A New Super Extension of Dirac Hierarchy
We derive a new super extension of the Dirac hierarchy associated with a matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.
In 1984, Kupershmidt  proposed a fermionic extension of the KdV equation with a Lax pair and a local super bi-Hamiltonian structure. Then, super integrable systems have received much attention with the development of integrable systems. Many experts and scholars do research on the topic. So far, many classical integrable hierarchies have been extended to the super ones by adding fermion fields, such as the super AKNS hierarchy [2, 3], the super Kaup-Newell hierarchy , the super Dirac hierarchy [2, 5], and the super Kadomtsev-Petviashvili (KP) hierarchy, and so on [6–10].
The Dirac hierarchy is based on the spectral problem and a super extension Dirac hierarchy can be constructed by the matrix super spectral  where , , and are fermion fields. It reduces to the spectral Dirac system (1) as .
In this paper, we consider a new matrix super spectral problem which generates a generalized super Dirac hierarchy with four Fermi variables. As we will show, this spectral problem takes the spectral Dirac system (1) and the super Dirac system (2) as special cases.
The paper is organized as follows. In the Section 2, we will construct a generalized super Dirac hierarchy related to the matrix super spectral problem and consider some special reductions. In Section 3, we prove the localness of the whole super soliton hierarchy. In Section 4, we present the super Hamiltonian structures for the generalized super Dirac hierarchy with the help of the super trace identity. The last section contains concluding remarks.
2. Super Extension Dirac Hierarchy
In this section, we will derive a generalized super Dirac hierarchy. To this end, we take a matrix super spectral problem where , , , , and are the commuting variables, which can be indicated by the degree (mod 2) as and , , , , and are the anticommuting variables, which can be indicated by the degree as . Here is assumed to be a constant spectral parameter.
We first solve the stationary zero-curvature equation where and each entry with the function of with , : Substituting (6) into (4) yields We put , and to be polynomials of : Substituting (8) into (7) and equating the coefficients of , we obtain Upon choosing the initial data then the recursion relations in (9) uniquely define a series of sets of differential polynomial functions in with respect to . The first two sets are as follows: From the recursion relations in (9), we can obtain the hereditary recursion operator which satisfies that where
Let satisfy the spectral problem (3) and an auxiliary problem where where denotes the polynomial part of .
The first nontrivial member in the hierarchy (17) is as follows: When and in (17), we can obtain the second-order nonlinear super integrable equations which are, respectively, reduced to the super Dirac equations or the famous Dirac equations when , , or .
We note that the recursion operator is an integrodifferential operator, but the generalized super Dirac system (17) are pure differential equations.
We put then, , , and satisfy the following equations: From (23), we obtain Integrating (24) with respect to , we get Substituting (22) into (25) and equating the coefficients of on both sides of the equation, we have We point that the left side of (27) involves an arbitrary constant . Based on (27) and (9), it is easy to find that , and are all differential polynomials in six variables , and . So the hierarchy of the super integrable equations (17) is pure differential equations.
4. The Super Hamiltonian Structures
Through direct calculations, we have Substituting the above results into the super trace identity (28) yields that Comparing the coefficients of on both sides of (31) gives rise to By employing the computing formula (29) on the constant , we obtain . Thus, we have It then follows that the super integrable hierarchy (17) possesses the following super Hamiltonian form: where the super Hamiltonian operator is given by
5. Concluding Remarks
In this paper, based on a matrix super spectral problem, we considered the related four Fermi component super Dirac-type systems and also proved the localness of the whole super soliton hierarchy. We obtained the super Hamiltonian structure and different reductions for the super integrable equations. Our computation reflects how to choose appropriate Lie algebras to generate soliton hierarchies , and the generating procedure can be applied to the other super soliton hierarchies. Let us notice that the super integrable hierarchy (17) allows for an arbitrary constant and such system is interesting since different specifications of lead to different super Dirac type equations. The nonlinearization approach for integrable systems is a powerful tool to generate finite-dimensional integrable Hamiltonian systems. The super integrable system (17) may admit nonlinearization. Moreover, the super Dirac system (17) may inherit various other integrable characteristics, such as first-degree time-dependent symmetries  and Bcklund transformation . In particular, it is of interest to study multi-integrable couplings and the corresponding super Hamiltonian structures of the super Dirac system (17) by the super variational identity . These related issues may be considered in further publication.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
This work was supported by Program for Liaoning Excellent Talents in University (Grant no. LJQ2011136), the Natural Science Foundation of China (Grant nos. 61372195 and 61304069), the Key Technologies R & D Program of Liaoning Province (Grant no. 2011224006), and Scientific Research Fund of Liaoning Provincial Education Department (Grant no. LGBS-1302).
W. X. Ma, “Integrable couplings and matrix loop algebra,” in Proceedings of the Nonlinear and Modern Mathe-Matical Physics, W. X. Ma and D. Kaup, Eds., vol. 1562 of AIP Conference Proceedings, pp. 105–122, American Institute of Physics, Melville, NY, USA, 2013.View at: Google Scholar