## Integrable Couplings: Generation, Hamiltonian Structures, Conservation Laws, and Applications

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# Super-Hamiltonian Structures and Conservation Laws of a New Six-Component Super-Ablowitz-Kaup-Newell-Segur Hierarchy

**Academic Editor:**Huanhe Dong

#### Abstract

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.

#### 1. Introduction

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 [4] 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 [5], the super-Dirac [9, 11, 12], and the super-Kadomtsev Petviashvili (KP) [13–15]. The super-AKNS hierarchy was first proposed in [6] based on the superalgebra sl(2,R). Extension of this work to other and higher dimensional superalgebras is given in [16]. 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 note that the recursion operator is an integrodifferential operator, but the generalized superintegrable system (23) is pure differential equations according to [12].

#### 4. Reductions

We now consider the possible reductions of our six-component super-AKNS hierarchy.

Assuming , hierarchy (23) reduces to the classical AKNS hierarchy [1]. Taking , , we can have the super-AKNS hierarchy [6, 9, 10].

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 [10] 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.

#### Acknowledgments

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).

#### References

- M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,”
*Studies in Applied Mathematics*, vol. 53, no. 4, pp. 249–315, 1974. View at: Google Scholar | MathSciNet - F. C. You and T. C. Xia, “Integrable couplings of the generalized AKNS hierarchy with an arbitrary function and its bi-Hamiltonian structure,”
*International Journal of Theoretical Physics*, vol. 46, no. 12, pp. 3159–3168, 2007. View at: Publisher Site | Google Scholar | MathSciNet - W. X. Ma, “Integrable couplings and matrix loop algebra,” in
*Nonlinear and Modern Mathematical 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 - V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,”
*Soviet Physics-JETP*, vol. 34, pp. 62–69, 1973. View at: Google Scholar - B. A. Kupershmidt, “A super Korteweg-de Vries equation: an integrable system,”
*Physics Letters A*, vol. 102, no. 5-6, pp. 213–215, 1984. View at: Publisher Site | Google Scholar | MathSciNet - M. Gürses and Ö. Oǧuz, “A super AKNS scheme,”
*Physics Letters A*, vol. 108, no. 9, pp. 437–440, 1985. View at: Publisher Site | Google Scholar | MathSciNet - M. Gürses and Ö. Oğuz, “A super soliton connection,”
*Letters in Mathematical Physics*, vol. 11, no. 3, pp. 235–246, 1986. View at: Publisher Site | Google Scholar | MathSciNet - Y. S. Li and L. N. Zhang, “Hamiltonian structure of the super evolution equation,”
*Journal of Mathematical Physics*, vol. 31, no. 2, pp. 470–475, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Ma, J. He, and Z. Qin, “A supertrace identity and its applications to superintegrable systems,”
*Journal of Mathematical Physics*, vol. 49, no. 3, Article ID 033511, 2008. View at: Publisher Site | Google Scholar | MathSciNet - F. C. You, “Nonlinear super integrable Hamiltonian couplings,”
*Journal of Mathematical Physics*, vol. 52, no. 12, Article ID 123510, 2011. View at: Publisher Site | Google Scholar | MathSciNet - F. C. You, “Nonlinear super integrable couplings of super Dirac hierarchy and its super Hamiltonian structures,”
*Communications in Theoretical Physics*, vol. 57, no. 6, pp. 961–966, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. Zhang, F. C. You, and Y. Zhao, “A new super extension of Dirac hierarchy,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 472101, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Q. P. Liu, “Darboux transformations for supersymmetric Korteweg-de Vries equations,”
*Letters in Mathematical Physics*, vol. 35, no. 2, pp. 115–122, 1995. View at: Publisher Site | Google Scholar | MathSciNet - X. B. Hu, “A powerful approach to generate new integrable systems,”
*Journal of Physics A: Mathematical and General*, vol. 27, no. 7, pp. 2497–2514, 1994. View at: Publisher Site | Google Scholar | MathSciNet - M. Gurses and O. Oguz, “A super soliton connection,”
*Letters in Mathematical Physics*, vol. 11, no. 3, pp. 235–246, 1986. View at: Publisher Site | Google Scholar | MathSciNet - M. Gurses, S. Salihoglu, and O. Oguz, “Integrable nonlinear partial differential equations on homogenous spaces,”
*International Journal of Modern Physics A*, vol. 5, no. 9, pp. 1801–1817, 1990. View at: Publisher Site | Google Scholar - X. B. Hu, “An approach to generate superextensions of integrable systems,”
*Journal of Physics A: Mathematical and General*, vol. 30, no. 2, pp. 619–632, 1997. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Fucai You et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.