Self-Consistent Sources and Conservation Laws for a Super Broer-Kaup-Kupershmidt Equation Hierarchy

Abstract

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 self-consistent sources, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy are obtained by us. Furthermore, we present the conservation laws for the super Broer-Kaup-Kupershmidt hierarchy. In the calculation process, extended Fermi quantities 𝑢₁ and 𝑢₂ play an important role; namely, 𝑢₁ and 𝑢₂ satisfy 𝑢²₁=𝑢²₂=0 and ?𝑢₁𝑢₂=−𝑢₂𝑢₁ in the whole paper. Furthermore, the operation between extended Fermi variables satisfies Grassmann algebra conditions.

2. A Super Soliton Hierarchy with Self-Consistent Sources

Based on a Lie superalgebra 𝐺, 𝑒₁=1000−10000,𝑒₂=,𝑒010000000₃=000100000,𝑒₄=0010000−10,𝑒₅=000001100 (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 𝜑₁𝜑₂𝜑_3𝑥()𝜑=𝑈𝑢,𝜆𝜑¹₂𝜑₃,𝑈(𝑢,𝜆)=𝑅₁+⁵_𝑖=1𝑢_𝑖𝑒_𝑖(𝜆),𝜑₁𝜑₂𝜑_3𝑡𝑛=𝑉_𝑛()𝜑𝑢,𝜆𝜑¹₂𝜑₃, (2) where??𝑢=(𝑢₁,…,𝑢_𝑠)^𝑇,𝑈_𝑛=𝑅₁+𝑢₁𝑒₁+⋯+𝑢₅𝑒₅,𝑢_𝑖(𝑛,𝑡)=𝑢_𝑖(𝑖=1,2,…,5),𝜑_𝑖=𝜑(𝑥,𝑡) are field variables defining 𝑥∈𝑅,𝑡∈𝑅;𝑒_𝑖=𝑒_𝑖(𝜆)∈𝑠𝑙̃(3) and 𝑅₁ is a pseudoregular element.

The compatibility of (2) gives rise to the well-known zero curvature equation as follows: 𝑈_𝑛𝑡−𝑉_𝑛𝑥+𝑈_𝑛,𝑉_𝑛=0,𝑛=1,2,…. (3) If an equation 𝑢_𝑡=𝐾(𝑢) (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 𝐻_𝑛 such that 𝑢_𝑡=𝐾(𝑢)=𝐽𝛿𝐻_𝑛+1,𝛿𝑢 (5) where 𝛿𝐻_𝑛𝛿𝑢=𝐿𝛿𝐻_𝑛−1𝛿𝑢=⋯=𝐿^𝑛𝛿𝐻₀,𝛿𝛿𝑢𝑛=1,2,…,=𝛿𝛿𝑢𝛿𝑢₁𝛿,…,𝛿𝑢₅^𝑇, (6) 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: 𝜑_1𝑗𝜑_2𝑗𝜑_3𝑗𝑥=𝑈𝑢,𝜆_𝑗𝜑_1𝑗𝜑_2𝑗𝜑_3𝑗=⁵_𝑖=1𝑢_𝑖𝑒_𝑖(𝜆)𝜑_1𝑗𝜑_2𝑗𝜑_3𝑗,𝜑_1𝑗𝜑_2𝑗𝜑_{3𝑗𝑡𝑛}=𝑉_𝑛𝑢,𝜆_𝑗𝜑₁𝜑₂𝜑₃=^𝑛_𝑚=0𝑉_𝑚(𝑢)𝜆_𝑗^𝑛−𝑚+Δ_𝑛𝑢,𝜆_𝑗𝜑₁𝜑₂𝜑₃. (7)

Based on the result in [30], we can show that the following equation: 𝛿𝐻_𝑘+𝛿𝑢^𝑁_𝑗=1𝛼_𝑗𝛿𝜆_𝑗𝛿𝑢=0 (8) holds true, where 𝛼_𝑗 are constants. Equation (8) determines a finite dimensional invariant set for the flows in (6).

From (7), we may know that 𝛿𝜆_𝑗𝛿𝑢_𝑖=13𝑆trΨ_𝑗𝜕𝑈𝑢,𝜆_𝑗𝜕𝑢_𝑖=13𝑆trΨ_𝑗𝑒_𝑖𝜆_𝑗,𝑖=1,2,…5, (9) where 𝑆tr denotes the trace of a matrix and Ψ_𝑗=𝜓_1𝑗𝜓_2𝑗−𝜓²_1𝑗𝜓_1𝑗𝜓_3𝑗𝜓²_2𝑗−𝜓_1𝑗𝜓_2𝑗𝜓_2𝑗𝜓_3𝑗𝜓_2𝑗𝜓_3𝑗−𝜓_1𝑗𝜓_3𝑗0. (10) From (8) and (9), a kind of super Hamiltonian soliton equation hierarchy with self-consistent sources is presented as follows: 𝑢_𝑛𝑡=𝐽𝛿𝐻_𝑛+1𝛿𝑢_𝑖+𝐽^𝑁_𝑗=1𝛼_𝑗𝛿𝜆_𝑗𝛿𝑢=𝐽𝐿^𝑛𝛿𝐻₁𝛿𝑢_𝑖+𝐽^𝑁_𝑗=1𝛼_𝑗𝛿𝜆_𝑗𝛿𝑢,𝑛=1,2,…. (11)

3. The Super Broer-Kaup-Kupershmidt Hierarchy with Self-Consistent Sources

The super Broer-Kaup-Kupershmidt spectral problem associated with the Lie super algebra is given in [29]: 𝜑_𝑥=𝑈𝜑,𝜑_𝑡=𝑉𝜑, (12) where 𝑈=𝜆+𝑟𝑠𝑢₁1−𝜆−𝑟𝑢₂𝑢₂−𝑢₁0,𝑉=𝐴𝐵𝜌𝐶−𝐴𝜎𝜎−𝜌0, (13) and??𝐴=∑_𝑚≥0𝐴_𝑚𝜆^−𝑚,𝐵=∑_𝑚≥0𝐵_𝑚𝜆^−𝑚,𝐶=∑_𝑚≥0𝐶_𝑚𝜆^−𝑚,𝜌=∑_𝑚≥0𝜌_𝑚𝜆^−𝑚, and??𝜎=∑_𝑚≥0𝜎_𝑚𝜆^−𝑚. As 𝑢₁ and 𝑢₂ are Fermi variables, they constitute Grassmann algebra. So, we have 𝑢₁𝑢₂=−𝑢₂𝑢₁,𝑢²₁=𝑢²₂=0.

Starting from the stationary zero curvature equation 𝑉_𝑥=[𝑈,𝑉], (14) we have 𝐴_𝑚𝑥=𝑠𝐶_𝑚+𝑢₁𝜎_𝑚−𝐵_𝑚+𝑢₂𝜌_𝑚,𝐵_𝑚𝑥=2𝐵_𝑚+1+2𝑟𝐵_𝑚−2𝑠𝐴_𝑚−2𝑢₁𝜌_𝑚,𝐶_𝑚𝑥=−2𝐶_𝑚+1−2𝑟𝐶_𝑚+2𝐴_𝑚+2𝑢₂𝜎_𝑚,𝜌_𝑚𝑥=𝜌_𝑚+1+𝑟𝜌_𝑚+𝑠𝜎_𝑚−𝑢₁𝐴_𝑚−𝑢₂𝐵_𝑚,𝜎_𝑚𝑥=−𝜎_𝑚+1−𝑟𝜎_𝑚+𝜌_𝑚−𝑢₁𝐶_𝑚+𝑢₂𝐴_𝑚,𝐵₀=𝐶₀=𝜌₀=𝜎₀𝐴=0,₀=1,𝐵₁=𝑠,𝐶₁𝜌=𝑟,₁=𝑢₁,𝜎₁=𝑢₂,𝐴₁=0,…. (15) Then we consider the auxiliary spectral problem 𝜑_𝑡𝑛=𝑉^(𝑛)𝜑=𝜆^𝑛𝑉₊𝜑, (16) where 𝑉^(𝑛)=^𝑛_𝑚=0𝐴_𝑚𝐵_𝑚𝜌_𝑚𝐶_𝑚−𝐴_𝑚𝜎_𝑚𝜎_𝑚−𝜌_𝑚0𝜆^𝑛−𝑚, (17) considering 𝑉^(𝑛)=𝑉₊^(𝑛)+Δ_𝑛,Δ_𝑛=−𝐶_𝑚+1𝑒₁. (18) Substituting (18) into the zero curvature equation 𝑈_𝑡𝑛−𝑉_𝑥^(𝑛)+𝑈,𝑉^(𝑛)=0, (19) we get the super Broer-Kaup-Kupershmidt hierarchy 𝑢_𝑡𝑛=𝑟𝑠𝑢₁𝑢_2𝑡=0𝜕00𝜕0𝑢₁−𝑢₂0𝑢₁10−20−𝑢₂−120−2𝐴_𝑛+1−𝐶_𝑛+12𝜎_𝑛+1−2𝜌_𝑛+1=𝐽−2𝐴_𝑛+1−𝐶_𝑛+12𝜎_𝑛+1−2𝜌_𝑛+1=𝐽𝑃_𝑛+1, (20) where 𝑃_𝑛+1=𝐿𝑃_𝑛,1𝐿=2𝜕−𝜕⁻¹𝑟𝜕−𝑠−𝜕⁻¹𝑠𝜕𝜕⁻¹𝑢₁1𝜕+2𝑢₁𝜕⁻¹𝑢₂1𝜕−2𝑢₂12−121𝜕−𝑟2𝑢₂0−𝑢₂2𝑢₁𝑢−𝑟−𝜕−1₁−𝑢₂𝜕2𝑠𝑢₂1𝑠+2𝑢₁𝑢₂.𝜕−𝑟 (21)

According to super trace identity on Lie super algebras, a direct calculation reads as 𝛿𝐻_𝑛=𝛿𝑢−2𝐴_𝑛+1,−𝐶_𝑛+1,2𝜎_𝑛+1,−2𝜌_𝑛+1^𝑇,𝐻_𝑛=2𝐴_𝑛+2𝑛+1𝑑𝑥,𝑛≥0. (22)

When we take 𝑛=2, the hierarchy (20) can be reduced to super nonlinear integrable couplings equations 𝑟_𝑡21=−2𝑟_𝑥𝑥+12𝑠_𝑥−2𝑟𝑟_𝑥+𝑢₁𝑢_2𝑥+𝑢₁𝑢_2𝑥𝑥,𝑠_𝑡2=12𝑠_𝑥𝑥−2(𝑟𝑠)_𝑥+2𝑢₁𝑢_1𝑥+2𝑠𝑢₂𝑢_2𝑥,𝑢_1𝑡2=𝑢_1𝑥𝑥−32𝑟_𝑥𝑢₁+12𝑠_𝑥𝑢₂−2𝑟𝑢_1𝑥+𝑠+𝑢₁𝑢₂𝑢_2𝑥,𝑢_2𝑡2=−𝑢_2𝑥𝑥−12𝑟_𝑥𝑢₂−2𝑟𝑢_2𝑥−𝑢_1𝑥. (23)

Next, we will construct the super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources. Consider the linear system 𝜑_1𝑗𝜑_2𝑗𝜑_3𝑗𝑥𝜑=𝑈𝜑^1𝑗_2𝑗𝜑_3𝑗,𝜑_1𝑗𝜑_2𝑗𝜑_3𝑗𝑡𝜑=𝑉𝜑^1𝑗_2𝑗𝜑_3𝑗. (24) From (8), for the system (12), we set 𝛿𝐻_𝑛=𝛿𝑢^𝑁_𝑗=1𝛿𝜆_𝑗𝛿𝑢 (25) and obtain the following 𝛿𝜆_𝑗/𝛿𝑢: ^𝑁_𝑗=1𝛿𝜆_𝑗=𝛿𝑢_𝑁𝑗=1𝑆trΨ_𝑗𝛿𝑈Ψ𝛿𝑞𝑆tr_𝑗𝛿𝑈Ψ𝛿𝑟𝑆tr_𝑗𝛿𝑈Ψ𝛿𝛼𝑆tr_𝑗𝛿𝑈=2Φ𝛿𝛽¹,Φ₂Φ₂,Φ₂−2Φ₂,Φ₃2Φ₁,Φ₃, (26) where Φ_𝑖=(𝜑_𝑖1,…,𝜑_𝑖𝑁)^𝑇,𝑖=1,2,3.

According to (11), the integrable super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is proposed as follows: 𝑢_𝑡𝑛=𝑟𝑠𝑢_1𝑡𝑢_{2𝑡𝑡𝑛}=𝐽−2𝐴_𝑛+1−𝐶_𝑛+12𝜎_𝑛+1−2𝜌_𝑛+12Φ+𝐽¹,Φ₂Φ₂,Φ₂−2Φ₂,Φ₃2Φ₁,Φ₃, (27) where Φ_𝑖=(𝜑_𝑖1,…,𝜑_𝑖𝑁)^𝑇,?𝑖=1,2,3,? satisfy 𝜑_1𝑗𝑥=(𝜆+𝑟)𝜑_1𝑗+𝑠𝜑_2𝑗+𝑢₁𝜑_3𝑗,𝜑_2𝑗𝑥=𝜑_1𝑗−(𝜆+𝑟)𝜑_2𝑗+𝑢₂𝜑_3𝑗,𝜑_3𝑗𝑥=𝑢₂𝜑_1𝑗−𝑢₁𝜑_2𝑗,𝑗=1,…,𝑁. (28) For 𝑛=2, we obtain the super Broer-Kaup-Kupershmidt equation with self-consistent sources as follows: 𝑟_𝑡21=−2𝑟_𝑥𝑥+12𝑠_𝑥−2𝑟𝑟_𝑥+𝑢₁𝑢_2𝑥+𝑢₁𝑢_2𝑥𝑥+𝜕^𝑁_𝑗=1𝜑²_2𝑗,𝑠_𝑡2=12𝑠_𝑥𝑥−2(𝑟𝑠)_𝑥+2𝑢₁𝑢_1𝑥+2𝑠𝑢₂𝑢_2𝑥+2𝜕^𝑁_𝑗=1𝜑_1𝑗𝜑_2𝑗−2𝑢₁^𝑁_𝑗=1𝜑_2𝑗𝜑_3𝑗−2𝑢₂^𝑁_𝑗=1𝜑_1𝑗𝜑_3𝑗,𝑢_1𝑡2=𝑢_1𝑥𝑥−32𝑟_𝑥𝑢₁+12𝑠_𝑥𝑢₂−2𝑟𝑢_1𝑥+𝑠+𝑢₁𝑢₂𝑢_2𝑥+𝑢₁^𝑁_𝑗=1𝜑²_2𝑗−^𝑁_𝑗=1𝜑_1𝑗𝜑_3𝑗,𝑢_2𝑡2=−𝑢_2𝑥𝑥−12𝑟_𝑥𝑢₂−2𝑟𝑢_2𝑥−𝑢_1𝑥−𝑢₂^𝑁_𝑗=1𝜑²_2𝑗+^𝑁_𝑗=1𝜑_2𝑗𝜑_3𝑗, (29) where Φ_𝑖=(𝜑_𝑖1,…,𝜑_𝑖𝑁)^𝑇,𝑖=1,2,3, satisfy 𝜑_1𝑗𝑥=(𝜆+𝑟)𝜑_1𝑗+𝑠𝜑_2𝑗+𝑢₁𝜑_3𝑗,𝜑_2𝑗𝑥=𝜑_1𝑗−(𝜆+𝑟)𝜑_2𝑗+𝑢₂𝜑_3𝑗,𝜑_3𝑗𝑥=𝑢₂𝜑_1𝑗−𝑢₁𝜑_2𝑗,𝑗=1,…,𝑁. (30)

4. Conservation Laws for the Super Broer-Kaup-Kupershmidt Hierarchy

In the following, we will construct conservation laws of the super Broer-Kaup-Kupershmidt hierarchy. We introduce the variables 𝜑𝐸=²𝜑₁𝜑,𝐾=³𝜑₁. (31) From (7) and (12), we have 𝐸_𝑥=1−2𝜆𝐸−2𝑟𝐸+𝑢₂𝐾−𝑠𝐸²−𝑢₁𝐾𝐸𝐾,_𝑥=𝑢₂−𝜆𝐾−𝑢₁𝐸−𝑟𝐾−𝑠𝐾𝐸−𝑢₁𝐾². (32) Expand 𝐸,??𝐾 in the power of 𝜆 as follows: 𝐸=^∞_𝑗=1𝑒_𝑗𝜆^−𝑗,𝐾=^∞_𝑗=1𝑘_𝑗𝜆^−𝑗. (33) Substituting (33) into (32) and comparing the coefficients of the same power of 𝜆, we obtain 𝑒₁=12,𝑘₁=𝑢₂,𝑒₂1=−2𝑘𝑟,₂=−𝑢_2𝑥−12𝑢₁−𝑟𝑢₂,𝑒₃=14𝑟_𝑥−12𝑢₂𝑢_2𝑥+12𝑟²−12𝑢₁𝑢₂−18𝑘𝑠,₃=𝑢_2𝑥𝑥+𝑟_𝑥𝑢₂+2𝑟𝑢_2𝑥+12𝑢_1𝑥+𝑟𝑢₁+𝑟²𝑢₂−12𝑠𝑢₂,… (34) and a recursion formula for 𝑒_𝑛 and 𝑘_𝑛𝑒_𝑛+11=−2𝑒_𝑛,𝑥−𝑟𝑒_𝑛+12𝑢₂𝑘_𝑛−12𝑠^𝑛−1_𝑙=1𝑒_𝑙𝑒_𝑛−𝑙−12𝑢₁^𝑛−1_𝑙=1𝑒_𝑙𝑘_𝑛−𝑙,𝑘_𝑛+1=−𝑘_𝑛,𝑥−𝑢₁𝑒_𝑛−𝑟𝑘_𝑛−𝑠^𝑛−1_𝑙=1𝑘_𝑙𝑒_𝑛−𝑙−𝑢₁^𝑛−1_𝑙=1𝑘_𝑙𝑘_𝑛−𝑙, (35) because of 𝜕𝜕𝑡𝜆+𝑟+𝑠𝐸+𝑢₁𝐾=𝜕,𝜕𝑥𝐴+𝐵𝐸+𝜌𝐾 (36) where 𝐴=𝑚₀𝜆²+𝑚₁1𝜆+2𝑚₀𝑠−𝑚₀𝑢₁𝑢₂,𝐵=𝑚₀1𝑠𝜆+2𝑚₀𝑠_𝑥−𝑚₀𝑟𝑠+𝑚₁𝑠,𝜌=𝑚₀𝑢₁𝜆+𝑚₀𝑢_1𝑥−𝑚₀𝑟𝑢₁+𝑚₁𝑢₁. (37)

Assume that 𝛿=𝜆+𝑟+𝑠𝐸+𝑢₁𝐾,𝜃=𝐴+𝐵𝐸+𝜌𝐾. Then (36) can be written as 𝛿_𝑡=𝜃_𝑥, 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+^∞_𝑗=1𝛿_𝑗𝜆^−𝑗,𝜃=𝑚₀𝜆²+𝑚₁𝜆+𝑚₀𝑠+^∞_𝑗=1𝜃_𝑗𝜆^−𝑗, (38) where 𝑚₀,𝑚₁ are constants of integration. The first two conserved densities and currents are read as follows: 𝛿₁=12𝑠+𝑢₁𝑢₂,𝜃₁=𝑚₀14𝑠_𝑥−𝑠𝑟−𝑢₁𝑢_2𝑥+𝑢₂𝑢_1𝑥−2𝑟𝑢₁𝑢₂+𝑚₁12𝑠+𝑢₁𝑢₂,𝛿₂1=−2𝑠𝑟−𝑢₁𝑢_2𝑥−𝑟𝑢₁𝑢₂,𝜃₂=𝑚₀14𝑠𝑟_𝑥−12𝑠𝑢₂𝑢_2𝑥+𝑠𝑟²−𝑠𝑢₁𝑢₂−18𝑠²−14𝑟𝑠_𝑥+𝑢₁𝑢_2𝑥𝑥+𝑟_𝑥𝑢₁𝑢₂+3𝑟𝑢₁𝑢_2𝑥+2𝑟²𝑢₁𝑢₂−𝑢_1𝑥𝑢_2𝑥−𝑟𝑢_1𝑥𝑢₂−𝑚₁12𝑠𝑟+𝑢₁𝑢_2𝑥+𝑟𝑢₁𝑢₂. (39) The recursion relation for 𝛿_𝑛 and 𝜃_𝑛 are 𝛿_𝑛=𝑠𝑒_𝑛+𝑢₁𝑘_𝑛,𝜃_𝑛=𝑚₀𝑠_𝑛+1+12𝑠_𝑥𝑒_𝑛−𝑟𝑠𝑒_𝑛+𝑢₁𝑘_𝑛+1+𝑢_1𝑥𝑘_𝑛−𝑟𝑢₁𝑘_𝑛+𝑚₁𝑠𝑒_𝑛+𝑢₁𝑘_𝑛, (40) where 𝑒_𝑛 and 𝑘_𝑛 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 Broer-Kaup-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).