Integrable Couplings: Generation, Hamiltonian Structures, Conservation Laws, and ApplicationsView this Special Issue
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Super-Hamiltonian Structures and Conservation Laws of a New Six-Component Super-Ablowitz-Kaup-Newell-Segur Hierarchy
A six-component super-Ablowitz-Kaup-Newell-Segur (-AKNS) hierarchy is proposed by the zero curvature equation associated with Lie superalgebras. Supertrace identity is used to furnish the super-Hamiltonian structures for the resulting nonlinear superintegrable hierarchy. Furthermore, we derive the infinite conservation laws of the first two nonlinear super-AKNS equations in the hierarchy by utilizing spectral parameter expansions. PACS: 02.30.Ik; 02.30.Jr; 02.20.Sv.
It is well known that many physically important integrable partial differential equations belong to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [1–3], such as the KdV equation, the mKdV equation, the nonlinear Schrödinger equation, the Sin-Gordon equation, and the mixed KdV-mKdV equation. The AKNS hierarchy is based on the Zakharov and Shabat  spectral problem It possesses Lax representation, Hamiltonian structures, and infinitely many conserved quantities and can be solved by the method of inverse scattering, Hirota method, Darboux transform, and others.
The superintegrable systems had already aroused strong interest in theoretical physics [5, 6], where the fermion fields are added and equally treated with the boson fields. Many classical integrable equations have been extended to the super ones by adding fermion fields, such as the super-AKNS [6–10], the super-KdV , the super-Dirac [9, 11, 12], and the super-Kadomtsev Petviashvili (KP) [13–15]. The super-AKNS hierarchy was first proposed in  based on the superalgebra sl(2,R). Extension of this work to other and higher dimensional superalgebras is given in . The super-AKNS matrix superspectral problem is where , , and are fermion fields. It reduces to the spectral AKNS’s system as .
In this paper, we consider a new matrix superspectral problem which generates a six-component super-AKNS hierarchy. As we will show this spectral problem takes the spectral AKNS’s system and super-AKNS’s system as special cases.
The paper is organized as follows. In Section 2, we will construct a six-component super-AKNS hierarchy related to the matrix superspectral problem. In Section 3, we present the super-Hamiltonian structures for the six-component super-AKNS hierarchy with the help of the supertrace identity. In Section 4, we consider some special reductions of the superintegrable hierarchy. In Section 5, we derive the infinite conservation laws for the associated hierarchy. The last section contains concluding remarks.
2. A Six-Component Super-AKNS Hierarchy
In this section, we will derive a new six-component super-AKNS hierarchy associated with the matrix spectral problem. Let be a commutative superalgebra over and a matrix loop superalgebra over with the nondegenerate Killing form. We take a matrix superspectral problem where and denote the partial derivatives with respect to and , is a potential consisting of commuting and anticommuting variables, are the commuting variables, which can be indicated by the degree as , and are the anticommuting variables, which can be indicated by the degree as . Here is assumed to be a constant spectral parameter (i.e., ).
Let us find the following temporal evolution equation associated with (3): where , and are commuting fields and , and are anticommuting fields. From the stationary zero curvature equation it gives rise to We put , and to be polynomial of : and substituting (7) into (6) and equating the coefficients of , we obtain Upon choosing the initial data then the recursion relations in (8) 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 (8), we can obtain the hereditary recursion operator which satisfies that where Taking here denotes the polynomial part of .
The compatibility conditions (i.e., zero curvature equation) of the matrix superspectral problems determine a new six-component super-AKNS integrable soliton hierarchy
3. The Super-Hamiltonian Structures
In this section, we will establish the super-Hamiltonian structure of the six-component super-AKNS hierarchy by supertrace identity [9, 17] where the constant is determined by Through direct calculations, we have Substituting the above results into the supertrace identity (17) yields that Comparing the coefficients of on both sides of (20) gives rise to By employing the computing formula (18) on the constant , we obtain . Thus we have It then follows that the superintegrable hierarchy (16) possesses the following super-Hamiltonian form: where the super-Hamiltonian operator is given by
We now consider the possible reductions of our six-component super-AKNS hierarchy.
When in (23), we obtain the first-order nonlinear superintegrable equations Taking in (23), we can obtain the second-order nonlinear superintegrable equations In particular, letting in (26), we have and taking , (26) becomes If we choose , , and , (26) can be reduced to the second-order super-AKNS equations  which is just the coupled nonlinear Schrödinger equations, also called Manakov equations as , , , , and .
5. Infinite Conservation Laws
In what follows, we will derive infinite conservation laws of (25) and (26). From the spectral problem (3), we can introduce the variables and then we obtain Next, we expand and as series of the spectral parameter , where are even, , and are odd, .
By substituting (33) into (32) and comparing the coefficients of , we raise the recursion formulas for and , We write below the first few terms of and : On the other hand, it is easy to see that which implies and then the form of the conservation law is with the assumption that , .
With regard to (25), we have Expanding and as then we have with the coefficients and , which are called conserved densities and currents, respectively. Then the first two conserved densities and currents of (25) read So the first two conservation laws of (25) are revealed as
For (26), one infers If we write and as the first two are The recursion relations for and are as follows: where and can be recursively calculated from (34). Then we display the first two conservation laws of (26) as where , and are defined in (46).
6. Concluding Remarks
In this paper, we proposed a six-component super-AKNS system from a matrix superspectral problem. We obtained the super-Hamiltonian structure and different reductions for the superintegrable equations. Infinitely many conservation laws were also considered. The obtained results supplement the existing theories on the superintegrable systems. The bosonization approach for supersymmetric systems is a powerful tool to generate exact solutions. The superintegrable system (23) may admit bosonization. This and other related issues may be considered in further publication.
Conflict of Interests
The authors declare that there is 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 (Grants nos. 11401392, 61372195, and 61304069), and the Key Technologies R & D Program of Liaoning Province (Grant no. 2011224006).
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