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

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 attractedmuch 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 u 1 and u 2 play an important role; namely, u 1 and u 2 satisfy u 1 = u 2

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][20][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][26][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 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 1 and 2 play an important role; namely, 1 and 2 satisfy 2 1 = 2 2 = 0 and 1 2 = − 2 1 2 Journal of Applied Mathematics in the whole paper. Furthermore, the operation between extended Fermi variables satisfies Grassmann algebra conditions.
The compatibility of (2) gives rise to the well-known zero curvature equation as follows: If an equation can be worked out through (3), we call (4) a super evolution equation. If there is a super Hamiltonian operator and a function such that where 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 distinct , = 1, 2, . . . , , the systems of (2) become as follows: Based on the result in [30], we can show that the following equation: holds true, where are constants. Equation (8) determines a finite dimensional invariant set for the flows in (6). From (7), we may know that where tr denotes the trace of a matrix and Ψ = ( From (8) and (9), a kind of super Hamiltonian soliton equation hierarchy with self-consistent sources is presented as follows:

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 From (7) and (12), we have Expand , in the power of as follows: Substituting (33) into (32) and comparing the coefficients of the same power of , we obtain Journal of Applied Mathematics 5 and a recursion formula for and because of Assume that = + + + 1 , = + + . Then (36) can be written as = , which is the right form of conservation laws. We expand and as series in powers of with the coefficients, which are called conserved densities and currents, respectively, where 0 , 1 are constants of integration. The first two conserved densities and currents are read as follows: The recursion relation for and are = + 1 , = 0 ( +1 + 1 2 − + 1 +1 + 1 − 1 ) where and can be calculated from (35). The infinitely many conservation laws of (20) can be easily obtained from (32)-(40), respectively.

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.