Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 913758, 6 pages
http://dx.doi.org/10.1155/2013/913758
Research Article
Self-Consistent Sources and Conservation Laws for
a Super Broer-Kaup-Kupershmidt Equation Hierarchy
Hanyu Wei1,2 and Tiecheng Xia2
1
2
Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou 466001, China
Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Hanyu Wei; weihanyu1982@163.com
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 with
self-consistent sources is established. Furthermore, we establish the infinitely many conservation laws for the integrable super
Broer-Kaup-Kupershmidt hierarchy. In the process of computation especially, Fermi variables also play an important role in super
integrable systems.
1. Introduction
Soliton theory has achieved great success during the last
decades; it is being applied to mathematics, physics, biology,
astrophysics, and other potential fields [1–12]. The diversity
and complexity of soliton theory enable investigators to do
research from different views, such as Hamiltonian structure,
self-consistent sources, conservation laws, and various solutions of soliton equations.
In recent years, with the development of integrable systems, super integrable systems have attracted much attention.
Many scholars and experts do research on the topic and
get lots of results. For example, in [13], Ma et al. gave
the supertrace identity based on Lie super algebras and
its application to super AKNS hierarchy and super Dirac
hierarchy, and to get their super Hamiltonian structures, Hu
gave an approach to generate superextensions of integrable
systems [14]. Afterwards, super Boussinesq hierarchy [15]
and super NLS-mKdV hierarchy [16] as well as their super
Hamiltonian structures are presented. The binary nonlinearization of the super classical Boussinesq hierarchy [17], the
Bargmann symmetry constraint, and binary nonlinearization
of the super Dirac systems were given [18].
Soliton equation with self-consistent sources is an important part in soliton theory. They are usually used to describe
interactions between different solitary waves, and they are
also relevant to some problems related to hydrodynamics,
solid state physics, plasma physics, and so forth. Some results
have 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 discussing the integrability for soliton hierarchy. An infinite
number of conservation laws for KdV equation were first
discovered by Miura et al. in 1968 [24], and then lots of
methods have been developed to find them. This may be
mainly due to the contributions of Wadati and others [25–
27]. Conservation laws also play an important role in mathematics and engineering as well. Many papers dealing with
symmetries and conservation laws were presented. The direct
construction method of multipliers for the conservation laws
was presented [28].
In this paper, starting from a Lie super algebra, isospectral
problems are designed. With the help of variational identity,
Yang got super Broer-Kaup-Kupershmidt hierarchy and its
Hamiltonian structure [29]. Then, based on the theory of selfconsistent sources, the self-consistent sources of super BroerKaup-Kupershmidt hierarchy are obtained by us. Furthermore, we present the conservation laws for the super BroerKaup-Kupershmidt hierarchy. In the calculation process,
extended Fermi quantities 𝑢1 and 𝑢2 play an important role;
namely, 𝑢1 and 𝑢2 satisfy 𝑢12 = 𝑢22 = 0 and 𝑢1 𝑢2 = −𝑢2 𝑢1
2
Journal of Applied Mathematics
in the whole paper. Furthermore, the operation between
extended Fermi variables satisfies Grassmann algebra conditions.
2. A Super Soliton Hierarchy with
Self-Consistent Sources
𝜑1𝑗
𝜑1𝑗
𝜑1𝑗
5
(𝜑2𝑗 ) = 𝑈 (𝑢, 𝜆 𝑗 ) (𝜑2𝑗 ) = ∑𝑢𝑖 𝑒𝑖 (𝜆) (𝜑2𝑗 ) ,
𝑖=1
𝜑3𝑗 𝑥
𝜑3𝑗
𝜑3𝑗
Based on a Lie superalgebra 𝐺,
1 0 0
𝑒1 = (0 −1 0) ,
0 0 0
0 0 0
𝑒3 = (1 0 0) ,
0 0 0
0 1 0
𝑒2 = (0 0 0) ,
0 0 0
0 0 1
𝑒4 = (0 0 0) ,
0 −1 0
0 0 0
𝑒5 = (0 0 1)
1 0 0
(1)
that is along with the communicative operation [𝑒1 , 𝑒2 ] =
2𝑒2 , [𝑒1 , 𝑒3 ] = −2𝑒3 , [𝑒2 , 𝑒3 ] = 𝑒1 , [𝑒1 , 𝑒4 ] = [𝑒2 , 𝑒5 ] =
𝑒4 , [𝑒1 , 𝑒5 ] = [𝑒4 , 𝑒3 ] = −𝑒5 , [𝑒4 , 𝑒5 ]+ = 𝑒1 , [𝑒4 , 𝑒4 ]+ = −2𝑒2 ,
and [𝑒5 , 𝑒5 ]+ = 2𝑒3 .
We consider an auxiliary linear problem
𝜑1
𝜑1
(𝜑2 ) = 𝑈 (𝑢, 𝜆) (𝜑2 ) ,
𝜑3 𝑥
𝜑3
then (4) possesses a super Hamiltonian equation. If so, we can
say that (4) has a super Hamiltonian structure.
According to (2), now we consider a new auxiliary linear
problem. For 𝑁 distinct 𝜆 𝑗 , 𝑗 = 1, 2, . . . , 𝑁, the systems of (2)
become as follows:
5
𝑈 (𝑢, 𝜆) = 𝑅1 + ∑𝑢𝑖 𝑒𝑖 (𝜆) ,
𝑖=1
𝜑1
𝜑1
(𝜑2 ) = 𝑉𝑛 (𝑢, 𝜆) (𝜑2 ) ,
𝜑3 𝑡
𝜑3
𝜑1𝑗
𝜑1
(𝜑2𝑗 ) = 𝑉𝑛 (𝑢, 𝜆 𝑗 ) (𝜑2 )
𝜑3
𝜑3𝑗 𝑡
𝑛
𝜑1
𝑛
𝜑
= [ ∑ 𝑉𝑚 (𝑢) 𝜆𝑛−𝑚
+
Δ
(𝑢,
𝜆
)]
(
2) .
𝑛
𝑗
𝑗
𝑚=0
𝜑3
Based on the result in [30], we can show that the following
equation:
𝛿𝐻𝑘 𝑁 𝛿𝜆 𝑗
+ ∑𝛼
=0
𝛿𝑢 𝑗=1 𝑗 𝛿𝑢
holds true, where 𝛼𝑗 are constants. Equation (8) determines a
finite dimensional invariant set for the flows in (6).
From (7), we may know that
𝜕𝑈 (𝑢, 𝜆 𝑗 )
1
= 𝑆 tr (Ψ𝑗
)
𝛿𝑢𝑖
3
𝜕𝑢𝑖
(2)
where 𝑢 = (𝑢1 , . . . , 𝑢𝑠 )𝑇 , 𝑈𝑛 = 𝑅1 +𝑢1 𝑒1 +⋅ ⋅ ⋅+𝑢5 𝑒5 , 𝑢𝑖 (𝑛, 𝑡) =
𝑢𝑖 (𝑖 = 1, 2, . . . , 5), 𝜑𝑖 = 𝜑(𝑥, 𝑡) are field variables defining
𝑥 ∈ 𝑅, 𝑡 ∈ 𝑅; 𝑒𝑖 = 𝑒𝑖 (𝜆) ∈ 𝑠̃𝑙(3) and 𝑅1 is a pseudoregular
element.
The compatibility of (2) gives rise to the well-known zero
curvature equation as follows:
𝑛 = 1, 2, . . . .
(3)
If an equation
𝑢𝑡 = 𝐾 (𝑢)
(5)
𝑛 = 1, 2, . . . ,
𝛿 𝑇
𝛿
𝛿
,...,
) ,
=(
𝛿𝑢
𝛿𝑢1
𝛿𝑢5
(9)
𝑖 = 1, 2, . . . 5,
where 𝑆 tr denotes the trace of a matrix and
2
𝜓1𝑗 𝜓2𝑗 −𝜓1𝑗
𝜓1𝑗 𝜓3𝑗
2
Ψ𝑗 = ( 𝜓2𝑗 −𝜓1𝑗 𝜓2𝑗 𝜓2𝑗 𝜓3𝑗 ) .
𝜓2𝑗 𝜓3𝑗 −𝜓1𝑗 𝜓3𝑗
0
(10)
From (8) and (9), a kind of super Hamiltonian soliton
equation hierarchy with self-consistent sources is presented
as follows:
𝑢𝑛𝑡 = 𝐽
𝑁
𝛿𝜆 𝑗
𝛿𝐻𝑛+1
+ 𝐽∑ 𝛼𝑗
𝛿𝑢𝑖
𝛿𝑢
𝑗=1
𝑛 𝛿𝐻1
= 𝐽𝐿
𝛿𝑢𝑖
𝑁
+ 𝐽∑ 𝛼𝑗
𝑗=1
𝛿𝜆 𝑗
𝛿𝑢
(11)
,
𝑛 = 1, 2, . . . .
3. The Super Broer-Kaup-Kupershmidt
Hierarchy with Self-Consistent Sources
where
𝛿𝐻
𝛿𝐻
𝛿𝐻𝑛
= 𝐿 𝑛−1 = ⋅ ⋅ ⋅ = 𝐿𝑛 0 ,
𝛿𝑢
𝛿𝑢
𝛿𝑢
1
= 𝑆 tr (Ψ𝑗 𝑒𝑖 𝜆 𝑗 ) ,
3
(4)
can be worked out through (3), we call (4) a super evolution
equation. If there is a super Hamiltonian operator 𝐽 and a
function 𝐻𝑛 such that
𝛿𝐻
𝑢𝑡 = 𝐾 (𝑢) = 𝐽 𝑛+1 ,
𝛿𝑢
(8)
𝛿𝜆 𝑗
𝑛
𝑈𝑛𝑡 − 𝑉𝑛𝑥 + [𝑈𝑛 , 𝑉𝑛 ] = 0,
(7)
(6)
The super Broer-Kaup-Kupershmidt spectral problem associated with the Lie super algebra is given in [29]:
𝜑𝑥 = 𝑈𝜑,
𝜑𝑡 = 𝑉𝜑,
(12)
Journal of Applied Mathematics
3
where
where
𝜆+𝑟
𝑠
𝑢1
𝑈 = ( 1 −𝜆 − 𝑟 𝑢2 ) ,
𝑢2
−𝑢1 0
𝐴 𝐵 𝜌
𝑉 = (𝐶 −𝐴 𝜎) ,
𝜎 −𝜌 0
(13)
and 𝐴 = ∑𝑚≥0 𝐴 𝑚 𝜆−𝑚 , 𝐵 = ∑𝑚≥0 𝐵𝑚 𝜆−𝑚 , 𝐶 =
∑𝑚≥0 𝐶𝑚 𝜆−𝑚 , 𝜌 = ∑𝑚≥0 𝜌𝑚 𝜆−𝑚 , and 𝜎 = ∑𝑚≥0 𝜎𝑚 𝜆−𝑚 . As
𝑢1 and 𝑢2 are Fermi variables, they constitute Grassmann
algebra. So, we have 𝑢1 𝑢2 = −𝑢2 𝑢1 , 𝑢12 = 𝑢22 = 0.
Starting from the stationary zero curvature equation
𝑉𝑥 = [𝑈, 𝑉] ,
𝑃𝑛+1 = 𝐿𝑃𝑛 ,
1
1
1
𝜕 − 𝜕−1 𝑟𝜕 −𝑠 − 𝜕−1 𝑠𝜕 𝜕−1 𝑢1 𝜕 + 𝑢1 𝜕−1 𝑢2 𝜕 − 𝑢2
2
2
2
1
1
1
0
− 𝜕−𝑟
𝑢
(
)
2
2
2 2
𝐿= (
).
−𝑢2
2𝑢1
−𝑟 − 𝜕
−1
1
2𝑠𝑢2
𝑠 + 𝑢1 𝑢2
𝜕−𝑟
( 𝑢1 − 𝑢2 𝜕
)
2
(21)
(14)
we have
According to super trace identity on Lie super algebras, a
direct calculation reads as
𝐴 𝑚𝑥 = 𝑠𝐶𝑚 + 𝑢1 𝜎𝑚 − 𝐵𝑚 + 𝑢2 𝜌𝑚 ,
𝐵𝑚𝑥 = 2𝐵𝑚+1 + 2𝑟𝐵𝑚 − 2𝑠𝐴 𝑚 − 2𝑢1 𝜌𝑚 ,
𝛿𝐻𝑛
𝑇
= (−2𝐴 𝑛+1 , −𝐶𝑛+1 , 2𝜎𝑛+1 , −2𝜌𝑛+1 ) ,
𝛿𝑢
𝐶𝑚𝑥 = −2𝐶𝑚+1 − 2𝑟𝐶𝑚 + 2𝐴 𝑚 + 2𝑢2 𝜎𝑚 ,
𝜌𝑚𝑥 = 𝜌𝑚+1 + 𝑟𝜌𝑚 + 𝑠𝜎𝑚 − 𝑢1 𝐴 𝑚 − 𝑢2 𝐵𝑚 ,
𝜎𝑚𝑥 = −𝜎𝑚+1 − 𝑟𝜎𝑚 + 𝜌𝑚 − 𝑢1 𝐶𝑚 + 𝑢2 𝐴 𝑚 ,
𝐵0 = 𝐶0 = 𝜌0 = 𝜎0 = 0,
𝐴 0 = 1,
𝐵1 = 𝑠,
𝜌1 = 𝑢1 ,
𝑛
𝜑𝑡𝑛 = 𝑉 𝜑 = (𝜆 𝑉)+ 𝜑,
(16)
𝐴 𝑚 𝐵𝑚 𝜌𝑚
𝑛
𝑉(𝑛) = ∑ ( 𝐶𝑚 −𝐴 𝑚 𝜎𝑚 ) 𝜆𝑛−𝑚 ,
𝑚=0
𝜎𝑚 −𝜌𝑚 0
(17)
𝑉(𝑛) = 𝑉+(𝑛) + Δ 𝑛 ,
(18)
1
1
𝑟𝑡2 = − 𝑟𝑥𝑥 + 𝑠𝑥 − 2𝑟𝑟𝑥 + (𝑢1 𝑢2 )𝑥 + (𝑢1 𝑢2𝑥 )𝑥 ,
2
2
1
𝑠𝑡2 = 𝑠𝑥𝑥 − 2(𝑟𝑠)𝑥 + 2𝑢1 𝑢1𝑥 + 2𝑠𝑢2 𝑢2𝑥 ,
2
where
𝑢1𝑡2
considering
Δ 𝑛 = −𝐶𝑚+1 𝑒1 .
Substituting (18) into the zero curvature equation
𝑈𝑡𝑛 − 𝑉𝑥(𝑛) + [𝑈, 𝑉(𝑛) ] = 0,
(19)
we get the super Broer-Kaup-Kupershmidt hierarchy
𝑟
𝑠
𝑢𝑡𝑛 = ( )
𝑢1
𝑢2 𝑡
0
𝜕
𝑛 ≥ 0.
𝐴 1 = 0, . . . .
Then we consider the auxiliary spectral problem
(𝑛)
(22)
When we take 𝑛 = 2, the hierarchy (20) can be reduced to
super nonlinear integrable couplings equations
𝐶1 = 𝑟,
𝜎1 = 𝑢2 ,
2𝐴 𝑛+2
𝐻𝑛 = ∫
𝑑𝑥,
𝑛+1
(15)
0
0
𝑢1 −𝑢2
−2𝐴 𝑛+1
1)
(0 𝑢
0 − ) ( −𝐶𝑛+1 )
=(
1
2𝜎𝑛+1
2
−2𝜌𝑛+1
1
0 −𝑢2 −
0
2
(
)
(20)
(23)
1
𝑢2𝑡2 = −𝑢2𝑥𝑥 − 𝑟𝑥 𝑢2 − 2𝑟𝑢2𝑥 − 𝑢1𝑥 .
2
Next, we will construct the super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources. Consider the
linear system
𝜑1𝑗
𝜑1𝑗
(𝜑2𝑗 ) = 𝑈 (𝜑2𝑗 ) ,
𝜑3𝑗 𝑥
𝜑3𝑗
𝜕
0
−2𝐴 𝑛+1
−𝐶𝑛+1
= 𝐽(
) = 𝐽𝑃𝑛+1 ,
2𝜎𝑛+1
−2𝜌𝑛+1
3
1
= 𝑢1𝑥𝑥 − 𝑟𝑥 𝑢1 + 𝑠𝑥 𝑢2 − 2𝑟𝑢1𝑥 + (𝑠 + 𝑢1 𝑢2 ) 𝑢2𝑥 ,
2
2
(24)
𝜑1𝑗
𝜑1𝑗
(𝜑2𝑗 ) = 𝑉 (𝜑2𝑗 ) .
𝜑3𝑗 𝑡
𝜑3𝑗
From (8), for the system (12), we set
𝑁 𝛿𝜆
𝛿𝐻𝑛
𝑗
=∑
𝛿𝑢
𝛿𝑢
𝑗=1
(25)
4
Journal of Applied Mathematics
3
1
𝑢1𝑡2 = 𝑢1𝑥𝑥 − 𝑟𝑥 𝑢1 + 𝑠𝑥 𝑢2 − 2𝑟𝑢1𝑥 + (𝑠 + 𝑢1 𝑢2 ) 𝑢2𝑥
2
2
and obtain the following 𝛿𝜆 𝑗 /𝛿𝑢:
𝑁
∑
𝑗=1
𝛿𝜆 𝑗
𝑁
𝑁
𝑗=1
𝑗=1
2
+ 𝑢1 ∑ 𝜑2𝑗
− ∑ 𝜑1𝑗 𝜑3𝑗 ,
𝛿𝑢
𝛿𝑈
)
𝛿𝑞
2 ⟨Φ1 , Φ2 ⟩
𝛿𝑈 )
(
(26)
(𝑆 tr (Ψ𝑗
))
⟨Φ2 , Φ2 ⟩
𝛿𝑟 )
𝑁 (
)
(
) (
)
= ∑(
=(
𝛿𝑈 )
(
(−2 ⟨Φ2 , Φ3 ⟩) ,
))
)
𝑗=1 (𝑆 tr (Ψ𝑗
(
𝛿𝛼 )
(
)
2 ⟨Φ1 , Φ3 ⟩
𝛿𝑈
𝑆 tr (Ψ𝑗
)
)
(
𝛿𝛽
(
)
𝑆 tr (Ψ𝑗
where Φ𝑖 = (𝜑𝑖1 , . . . , 𝜑𝑖𝑁)𝑇 , 𝑖 = 1, 2, 3.
According to (11), the integrable super Broer-KaupKupershmidt hierarchy with self-consistent sources is proposed as follows:
𝑁
𝑁
1
2
𝑢2𝑡2 = −𝑢2𝑥𝑥 − 𝑟𝑥 𝑢2 − 2𝑟𝑢2𝑥 − 𝑢1𝑥 − 𝑢2 ∑ 𝜑2𝑗
+ ∑ 𝜑2𝑗 𝜑3𝑗 ,
2
𝑗=1
𝑗=1
(29)
where Φ𝑖 = (𝜑𝑖1 , . . . , 𝜑𝑖𝑁)𝑇 , 𝑖 = 1, 2, 3, satisfy
𝜑1𝑗𝑥 = (𝜆 + 𝑟) 𝜑1𝑗 + 𝑠𝜑2𝑗 + 𝑢1 𝜑3𝑗 ,
𝜑2𝑗𝑥 = 𝜑1𝑗 − (𝜆 + 𝑟) 𝜑2𝑗 + 𝑢2 𝜑3𝑗 ,
𝜑3𝑗𝑥 = 𝑢2 𝜑1𝑗 − 𝑢1 𝜑2𝑗 ,
(30)
𝑗 = 1, . . . , 𝑁.
4. Conservation Laws for the Super
Broer-Kaup-Kupershmidt Hierarchy
𝑟
𝑠
𝑢𝑡𝑛 = ( )
𝑢1𝑡
𝑢2𝑡 𝑡
In the following, we will construct conservation laws of the
super Broer-Kaup-Kupershmidt hierarchy. We introduce the
variables
𝑛
2 ⟨Φ1 , Φ2 ⟩
(27)
⟨Φ2 , Φ2 ⟩
−2𝐴 𝑛+1
)
(
−𝐶𝑛+1
)
= 𝐽(
) + 𝐽(
(−2 ⟨Φ2 , Φ3 ⟩) ,
2𝜎𝑛+1
−2𝜌𝑛+1
2 ⟨Φ1 , Φ3 ⟩
)
(
𝐸=
𝜑2
,
𝜑1
𝐾=
𝜑3
.
𝜑1
(31)
From (7) and (12), we have
𝐸𝑥 = 1 − 2𝜆𝐸 − 2𝑟𝐸 + 𝑢2 𝐾 − 𝑠𝐸2 − 𝑢1 𝐸𝐾,
where Φ𝑖 = (𝜑𝑖1 , . . . , 𝜑𝑖𝑁)𝑇 , 𝑖 = 1, 2, 3, satisfy
𝐾𝑥 = 𝑢2 − 𝜆𝐾 − 𝑢1 𝐸 − 𝑟𝐾 − 𝑠𝐾𝐸 − 𝑢1 𝐾2 .
𝜑1𝑗𝑥 = (𝜆 + 𝑟) 𝜑1𝑗 + 𝑠𝜑2𝑗 + 𝑢1 𝜑3𝑗 ,
(32)
Expand 𝐸, 𝐾 in the power of 𝜆 as follows:
𝜑2𝑗𝑥 = 𝜑1𝑗 − (𝜆 + 𝑟) 𝜑2𝑗 + 𝑢2 𝜑3𝑗 ,
(28)
𝜑3𝑗𝑥 = 𝑢2 𝜑1𝑗 − 𝑢1 𝜑2𝑗 ,
∞
𝐸 = ∑𝑒𝑗 𝜆−𝑗 ,
𝑗=1
∞
𝐾 = ∑𝑘𝑗 𝜆−𝑗 .
𝑗=1
(33)
𝑗 = 1, . . . , 𝑁.
For 𝑛 = 2, we obtain the super Broer-Kaup-Kupershmidt
equation with self-consistent sources as follows:
𝑁
1
1
2
𝑟𝑡2 = − 𝑟𝑥𝑥 + 𝑠𝑥 − 2𝑟𝑟𝑥 + (𝑢1 𝑢2 )𝑥 + (𝑢1 𝑢2𝑥 )𝑥 + 𝜕∑ 𝜑2𝑗
,
2
2
𝑗=1
𝑠𝑡2 =
𝑁
1
𝑠𝑥𝑥 − 2(𝑟𝑠)𝑥 + 2𝑢1 𝑢1𝑥 + 2𝑠𝑢2 𝑢2𝑥 + 2𝜕∑ 𝜑1𝑗 𝜑2𝑗
2
𝑗=1
𝑁
𝑁
𝑗=1
𝑗=1
− 2𝑢1 ∑ 𝜑2𝑗 𝜑3𝑗 − 2𝑢2 ∑𝜑1𝑗 𝜑3𝑗 ,
Substituting (33) into (32) and comparing the coefficients of
the same power of 𝜆, we obtain
1
𝑒1 = ,
2
𝑘1 = 𝑢2 ,
1
𝑒2 = − 𝑟,
2
1
𝑘2 = −𝑢2𝑥 − 𝑢1 − 𝑟𝑢2 ,
2
1
1
1
1
1
𝑒3 = 𝑟𝑥 − 𝑢2 𝑢2𝑥 + 𝑟2 − 𝑢1 𝑢2 − 𝑠,
4
2
2
2
8
1
1
𝑘3 = 𝑢2𝑥𝑥 + 𝑟𝑥 𝑢2 + 2𝑟𝑢2𝑥 + 𝑢1𝑥 + 𝑟𝑢1 + 𝑟2 𝑢2 − 𝑠𝑢2 , . . .
2
2
(34)
Journal of Applied Mathematics
5
and a recursion formula for 𝑒𝑛 and 𝑘𝑛
The recursion relation for 𝛿𝑛 and 𝜃𝑛 are
1
1
1 𝑛−1
1 𝑛−1
𝑒𝑛+1 = − 𝑒𝑛,𝑥 − 𝑟𝑒𝑛 + 𝑢2 𝑘𝑛 − 𝑠 ∑ 𝑒𝑙 𝑒𝑛−𝑙 − 𝑢1 ∑ 𝑒𝑙 𝑘𝑛−𝑙 ,
2
2
2 𝑙=1
2 𝑙=1
𝑛−1
𝑛−1
𝑙=1
𝑙=1
𝑘𝑛+1 = −𝑘𝑛,𝑥 − 𝑢1 𝑒𝑛 − 𝑟𝑘𝑛 − 𝑠 ∑ 𝑘𝑙 𝑒𝑛−𝑙 − 𝑢1 ∑ 𝑘𝑙 𝑘𝑛−𝑙 ,
𝛿𝑛 = 𝑠𝑒𝑛 + 𝑢1 𝑘𝑛 ,
1
𝜃𝑛 = 𝑚0 (𝑠𝑛+1 + 𝑠𝑥 𝑒𝑛 − 𝑟𝑠𝑒𝑛 + 𝑢1 𝑘𝑛+1 + 𝑢1𝑥 𝑘𝑛 − 𝑟𝑢1 𝑘𝑛 )
2
+ 𝑚1 (𝑠𝑒𝑛 + 𝑢1 𝑘𝑛 ) ,
(40)
(35)
because of
𝜕
𝜕
[𝜆 + 𝑟 + 𝑠𝐸 + 𝑢1 𝐾] =
[𝐴 + 𝐵𝐸 + 𝜌𝐾] ,
𝜕𝑡
𝜕𝑥
(36)
where
1
𝐴 = 𝑚0 𝜆2 + 𝑚1 𝜆 + 𝑚0 𝑠 − 𝑚0 𝑢1 𝑢2 ,
2
1
𝐵 = 𝑚0 𝑠𝜆 + 𝑚0 𝑠𝑥 − 𝑚0 𝑟𝑠 + 𝑚1 𝑠,
2
(37)
𝜌 = 𝑚0 𝑢1 𝜆 + 𝑚0 𝑢1𝑥 − 𝑚0 𝑟𝑢1 + 𝑚1 𝑢1 .
Assume that 𝛿 = 𝜆 + 𝑟 + 𝑠𝐸 + 𝑢1 𝐾, 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 of
𝜆 with the coefficients, which are called conserved densities
and currents, respectively,
∞
𝛿 = 𝜆 + 𝑟 + ∑𝛿𝑗 𝜆−𝑗 ,
𝑗=1
2
∞
(38)
−𝑗
𝜃 = 𝑚0 𝜆 + 𝑚1 𝜆 + 𝑚0 𝑠 + ∑𝜃𝑗 𝜆 ,
where 𝑒𝑛 and 𝑘𝑛 can 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 equation hierarchy. With the help of variational identity, the
Hamiltonian structure can also be presented. Based on Lie
super algebra, the self-consistent sources of super BroerKaup-Kupershmidt hierarchy can be obtained. It enriched the
content of self-consistent sources of super soliton hierarchy.
Finally, we also get the conservation laws of the super
Broer-Kaup-Kupershmidt hierarchy. It is worth to note that
the coupling terms of super integrable hierarchies involve
fermi variables; they satisfy the Grassmann algebra which is
different from the ordinary one.
Acknowledgments
This project is supported by the National Natural Science
Foundation of China (Grant nos. 11271008, 61072147, and
11071159), the First-Class Discipline of Universities in Shanghai, and the Shanghai University Leading Academic Discipline Project (Grant no. A13-0101-12-004).
𝑗=1
where 𝑚0 , 𝑚1 are constants of integration. The first two
conserved densities and currents are read as follows:
1
𝛿1 = 𝑠 + 𝑢1 𝑢2 ,
2
1
𝜃1 = 𝑚0 ( 𝑠𝑥 − 𝑠𝑟 − 𝑢1 𝑢2𝑥 + 𝑢2 𝑢1𝑥 − 2𝑟𝑢1 𝑢2 )
4
1
+ 𝑚1 ( 𝑠 + 𝑢1 𝑢2 ) ,
2
1
𝛿2 = − 𝑠𝑟 − 𝑢1 𝑢2𝑥 − 𝑟𝑢1 𝑢2 ,
2
1
1
1
𝜃2 = 𝑚0 ( 𝑠𝑟𝑥 − 𝑠𝑢2 𝑢2𝑥 + 𝑠𝑟2 − 𝑠𝑢1 𝑢2 − 𝑠2
4
2
8
1
− 𝑟𝑠𝑥 + 𝑢1 𝑢2𝑥𝑥 + 𝑟𝑥 𝑢1 𝑢2 + 3𝑟𝑢1 𝑢2𝑥
4
+2𝑟2 𝑢1 𝑢2 − 𝑢1𝑥 𝑢2𝑥 − 𝑟𝑢1𝑥 𝑢2 )
1
− 𝑚1 ( 𝑠𝑟 + 𝑢1 𝑢2𝑥 + 𝑟𝑢1 𝑢2 ) .
2
(39)
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