ aq Þ B mþ1 ¼B @ aC mþ1  qmþ1 tm

1 C C C; A

m P 0:

ð3:9Þ

rmþ1  aC mþ1  qmþ1 þ bC mþ1

When m ¼ 2, the resulting system leads to the first non-trivial system

8 ut2 ¼  12 uxx  14 v xx  2uux  12 ðuv Þx þ 12 ðabÞxx þ ðuabÞx  ½ðax  bx Þbx ; > > > < v t2 ¼ 12 v xx  2ðuv Þx  32 vv x þ ðv abÞx  aaxx þ 2uðaax þ bbx þ ax aÞ; 1 3 3 1 1 3 1 > > at2 ¼ axx  2 ðuaÞx  2 ux a  2 ðv aÞx þ 2 ðv bÞx þ 2 v bx þ 4 v x a þ 2 ax b þ abð2ax  bx Þ; > : bt2 ¼ axx  bxx  ðubÞx  12 ðv bÞx þ 12 ux b þ 14 v x b  12 bðabÞx þ uða  bÞx ; and, obviously reduces to Eq. (1.6) when a ¼ b ¼ 0. In what follows, we would like to construct the super-Hamiltonian structure for the superintegrable system (3.9). To this end, we apply the super-trace identity [28,29]

d dU

Z

  @ strðadV~ ad@ U=@k Þdx ¼ ke ke strðadV~ ad@ U=@U Þ: ~ ~ @k

The substitution of the selection [28]

~ U ~ k Þ ¼ 3ð2A þ v c þ 2arÞ; strðadV~ ad@ U=@k Þ ¼ 3strðV; ~ Þ ¼ 6A; strðadV~ ad@ U=@ strðadV~ ad@ U=@u ~ ~ v Þ ¼ 3kC; strðadV~ ad@ U=@ Þ ¼ 3ð2kqÞ; ~ a Þ ¼ 3ð2krÞ; strðadV ~ ~ ad@ U=@b

ð3:10Þ

1506

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

into the super-trace identity (3.10) gives rise to

d dU

Z

1 0 2A  B C @ B kC C ð2A þ v C þ 2arÞdx ¼ ke ke B C: @ 2kr A @k 2kq m1

Comparing the coefficients of k

yields

0 d dU

Z

2Am

1

B C mþ1 C B C ð2Amþ1 þ v C mþ1 þ 2armþ1 Þdx ¼ ðe  mÞB C: @ 2rmþ1 A 2qmþ1

It is easy to verify that

e ¼ 0 if we simply set m ¼ 0. That is

1 2Am C B 2Amþ1  v C mþ1  2armþ1 B C mþ1 C dx ¼ B C; @ 2rmþ1 A m 0

d dU

Z

m P 1:

2qmþ1 Then, the super system (3.9) can be rewritten in its super-Hamiltonian form as follows:

Utm ¼ Km ðUÞ ¼ J

dHm ; dU

m P 1;

ð3:11Þ

in which the super-Hamiltonian operator J and the super-Hamiltonian functionals Hm are given by

01 2

@

B @ B J¼B @ 0

0

@

0

1

a b  a C C C; 0  12 A

0 a

b  a  12  12 Z 2Amþ1  v C mþ1  2armþ1 Hm ¼ dx; m

ð3:12Þ

0

m P 1:

If we set

dHmþ1 dHm ¼W ; dU dU from recursion relation (3.6), we have the hereditary recursion operator

0

W11 BW B 21 W¼B @ W31 W41

W12 W22 W32 W42

W13 W23 W33 W43

1

W14 W24 C C C; W34 A W44

with

@ 2

W11 ¼ @ 1 u@  ; W12 ¼ @ 1 v @ þ v ;

a

b W14 ¼ @ 1 b@ þ ; 2 2   1 1 @ 1 @ u@  ; W22 ¼ ð@ 1 v @ þ v þ @ þ 2uÞ; ¼ 2 2 2   1  1 a b 1 b ; @ 1 b@ þ ¼  @ a@ þ þ ; W24 ¼ 2 2 2 2 2   @ ; W32 ¼ bð@ 1 v @  v Þ  a@  2ua; ¼ b @ 1 u@ þ 2 3 ¼ b@ 1 a@  ab þ @ þ u þ v ; W34 ¼ b@ 1 b@  u þ @; 2  @ 1  @ u@  b@; W42 ¼ að@ 1 v @ þ v Þ þ 2bv ; ¼a 2 1 ¼ a@ 1 a@  v þ ab; W44 ¼ a@ 1 b@  ab  @ þ u: 2

W13 ¼ @ 1 a@  ; W21 W23 W31 W33 W41 W43

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

1507

Then, the super-Geng soliton hierarchy (3.9) has the following super bi-Hamiltonian structure:

Utm ¼ Km ðUÞ ¼ J

dHm dHm1 ¼M ; dU dU

m P 1;

where the second compatible operator reads [24]

0

0

B B u@ þ @ 2 B 2 M ¼ JW ¼ B b B @ @ 2 0

2

 12 @b

@u  @2

ðv @ þ @ v Þ  a2 @  ua  v b ub þ 2b @

@ 2

0

1 C

a  ua  v b ub  2@ b C C C: v  ab 1 ð@  uÞ C A 2 2  12 ð@ þ uÞ

0

Proposition 2. The super-Hamiltonian functions fHm g1 m¼1 defined by (3.12) forms an infinite set of commuting conserved quantities of the hierarchy (3.9), and fHm g1 m¼1 are in involution in pairs with respect to the super-Poisson bracket (3.1). The hierarchy (3.9) possesses infinitely many commuting symmetries fKm g1 m¼1 . So, the system (3.9) is a bi-Hamiltonian superintegrable system.

4. Concluding remarks By considering an isospectral problem, a hierarchy of soliton equations (briefly called Geng equation) are derived. Two hierarchies of integrable and superintegrable systems, as Geng’s integrable extensions, are deduced. From largeness point of view of potentials and dimensions, the Geng hierarchy is extended to associate with a 4  4 matrix problem and involve four potentials. By virtue of semi-direct sum Lie algebras, the integrable larger system of Geng hierarchy is established. On the second part, a super extension of Geng hierarchy is derived with super-Lie algebras. By using of generalized trace identity and super-trace identity, the Hamiltonian and super-Hamiltonian structures are constructed for the extended systems. We should mention that, the first extension is actually the integrable coupling system of Geng equation. So, there will be an open problem what is the coupling and super coupling system for the hierarchy (3.9). In addition, what is the scheme for the discrete systems by the present technique? How to establish the Darboux and Bäcklund transformation for the superintegrable system (3.9) [32]? These problems will be investigated in another occasions. Acknowledgement We are grateful to the referee for helpful remarks. The first author (H.X. Yang) is very grateful to Prof. W.X. Ma for many helpful discussions and great encouragement. This work was in part supported by the Nature Science Foundation of Shandong Province (Grant No. Q2006A04) and the Domestic Visiting Scholar Project for Young-Outstanding Teachers in College of Shandong Province. X.X. Xu thanks also the Science and Technology Plan Projects of the Educational Department of Shandong Province (Grant No. J08LI08). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolation Equations and Inverse Scattering, Cambridge University Press, England, 1992. C. Gu, Soliton Theory and its Application, Springer, Berlin, 1995. D. Chen, Soliton Introduction, Science Press, Beijing, 2006. G. Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys. 30 (1989) 330–338. C. Cao, X. Geng, Nonconforcal generator of involutive systems and three associated soliton hierarchies, J. Math. Phys. 32 (1991) 2323–2328. W. Ma, A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. Ann. Math. 13 (A) (1992) 115–123 (in Chinese). X. Hu, Three kinds of nonlinear evolution equations and their Hamiltonian structure, Northeast. Math. J. 6 (1990) 187–194. X. Geng, A bargmann system and a Neumann system, Acta Math. Sci. 13 (1993) 80–84. X. Xu, A generalized Wadati–Konno–Ichikawa hierarchy and new finite-dimensional integrable systems, Phys. Lett. A 301 (2002) 250–262. Y. Sun, D. Chen, A Liouville integrable hierarchy, symmetry constraint, new finite-dimensional integrable systems, involutive solution and expanding integrable models, Chaos, Solitons Fractals 29 (2006) 978–987. W. Ma, B. Fuchssteiner, Integrable theory of the perturbation equations, Chaos, Soliton Fractals 7 (1996) 1227–1250. W. Ma, Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. Lett. A 316 (2003) 72–76. W. Ma, Integrable couplings of vector AKNS soliton equations, J. Math. Phys. 46 (2005) 033507–033519. W. Ma, X. Xu, Y. Zhang, Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. Lett. A 351 (2006) 125–130. W. Ma, X. Xu, Y. Zhang, Semi-direct sums of Lie algebras and discrete integrable couplings, J. Math. Phys. 47 (2006) 053501–053516. W. Ma, M. Chen, Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A: Math. Gen. 39 (2006) 10787–10801. X. Xu, H. Yang, W. Cao, A 4  4 Matrix eigenvalue problem, related hierarchy of Lax integrable equations and their Hamiltonian structures, Int. J. Mod. Phys. Lett. B 22 (2008) 4027–4040. L. Ma, X. Xu, A hierarchy of integrable nonlinear lattice equations and its two integrable coupling systems, Appl. Math. Comput. 203 (2008) 7–13. Y. Zhang, E. Fan, An approach for generating enlarging integrable systems, Phys. Lett. A 365 (2007) 89–96. Y. Zhang, H. Zhang, A direct method for integrable couplings of TD hierarchy, J. Math. Phys. 1 (2002) 466–472.

1508

H.-X. Yang et al. / Applied Mathematics and Computation 217 (2010) 1497–1508

[21] H. Yang, X. Xu, Y. Sun, H. Ding, Integrable relativistic Toda type lattice hierarchies associated coupling systems and the Darboux transformation, J Phys. A: Math. Gen. 39 (2006) 3933–3947. [22] S. Stanciu, Supersymmetric integrable hierarchies and string theory, . [23] A.P. Fordy, Quantum super-integrable systems as exactly solvable models, symmetry, integrability and geometry: methods and applications, SIGMA 3 (2007) 025–034. [24] Q. Liu, X. Hu, M. Zhang, Supersymmetric modified Korteweg-de Vries equation: bilinear approach, Nonlinearity 18 (2005) 1597–1603. [25] Q. Liu, M. Manas, Darboux transformation for supersymmetric KP hierarchies, Phys. Lett. B 485 (2000) 293–300. [26] Y. Li, L. Zhang, Hamiltonian structure of the super evolution equation, J. Math. Phys. 31 (1990) 470–475. [27] Q. Liu, Bi-Hamiltonian structure of a generalized super Korteweg-de Vries equation, J. Phys. A: Math. Gen. 26 (1993) L1239–L1242. [28] W. Ma, A supertrace identity and ite applications to super-integrable systems, J. Math. Phys. 49 (2008) 033511–033513. [29] X. Hu, An approach to generate superextensions of integrable systems, J. Phys. A: Math. Gen. 30 (1997) 619–632. [30] X. Hu, A powerful approach to generate new integrable systems, J. Phys. A: Math. Gen. 27 (1994) 2497–2514. [31] H. Sun, Q. Han, Lie Algebras and Lie Superalgebras and Their Applications in Physics, Peking University Press, Beijing, 1999. [32] A.R. Chowdhury, S. Roy, On the Bäcklund transformation and Hamiltonian properties of super evaluation equations, J. Math. Phys. 27 (1986) 2464– 2468.