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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 416520, 10 pages
A Matrix Lie Superalgebra and Its Applications
1School of Information Engineering, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
2School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
3School of Science, Ningbo University, Ningbo, Zhejiang 315211, China
Received 24 August 2013; Accepted 30 November 2013
Academic Editor: Xing Biao Hu
Copyright © 2013 Jingwei Han 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.
A matrix Lie superalgebra is established. As its applications, multicomponent super Ablowitz-Kaup-Newell-Segur (AKNS) equations and multicomponent super Dirac equations are constructed. By making use of supertrace identity, their super-Hamiltonian structures are presented, respectively.
It is well-known that by making use of the trace identity, many integrable equations can be written as the Hamiltonian forms, which was first proposed by Tu in . The key ideas include the following three aspects. Firstly, from the finite-dimensional Lie algebra Tu constructs a loop Lie algebra where . Secondly, an iso-spectral problem () is considered, where is a spectral parameter and is a potential. By solving the adjoint representation equation and zero curvature equations where plus denotes the choice of the nonnegative power of , Tu obtains integrable equations. Lastly, by using the trace identity where is a undetermined constant, the obtained integrable equations can be written as the Hamiltonian form. Some well-known integrable Hamiltonian equations have been obtained by this method, such as AKNS equations, Kaup-Newll (KN) equations, and Wadati-Konno-Ichikawa (WKI) equations.
This method for constructing Hamiltonian structures of integrable equations was extended to superintegrable equations by Hu in , where the supertrace identity was first proposed by Hu in  and proved by Ma et al. in . Similarly, many integrable super-Hamiltonian equations have also been constructed, such as super AKNS equations, super Dirac equations , super coupled Korteweg-de Vries (cKdV) equations [5, 6], and super KN equations [7, 8].
It is a valuable generalization from one component to multicomponent in soliton equations because multicomponent soliton equations possess more complex structure and become more extensive than one-component ones. In physics, multicomponent integrable system is widely applied. For example, mixed N-coupled nonlinear Schr####^~^~^~^~^~^####xf6;dinger equations (NLS), two-component Bose-Einstein condensates (BECs), and coupled Schr####^~^~^~^~^~^####xf6;dinger equation are, respectively, discussed in [9####^~^~^~^~^~^####x2013;11], and they obtain many new results for these multicomponent equations. In mathematics, the inverse scattering method provides us a powerful tool to find multicomponent extensions of one-component soliton equations, like multicomponent AKNS equations [12, 13], multicomponent NLS equation , and so on [15####^~^~^~^~^~^####x2013;18]. Moreover, from the point of mathematics, we find that multicomponent generalizations mainly include the following aspects:(1)symmetric space [14, 19, 20],(2)matrix algebra [13, 21####^~^~^~^~^~^####x2013;23],(3)soliton hierarchy associated with a matrix pseudodifferential operator [24, 25].
Based on results from the above analysis, some questions are listed as follows.(1)Do multicomponent superintegrable equations exist?(2)If they exist, how can the multicomponent superintegrable equations be constructed?(3)By making use of the supertrace identity (7), can multicomponent superintegrable equations be written as the super-Hamiltonian form?
The purpose of this paper is to answer these questions. The paper is organized as follows. In the next section, we construct a matrix Lie superalgebra . As its applications, multicomponent super AKNS equations and multicomponent super Dirac equations are, respectively, constructed in Sections 3 and 4. Their super bi-Hamiltonian forms are also, respectively, constructed in these sections. Some conclusions and discussions are listed in the last section.
2. A Matrix Lie Superalgebra
Let us start with the following linear space : where is a unit matrix, is a zero matrix, is even, is odd, is the super Lie bracket, and denotes the parity of the arbitrary element .
It is easy to prove that the linear space is a matrix Lie superalgebra . The corresponding loop superalgebra is presented as
3. Multicomponent Super AKNS Hierarchy and Its Super-Hamiltonian Structure
In this section, we will derive multicomponent super AKNS hierarchy by the above matrix Lie superalgebra, and further, its super-Hamiltonian structure will be constructed.
Let us consider the following spectral problem: which can be written in the following matrix form: where , , , , , , (). Note that . Taking where ,,,, and, the adjoint representation equation (4) gives where . Let , , , , and and then (13) becomes where . Moreover, we find that (14) can be written as the following recursive form: where the recursive operator is given by where
After a direct calculation, we obtain that . Choosing and constant of integration to be zero, the first few terms are listed as follows: where .
Then, let us consider the spectral problem (11) with the following auxiliary spectral problem: The compatibility conditions of (11) and (19), that is, the zero curvature equations (5) give an infinite hierarchy of nonlinear partial differential equations: or where ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, , , , with . When , (21) is equivalent to the super AKNS soliton hierarchy [27####^~^~^~^~^~^####x2013;30], and thus (21) is called multicomponent super AKNS hierarchy.
In what follows, a super-Hamiltonian structure of (21) is derived by means of the supertrace identity (7). To this end, the following quantities are needed: Thus, the supertrace identity (7) gives the following equality: Equating the coefficients of on two sides of the above equality, we have By taking , we obtain that the constant . Thus, we have where
Therefore, (21) can be written in the following super bi-Hamiltonian form: where is the supersymplectic operator.
Example 1. Let in (21), and we have which is the first nonlinear two-component super AKNS equations.
4. MultiComponent Super Dirac Hierarchy and Its Super-Hamiltonian Structure
As another application of the matrix Lie superalgebra, multicomponent super Dirac hierarchy will be constructed, and further, its super-Hamiltonian structure will also be obtained.
Let us consider the following spectral problem: which can be written in the following matrix form: where , , , , ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, and ####^~^~^~^~^~^####x2009; (). . Solving the adjoint representation equation (4), where with , , , , and , we have where . Let , , , and and then we have where , which can be written as a recursive form: where the recursive operator is given by with
It is easy to verify that . Choosing and constant of integration to be zero, the first few terms can be worked out as follows:
In what follows, the spectral problem (29) is associated with the auxiliary spectral problem: whereThe compatibility condition between (29) and (37) gives the following super nonlinear soliton hierarchy: where . By denoting , , , , and , with , and (39) can be written as follows: Similarly, when , (40) is equivalent to the super Dirac soliton hierarchy [4, 30], and thus (40) is called multicomponent super Dirac hierarchy.
To derive super bi-Hamiltonian structure of (40), we need to use the supertrace identity (7). To this end, we firstly obtain the following equalities: Thus, the supertrace identity (7) becomes the following equality: Equating the coefficients of in the above equality, we have Let , and we have . Thus, we obtain Therefore, (39) or (40) can be written in the following super bi-Hamiltonian structure: where is the supersymplectic operator.
Example 2. Let , in (45), and we obtain two-component super Dirac equations:
5. Conclusions and Discussions
Starting from the matrix Lie superalgebra , we have, respectively, constructed multicomponent super AKNS equations (21) and multicomponent super Dirac equations (40). By making use of the supertrace identity (7), (21), and (40) have been rewritten as integrable super-Hamiltonian forms (26) and (45), respectively. Moreover, we believe that many multicomponent superintegrable equations can also be constructed, which may be helpful to many physical and mathematical researchers in their future work.
This work is supported by the National Natural Science Foundation of China under Grant nos. 10971109, 11001069, 11271210, and 61273077 and Zhejiang Provincial Natural Science Foundation of China under Grant nos. LQ12A01002 and LQ12A01003.
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