research article self-consistent sources and conservation...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 913758, 6 pageshttp://dx.doi.org/10.1155/2013/913758

Research ArticleSelf-Consistent Sources and Conservation Laws fora Super Broer-Kaup-Kupershmidt Equation Hierarchy

Hanyu Wei1,2 and Tiecheng Xia2

1 Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou 466001, China2Department of Mathematics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Hanyu Wei; [email protected]

Received 14 February 2013; Accepted 2 June 2013

Academic Editor: Yongkun Li

Copyright © 2013 H. Wei and T. Xia. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the matrix Lie superalgebras and supertrace identity, the integrable super Broer-Kaup-Kupershmidt hierarchy withself-consistent sources is established. Furthermore, we establish the infinitely many conservation laws for the integrable superBroer-Kaup-Kupershmidt hierarchy. In the process of computation especially, Fermi variables also play an important role in superintegrable systems.

1. Introduction

Soliton theory has achieved great success during the lastdecades; it is being applied to mathematics, physics, biology,astrophysics, and other potential fields [1–12]. The diversityand complexity of soliton theory enable investigators to doresearch from different views, such as Hamiltonian structure,self-consistent sources, conservation laws, and various solu-tions of soliton equations.

In recent years, with the development of integrable sys-tems, super integrable systems have attractedmuch attention.Many scholars and experts do research on the topic andget lots of results. For example, in [13], Ma et al. gavethe supertrace identity based on Lie super algebras andits application to super AKNS hierarchy and super Dirachierarchy, and to get their super Hamiltonian structures, Hugave an approach to generate superextensions of integrablesystems [14]. Afterwards, super Boussinesq hierarchy [15]and super NLS-mKdV hierarchy [16] as well as their superHamiltonian structures are presented. The binary nonlin-earization of the super classical Boussinesq hierarchy [17], theBargmann symmetry constraint, and binary nonlinearizationof the super Dirac systems were given [18].

Soliton equation with self-consistent sources is an impor-tant part in soliton theory. They are usually used to describeinteractions between different solitary waves, and they are

also relevant to some problems related to hydrodynamics,solid state physics, plasma physics, and so forth. Some resultshave been obtained by some authors [19–21]. Very recently,self-consistent sources for super CKdV equation hierarchy[22] and super G-J hierarchy are presented [23].

The conservation laws play an important role in dis-cussing the integrability for soliton hierarchy. An infinitenumber of conservation laws for KdV equation were firstdiscovered by Miura et al. in 1968 [24], and then lots ofmethods have been developed to find them. This may bemainly due to the contributions of Wadati and others [25–27]. Conservation laws also play an important role in math-ematics and engineering as well. Many papers dealing withsymmetries and conservation laws were presented.The directconstruction method of multipliers for the conservation lawswas presented [28].

In this paper, starting from a Lie super algebra, isospectralproblems are designed. With the help of variational identity,Yang got super Broer-Kaup-Kupershmidt hierarchy and itsHamiltonian structure [29].Then, based on the theory of self-consistent sources, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy are obtained by us. Further-more, we present the conservation laws for the super Broer-Kaup-Kupershmidt hierarchy. In the calculation process,extended Fermi quantities 𝑢

1and 𝑢

2play an important role;

namely, 𝑢1and 𝑢

2satisfy 𝑢2

1= 𝑢2

2= 0 and 𝑢

1𝑢2

= −𝑢2𝑢1

2 Journal of Applied Mathematics

in the whole paper. Furthermore, the operation betweenextended Fermi variables satisfies Grassmann algebra condi-tions.

2. A Super Soliton Hierarchy withSelf-Consistent Sources

Based on a Lie superalgebra 𝐺,

𝑒1= (

1 0 0

0 −1 0

0 0 0

) , 𝑒2= (

0 1 0

0 0 0

0 0 0

) ,

𝑒3= (

0 0 0

1 0 0

0 0 0

) , 𝑒4= (

0 0 1

0 0 0

0 −1 0

) , 𝑒5= (

0 0 0

0 0 1

1 0 0

)

(1)

that is along with the communicative operation [𝑒1, 𝑒2] =

2𝑒2, [𝑒1, 𝑒3] = −2𝑒

3, [𝑒2, 𝑒3] = 𝑒

1, [𝑒1, 𝑒4] = [𝑒

2, 𝑒5] =

𝑒4, [𝑒1, 𝑒5] = [𝑒

4, 𝑒3] = −𝑒

5, [𝑒4, 𝑒5]+= 𝑒1, [𝑒4, 𝑒4]+= −2𝑒

2,

and [𝑒5, 𝑒5]+= 2𝑒3.

We consider an auxiliary linear problem

(

𝜑1

𝜑2

𝜑3

)

𝑥

= 𝑈 (𝑢, 𝜆)(

𝜑1

𝜑2

𝜑3

) , 𝑈 (𝑢, 𝜆) = 𝑅1+

5

∑

𝑖=1

𝑢𝑖𝑒𝑖(𝜆) ,

(

𝜑1

𝜑2

𝜑3

)

𝑡𝑛

= 𝑉𝑛(𝑢, 𝜆)(

𝜑1

𝜑2

𝜑3

) ,

(2)

where 𝑢 = (𝑢1, . . . , 𝑢

𝑠)𝑇, 𝑈𝑛= 𝑅1+𝑢1𝑒1+⋅ ⋅ ⋅+𝑢

5𝑒5, 𝑢𝑖(𝑛, 𝑡) =

𝑢𝑖(𝑖 = 1, 2, . . . , 5), 𝜑

𝑖= 𝜑(𝑥, 𝑡) are field variables defining

𝑥 ∈ 𝑅, 𝑡 ∈ 𝑅; 𝑒𝑖= 𝑒𝑖(𝜆) ∈ 𝑠𝑙(3) and 𝑅

1is a pseudoregular

element.The compatibility of (2) gives rise to the well-known zero

curvature equation as follows:

𝑈𝑛𝑡

− 𝑉𝑛𝑥

+ [𝑈𝑛, 𝑉𝑛] = 0, 𝑛 = 1, 2, . . . . (3)

If an equation

𝑢𝑡= 𝐾 (𝑢) (4)

can be worked out through (3), we call (4) a super evolutionequation. If there is a super Hamiltonian operator 𝐽 and afunction𝐻

𝑛such that

𝑢𝑡= 𝐾 (𝑢) = 𝐽

𝛿𝐻𝑛+1

𝛿𝑢

, (5)

where

𝛿𝐻𝑛

𝛿𝑢

= 𝐿

𝛿𝐻𝑛−1

𝛿𝑢

= ⋅ ⋅ ⋅ = 𝐿𝑛 𝛿𝐻0

𝛿𝑢

,

𝑛 = 1, 2, . . . ,

𝛿

𝛿𝑢

= (

𝛿

𝛿𝑢1

, . . . ,

𝛿

𝛿𝑢5

)

𝑇

,

(6)

then (4) possesses a superHamiltonian equation. If so, we cansay that (4) has a super Hamiltonian structure.

According to (2), now we consider a new auxiliary linearproblem. For𝑁 distinct 𝜆

𝑗, 𝑗 = 1, 2, . . . , 𝑁, the systems of (2)

become as follows:

(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

)

𝑥

= 𝑈 (𝑢, 𝜆𝑗)(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

) =

5

∑

𝑖=1

𝑢𝑖𝑒𝑖(𝜆)(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

) ,

(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

)

𝑡𝑛

= 𝑉𝑛(𝑢, 𝜆𝑗)(

𝜑1

𝜑2

𝜑3

)

= [

𝑛

∑

𝑚=0

𝑉𝑚(𝑢) 𝜆𝑛−𝑚

𝑗+ Δ𝑛(𝑢, 𝜆𝑗)](

𝜑1

𝜑2

𝜑3

) .

(7)

Based on the result in [30], we can show that the followingequation:

𝛿𝐻𝑘

𝛿𝑢

+

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

= 0 (8)

holds true, where 𝛼𝑗are constants. Equation (8) determines a

finite dimensional invariant set for the flows in (6).From (7), we may know that

𝛿𝜆𝑗

𝛿𝑢𝑖

=

1

3

𝑆 tr(Ψ𝑗

𝜕𝑈 (𝑢, 𝜆𝑗)

𝜕𝑢𝑖

)

=

1

3

𝑆 tr (Ψ𝑗𝑒𝑖𝜆𝑗) , 𝑖 = 1, 2, . . . 5,

(9)

where 𝑆 tr denotes the trace of a matrix and

Ψ𝑗= (

𝜓1𝑗𝜓2𝑗

−𝜓2

1𝑗𝜓1𝑗𝜓3𝑗

𝜓2

2𝑗−𝜓1𝑗𝜓2𝑗

𝜓2𝑗𝜓3𝑗

𝜓2𝑗𝜓3𝑗

−𝜓1𝑗𝜓3𝑗

0

) . (10)

From (8) and (9), a kind of super Hamiltonian solitonequation hierarchy with self-consistent sources is presentedas follows:

𝑢𝑛𝑡

= 𝐽

𝛿𝐻𝑛+1

𝛿𝑢𝑖

+ 𝐽

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

= 𝐽𝐿𝑛 𝛿𝐻1

𝛿𝑢𝑖

+ 𝐽

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

, 𝑛 = 1, 2, . . . .

(11)

3. The Super Broer-Kaup-KupershmidtHierarchy with Self-Consistent Sources

The super Broer-Kaup-Kupershmidt spectral problem asso-ciated with the Lie super algebra is given in [29]:

𝜑𝑥= 𝑈𝜑, 𝜑

𝑡= 𝑉𝜑, (12)

Journal of Applied Mathematics 3

where

𝑈 = (

𝜆 + 𝑟 𝑠 𝑢1

1 −𝜆 − 𝑟 𝑢2

𝑢2

−𝑢1

0

) , 𝑉 = (

𝐴 𝐵 𝜌

𝐶 −𝐴 𝜎

𝜎 −𝜌 0

) , (13)

and 𝐴 = ∑𝑚≥0

𝐴𝑚𝜆−𝑚

, 𝐵 = ∑𝑚≥0

𝐵𝑚𝜆−𝑚

, 𝐶 =

∑𝑚≥0

𝐶𝑚𝜆−𝑚

, 𝜌 = ∑𝑚≥0

𝜌𝑚𝜆−𝑚, and 𝜎 = ∑

𝑚≥0𝜎𝑚𝜆−𝑚. As

𝑢1and 𝑢

2are Fermi variables, they constitute Grassmann

algebra. So, we have 𝑢1𝑢2= −𝑢2𝑢1, 𝑢2

1= 𝑢2

2= 0.

Starting from the stationary zero curvature equation𝑉𝑥= [𝑈,𝑉] , (14)

we have𝐴𝑚𝑥

= 𝑠𝐶𝑚+ 𝑢1𝜎𝑚− 𝐵𝑚+ 𝑢2𝜌𝑚,

𝐵𝑚𝑥

= 2𝐵𝑚+1

+ 2𝑟𝐵𝑚− 2𝑠𝐴

𝑚− 2𝑢1𝜌𝑚,

𝐶𝑚𝑥

= −2𝐶𝑚+1

− 2𝑟𝐶𝑚+ 2𝐴𝑚+ 2𝑢2𝜎𝑚,

𝜌𝑚𝑥

= 𝜌𝑚+1

+ 𝑟𝜌𝑚+ 𝑠𝜎𝑚− 𝑢1𝐴𝑚− 𝑢2𝐵𝑚,

𝜎𝑚𝑥

= −𝜎𝑚+1

− 𝑟𝜎𝑚+ 𝜌𝑚− 𝑢1𝐶𝑚+ 𝑢2𝐴𝑚,

𝐵0= 𝐶0= 𝜌0= 𝜎0= 0,

𝐴0= 1, 𝐵

1= 𝑠, 𝐶

1= 𝑟,

𝜌1= 𝑢1, 𝜎

1= 𝑢2, 𝐴

1= 0, . . . .

(15)

Then we consider the auxiliary spectral problem

𝜑𝑡𝑛

= 𝑉(𝑛)

𝜑 = (𝜆𝑛𝑉)+𝜑, (16)

where

𝑉(𝑛)

=

𝑛

∑

𝑚=0

(

𝐴𝑚

𝐵𝑚

𝜌𝑚

𝐶𝑚

−𝐴𝑚

𝜎𝑚

𝜎𝑚

−𝜌𝑚

0

)𝜆𝑛−𝑚

, (17)

considering

𝑉(𝑛)

= 𝑉(𝑛)

++ Δ𝑛, Δ𝑛= −𝐶𝑚+1

𝑒1. (18)

Substituting (18) into the zero curvature equation

𝑈𝑡𝑛

− 𝑉(𝑛)

𝑥+ [𝑈,𝑉

(𝑛)] = 0, (19)

we get the super Broer-Kaup-Kupershmidt hierarchy

𝑢𝑡𝑛

= (

𝑟

𝑠

𝑢1

𝑢2

)

𝑡

=(

(

(

0 𝜕 0 0

𝜕 0 𝑢1

−𝑢2

0 𝑢1

0 −

1

2

0 −𝑢2

−

1

2

0

)

)

)

(

−2𝐴𝑛+1

−𝐶𝑛+1

2𝜎𝑛+1

−2𝜌𝑛+1

)

= 𝐽(

−2𝐴𝑛+1

−𝐶𝑛+1

2𝜎𝑛+1

−2𝜌𝑛+1

) = 𝐽𝑃𝑛+1

,

(20)

where

𝑃𝑛+1= 𝐿𝑃𝑛,

𝐿=((

(

1

2𝜕 − 𝜕−1

𝑟𝜕 −𝑠 − 𝜕−1

𝑠𝜕 𝜕−1

𝑢1𝜕 +1

2𝑢1𝜕−1

𝑢2𝜕 −1

2𝑢2

1

2−1

2𝜕 − 𝑟

1

2𝑢2

0

−𝑢2

2𝑢1

−𝑟 − 𝜕 −1

𝑢1− 𝑢2𝜕 2𝑠𝑢

2𝑠 +1

2𝑢1𝑢2

𝜕 − 𝑟

))

)

.

(21)

According to super trace identity on Lie super algebras, adirect calculation reads as

𝛿𝐻𝑛

𝛿𝑢

= (−2𝐴𝑛+1

, −𝐶𝑛+1

, 2𝜎𝑛+1

, −2𝜌𝑛+1

)𝑇

,

𝐻𝑛= ∫

2𝐴𝑛+2

𝑛 + 1

𝑑𝑥, 𝑛 ≥ 0.

(22)

When we take 𝑛 = 2, the hierarchy (20) can be reduced tosuper nonlinear integrable couplings equations

𝑟𝑡2

= −

1

2

𝑟𝑥𝑥

+

1

2

𝑠𝑥− 2𝑟𝑟𝑥+ (𝑢1𝑢2)𝑥+ (𝑢1𝑢2𝑥)𝑥,

𝑠𝑡2

=

1

2

𝑠𝑥𝑥

− 2(𝑟𝑠)𝑥+ 2𝑢1𝑢1𝑥

+ 2𝑠𝑢2𝑢2𝑥,

𝑢1𝑡2

= 𝑢1𝑥𝑥

−

3

2

𝑟𝑥𝑢1+

1

2

𝑠𝑥𝑢2− 2𝑟𝑢

1𝑥

+ (𝑠 + 𝑢1𝑢2) 𝑢2𝑥,

𝑢2𝑡2

= −𝑢2𝑥𝑥

−

1

2

𝑟𝑥𝑢2− 2𝑟𝑢

2𝑥− 𝑢1𝑥.

(23)

Next, we will construct the super Broer-Kaup-Kuper-shmidt hierarchy with self-consistent sources. Consider thelinear system

(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

)

𝑥

= 𝑈(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

) ,

(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

)

𝑡

= 𝑉(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

) .

(24)

From (8), for the system (12), we set

𝛿𝐻𝑛

𝛿𝑢

=

𝑁

∑

𝑗=1

𝛿𝜆𝑗

𝛿𝑢

(25)


and obtain the following 𝛿𝜆𝑗/𝛿𝑢:

𝑁

∑

𝑗=1

𝛿𝜆𝑗

𝛿𝑢

=

𝑁

∑

𝑗=1

(

(

(

(

(

(

(

(

(

(

𝑆 tr(Ψ𝑗

𝛿𝑈

𝛿𝑞

)

𝑆 tr(Ψ𝑗

𝛿𝑈

𝛿𝑟

)

𝑆 tr(Ψ𝑗

𝛿𝑈

𝛿𝛼

)

𝑆 tr(Ψ𝑗

𝛿𝑈

𝛿𝛽

)

)

)

)

)

)

)

)

)

)

)

=

(

(

(

(

2⟨Φ1, Φ2⟩

⟨Φ2, Φ2⟩

−2 ⟨Φ2, Φ3⟩

2 ⟨Φ1, Φ3⟩

)

)

)

)

,

(26)

where Φ𝑖= (𝜑𝑖1, . . . , 𝜑

𝑖𝑁)𝑇, 𝑖 = 1, 2, 3.

According to (11), the integrable super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is pro-posed as follows:

𝑢𝑡𝑛

= (

𝑟

𝑠

𝑢1𝑡

𝑢2𝑡

)

𝑡𝑛

= 𝐽(

−2𝐴𝑛+1

−𝐶𝑛+1

2𝜎𝑛+1

−2𝜌𝑛+1

) + 𝐽

(

(

(

(

2⟨Φ1, Φ2⟩

⟨Φ2, Φ2⟩

−2 ⟨Φ2, Φ3⟩

2 ⟨Φ1, Φ3⟩

)

)

)

)

,

(27)


𝑖𝑁)𝑇, 𝑖 = 1, 2, 3, satisfy

𝜑1𝑗𝑥

= (𝜆 + 𝑟) 𝜑1𝑗

+ 𝑠𝜑2𝑗

+ 𝑢1𝜑3𝑗,

𝜑2𝑗𝑥

= 𝜑1𝑗

− (𝜆 + 𝑟) 𝜑2𝑗

+ 𝑢2𝜑3𝑗,

𝜑3𝑗𝑥

= 𝑢2𝜑1𝑗

− 𝑢1𝜑2𝑗,

𝑗 = 1, . . . , 𝑁.

(28)

For 𝑛 = 2, we obtain the super Broer-Kaup-Kupershmidtequation with self-consistent sources as follows:

𝑟𝑡2

= −

1

2

𝑟𝑥𝑥

+

1

2

𝑠𝑥− 2𝑟𝑟𝑥+ (𝑢1𝑢2)𝑥+ (𝑢1𝑢2𝑥)𝑥+ 𝜕

𝑁

∑

𝑗=1

𝜑2

2𝑗,

𝑠𝑡2

=

1

2

𝑠𝑥𝑥

− 2(𝑟𝑠)𝑥+ 2𝑢1𝑢1𝑥

+ 2𝑠𝑢2𝑢2𝑥

+ 2𝜕

𝑁

∑

𝑗=1

𝜑1𝑗𝜑2𝑗

− 2𝑢1

𝑁

∑

𝑗=1

𝜑2𝑗𝜑3𝑗

− 2𝑢2

𝑁

∑

𝑗=1

𝜑1𝑗𝜑3𝑗,

𝑢1𝑡2

= 𝑢1𝑥𝑥

−

3

2

𝑟𝑥𝑢1+

1

2

𝑠𝑥𝑢2− 2𝑟𝑢

1𝑥

+ (𝑠 + 𝑢1𝑢2) 𝑢2𝑥

+ 𝑢1

𝑁

∑

𝑗=1

𝜑2

2𝑗−

𝑁

∑

𝑗=1

𝜑1𝑗𝜑3𝑗,

𝑢2𝑡2

= −𝑢2𝑥𝑥

−

1

2

𝑟𝑥𝑢2− 2𝑟𝑢

2𝑥− 𝑢1𝑥

− 𝑢2

𝑁

∑

𝑗=1

𝜑2

2𝑗+

𝑁

∑

𝑗=1

𝜑2𝑗𝜑3𝑗,

(29)


𝑖𝑁)𝑇, 𝑖 = 1, 2, 3, satisfy

𝜑1𝑗𝑥

= (𝜆 + 𝑟) 𝜑1𝑗

+ 𝑠𝜑2𝑗

+ 𝑢1𝜑3𝑗,

𝜑2𝑗𝑥

= 𝜑1𝑗

− (𝜆 + 𝑟) 𝜑2𝑗

+ 𝑢2𝜑3𝑗,

𝜑3𝑗𝑥

= 𝑢2𝜑1𝑗

− 𝑢1𝜑2𝑗,

𝑗 = 1, . . . , 𝑁.

(30)

4. Conservation Laws for the SuperBroer-Kaup-Kupershmidt Hierarchy

In the following, we will construct conservation laws of thesuper Broer-Kaup-Kupershmidt hierarchy. We introduce thevariables

𝐸 =

𝜑2

𝜑1

, 𝐾 =

𝜑3

𝜑1

. (31)

From (7) and (12), we have

𝐸𝑥= 1 − 2𝜆𝐸 − 2𝑟𝐸 + 𝑢

2𝐾 − 𝑠𝐸

2− 𝑢1𝐸𝐾,

𝐾𝑥= 𝑢2− 𝜆𝐾 − 𝑢

1𝐸 − 𝑟𝐾 − 𝑠𝐾𝐸 − 𝑢

1𝐾2.

(32)

Expand 𝐸, 𝐾 in the power of 𝜆 as follows:

𝐸 =

∞

∑

𝑗=1

𝑒𝑗𝜆−𝑗, 𝐾 =

∞

∑

𝑗=1

𝑘𝑗𝜆−𝑗. (33)

Substituting (33) into (32) and comparing the coefficients ofthe same power of 𝜆, we obtain

𝑒1=

1

2

, 𝑘1= 𝑢2, 𝑒

2= −

1

2

𝑟,

𝑘2= −𝑢2𝑥

−

1

2

𝑢1− 𝑟𝑢2,

𝑒3=

1

4

𝑟𝑥−

1

2

𝑢2𝑢2𝑥

+

1

2

𝑟2−

1

2

𝑢1𝑢2−

1

8

𝑠,

𝑘3= 𝑢2𝑥𝑥

+ 𝑟𝑥𝑢2+ 2𝑟𝑢

2𝑥+

1

2

𝑢1𝑥

+ 𝑟𝑢1+ 𝑟2𝑢2−

1

2

𝑠𝑢2, . . .

(34)

Journal of Applied Mathematics 5

and a recursion formula for 𝑒𝑛and 𝑘𝑛

𝑒𝑛+1

= −

1

2

𝑒𝑛,𝑥

− 𝑟𝑒𝑛+

1

2

𝑢2𝑘𝑛−

1

2

𝑠

𝑛−1

∑

𝑙=1

𝑒𝑙𝑒𝑛−𝑙

−

1

2

𝑢1

𝑛−1

∑

𝑙=1

𝑒𝑙𝑘𝑛−𝑙

,

𝑘𝑛+1

= −𝑘𝑛,𝑥

− 𝑢1𝑒𝑛− 𝑟𝑘𝑛− 𝑠

𝑛−1

∑

𝑙=1

𝑘𝑙𝑒𝑛−𝑙

− 𝑢1

𝑛−1

∑

𝑙=1

𝑘𝑙𝑘𝑛−𝑙

,

(35)

because of

𝜕

𝜕𝑡

[𝜆 + 𝑟 + 𝑠𝐸 + 𝑢1𝐾] =

𝜕

𝜕𝑥

[𝐴 + 𝐵𝐸 + 𝜌𝐾] , (36)

where

𝐴 = 𝑚0𝜆2+ 𝑚1𝜆 +

1

2

𝑚0𝑠 − 𝑚0𝑢1𝑢2,

𝐵 = 𝑚0𝑠𝜆 +

1

2

𝑚0𝑠𝑥− 𝑚0𝑟𝑠 + 𝑚

1𝑠,

𝜌 = 𝑚0𝑢1𝜆 + 𝑚

0𝑢1𝑥

− 𝑚0𝑟𝑢1+ 𝑚1𝑢1.

(37)

Assume that 𝛿 = 𝜆 + 𝑟 + 𝑠𝐸 + 𝑢1𝐾, 𝜃 = 𝐴 + 𝐵𝐸 + 𝜌𝐾.

Then (36) can be written as 𝛿𝑡= 𝜃𝑥, which is the right form of

conservation laws. We expand 𝛿 and 𝜃 as series in powers of𝜆 with the coefficients, which are called conserved densitiesand currents, respectively,

𝛿 = 𝜆 + 𝑟 +

∞

∑

𝑗=1

𝛿𝑗𝜆−𝑗,

𝜃 = 𝑚0𝜆2+ 𝑚1𝜆 + 𝑚

0𝑠 +

∞

∑

𝑗=1

𝜃𝑗𝜆−𝑗,

(38)

where 𝑚0, 𝑚1are constants of integration. The first two

conserved densities and currents are read as follows:

𝛿1=

1

2

𝑠 + 𝑢1𝑢2,

𝜃1= 𝑚0(

1

4

𝑠𝑥− 𝑠𝑟 − 𝑢

1𝑢2𝑥

+ 𝑢2𝑢1𝑥

− 2𝑟𝑢1𝑢2)

+ 𝑚1(

1

2

𝑠 + 𝑢1𝑢2) ,

𝛿2= −

1

2

𝑠𝑟 − 𝑢1𝑢2𝑥

− 𝑟𝑢1𝑢2,

𝜃2= 𝑚0(

1

4

𝑠𝑟𝑥−

1

2

𝑠𝑢2𝑢2𝑥

+ 𝑠𝑟2− 𝑠𝑢1𝑢2−

1

8

𝑠2

−

1

4

𝑟𝑠𝑥+ 𝑢1𝑢2𝑥𝑥

+ 𝑟𝑥𝑢1𝑢2+ 3𝑟𝑢

1𝑢2𝑥

+2𝑟2𝑢1𝑢2− 𝑢1𝑥𝑢2𝑥

− 𝑟𝑢1𝑥𝑢2)

− 𝑚1(

1

2

𝑠𝑟 + 𝑢1𝑢2𝑥

+ 𝑟𝑢1𝑢2) .

(39)

The recursion relation for 𝛿𝑛and 𝜃𝑛are

𝛿𝑛= 𝑠𝑒𝑛+ 𝑢1𝑘𝑛,

𝜃𝑛= 𝑚0(𝑠𝑛+1

+

1

2

𝑠𝑥𝑒𝑛− 𝑟𝑠𝑒𝑛+ 𝑢1𝑘𝑛+1

+ 𝑢1𝑥𝑘𝑛− 𝑟𝑢1𝑘𝑛)

+ 𝑚1(𝑠𝑒𝑛+ 𝑢1𝑘𝑛) ,

(40)

where 𝑒𝑛and 𝑘

𝑛can be calculated from (35). The infinitely

many conservation laws of (20) can be easily obtained from(32)–(40), respectively.

5. Conclusions

Starting from Lie super algebras, we may get super equa-tion hierarchy. With the help of variational identity, theHamiltonian structure can also be presented. Based on Liesuper algebra, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy can be obtained. It enriched thecontent of self-consistent sources of super soliton hierarchy.Finally, we also get the conservation laws of the superBroer-Kaup-Kupershmidt hierarchy. It is worth to note thatthe coupling terms of super integrable hierarchies involvefermi variables; they satisfy the Grassmann algebra which isdifferent from the ordinary one.

Acknowledgments

This project is supported by the National Natural ScienceFoundation of China (Grant nos. 11271008, 61072147, and11071159), the First-Class Discipline of Universities in Shang-hai, and the Shanghai University Leading Academic Disci-pline Project (Grant no. A13-0101-12-004).

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