A New Super Extension of Dirac Hierarchy

and Applied Analysis 3


Introduction
In 1984, Kupershmidt [1] 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 [4], the super Dirac hierarchy [2,5], and the super Kadomtsev-Petviashvili (KP) hierarchy, and so on [6][7][8][9][10].
The Dirac hierarchy is based on the spectral problem and a super extension Dirac hierarchy can be constructed by the matrix super spectral [2] ( where  3 , , and  are fermion fields.It reduces to the spectral Dirac system (1) as  =  = 0.
In this paper, we consider a new 3 × 3 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 3 × 3 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.

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 , , ,  1 , and  2 are the commuting variables, which can be indicated by the degree (mod 2)  as () = () = () = ( 1 ) = ( 2 ) = 0 and  1 ,  2 ,  1 ,  2 , and  3 are the anticommuting variables, which can be indicated by the degree  as ( Here  is assumed to be a constant spectral parameter.

Abstract and Applied Analysis
We first solve the stationary zero-curvature equation where 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   .
Then, the compatibility condition of ( 3) and ( 14) yields the zero-curvature equation which is equivalent to a hierarchy of generalized super Dirac equations ) The first nontrivial member in the hierarchy ( 17) is as follows: When  = 2 and  0 = 0 in (17), we can obtain the secondorder nonlinear super integrable equations which are, respectively, reduced to the super Dirac equations or the famous Dirac equations when

Concluding Remarks
In this paper, based on a 3 × 3 matrix super spectral problem, we considered the related four Fermi component super Diractype 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 [12], 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  0 and such system is interesting since different specifications of  0 lead to different super Dirac type equations.The nonlinearization approach for integrable systems is a powerful tool to generate finitedimensional 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 [13] and B ä cklund transformation [14].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 [15].These related issues may be considered in further publication.