A Matrix Lie Superalgebra and Its Applications

Abstract

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.

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Jingwei Han 1 and Jing Yu 2 and Jingsong He 3

Academic Editor:Xing Biao Hu

1, School of Information Engineering, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
2, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
3, School of Science, Ningbo University, Ningbo, Zhejiang 315211, China

Received 24 August 2013; Accepted 30 November 2013

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.

1. Introduction

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 [1]. The key ideas include the following three aspects. Firstly, from the finite-dimensional Lie algebra [figure omitted; refer to PDF] Tu constructs a loop Lie algebra [figure omitted; refer to PDF] where _{E i} ( n ) = _{E i} [ecedil]7; ^{λ n} = _{E i} ^{λ n} . Secondly, an iso-spectral problem ( _{λ t n} = 0 ) [figure omitted; refer to PDF] is considered, where λ is a spectral parameter and u is a potential. By solving the adjoint representation equation [figure omitted; refer to PDF] and zero curvature equations [figure omitted; refer to PDF] where plus denotes the choice of the nonnegative power of λ , Tu obtains integrable equations. Lastly, by using the trace identity [figure omitted; refer to PDF] where γ 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 [2], where the supertrace identity [figure omitted; refer to PDF] was first proposed by Hu in [3] and proved by Ma et al. in [4]. Similarly, many integrable super-Hamiltonian equations have also been constructed, such as super AKNS equations, super Dirac equations [4], 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 Schrödinger equations (NLS), two-component Bose-Einstein condensates (BECs), and coupled Schrödinger equation are, respectively, discussed in [9-11], 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 [14], and so on [15-18]. 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, 21-23],

(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 A ( 2 m - 1 , m - 1 ) [26]. 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 G = { _{e 1} , _{e 2} , _{e 3} , _{e 4} , _{e 5} } : [figure omitted; refer to PDF] where E is a m × m unit matrix, O is a m × m zero matrix, _{G 0} = { _{e 1} , _{e 2} , _{e 3} } is even, _{G 1} = { _{e 4} , _{e 5} } is odd, [ a , b } = a b - ( - 1 ^{) p ( a ) p ( b )} b a is the super Lie bracket, and p ( f ) denotes the parity of the arbitrary element f .

It is easy to prove that the linear space G is a matrix Lie superalgebra A ( 2 m - 1 , m - 1 ) . The corresponding loop superalgebra G ~ is presented as [figure omitted; refer to PDF]

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: [figure omitted; refer to PDF] which can be written in the following matrix form: [figure omitted; refer to PDF] where v = diag ( _{v 1} , ... , _{v m} ) , w = diag ( _{w 1} , ... , _{w m} ) , α = diag ( _{α 1} , ... , _{α m} ) , β = diag ( _{β 1} , ... , _{β m} ) , u = ( v , w , α , β ^{) T} , p ( λ ) = p ( _{v j} ) = p ( _{w j} ) = 0 , and p ( _{α j} ) = p ( _{β j} ) = 1 ( 1 ...4; j ...4; m ). Note that U ( u , λ ) ∈ A ( 2 m - 1 , m - 1 ) . Taking [figure omitted; refer to PDF] where a = diag ( _{a 1} , ... , _{a m} ) , b = diag ( _{b 1} , ... , _{b m} ) , c = diag ( _{c 1} , ... , _{c m} ) , ρ = diag ( _{ρ 1} , ... , _{ρ m} ) , and δ = diag ( _{δ 1} , ... , _{δ m} ) , the adjoint representation equation (4) gives [figure omitted; refer to PDF] where 1 ...4; k ...4; m . Let _{a k} = _{∑ j ...5; 0} ^{a k ( j ) λ - j} , _{b k} = _{∑ j ...5; 0} ^{b k ( j ) λ - j} , _{c k} = _{∑ j ...5; 0} ^{c k ( j ) λ - j} , _{ρ k} = _{∑ j ...5; 0} ^{ρ k ( j ) λ - j} , and _{δ k} = _{∑ j ...5; 0} ^{δ k ( j ) λ - j} , and then (13) becomes [figure omitted; refer to PDF] where 1 ...4; k ...4; m . Moreover, we find that (14) can be written as the following recursive form: [figure omitted; refer to PDF] where the recursive operator is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

After a direct calculation, we obtain that ^{a k , x ( 0 )} = 0 . Choosing ^{a k ( 0 )} = - 1 ( 1 ...4; k ...4; m ) and constant of integration to be zero, the first few terms are listed as follows: [figure omitted; refer to PDF] where 1 ...4; k ...4; m .

Then, let us consider the spectral problem (11) with the following auxiliary spectral problem: [figure omitted; refer to PDF] The compatibility conditions of (11) and (19), that is, the zero curvature equations (5) give an infinite hierarchy of nonlinear partial differential equations: [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where ^{a ( j )} = diag ( ^{a 1 ( j )} , ... , ^{a m ( j )} ) , ^{b ( j )} = diag ( ^{b 1 ( j )} , ... , ^{b m ( j )} ) , ^{c ( j )} = diag ( ^{c 1 ( j )} , ... , ^{c m ( j )} ) , ^{ρ ( j )} = diag ( ^{ρ 1 ( j )} , ... , ^{ρ m ( j )} ) , ^{δ ( j )} = diag ( ^{δ 1 ( j )} , ... , ^{δ m ( j )} ) , with j ...5; 0 . When m = 1 , (21) is equivalent to the super AKNS soliton hierarchy [27-30], 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: [figure omitted; refer to PDF] Thus, the supertrace identity (7) gives the following equality: [figure omitted; refer to PDF] Equating the coefficients of ^{λ - j - 2} on two sides of the above equality, we have [figure omitted; refer to PDF] By taking j = 0 , we obtain that the constant γ = 0 . Thus, we have [figure omitted; refer to PDF] where _{H j} = ∫ ... ( 2 / ( j + 1 ) ) ^{∑ k = 1 m a k ( j + 2 )} d x .

Therefore, (21) can be written in the following super bi-Hamiltonian form: [figure omitted; refer to PDF] where J = ( 0 - 2 E 0 0 2 E 0 0 0 0 0 0 ( 1 / 2 ) E 0 0 ( 1 / 2 ) E 0 ) is the supersymplectic operator.

Example 1.

Let m = 2 , n = 2 in (21), and we have [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] which can be written in the following matrix form: [figure omitted; refer to PDF] where r = diag ( _{r 1} , ... , _{r m} ) , s = diag ( _{s 1} , ... , _{s m} ) , α = diag ( _{α 1} , ... , _{α m} ) , β = diag ( _{β 1} , ... , _{β m} ) , u ~ = ( r , s , α , β ^{) T} , p ( λ ) = p ( _{r i} ) = p ( _{s i} ) = 0 , and p ( _{α i} ) = p ( _{β i} ) = 1 ( 1 ...4; i ...4; m ). U ( u ~ , λ ) ∈ A ( 2 m - 1 , m - 1 ) . Solving the adjoint representation equation (4), where [figure omitted; refer to PDF] with A = diag ( _{A 1} , ... , _{A m} ) , B = diag ( _{B 1} , ... , _{B m} ) , C = diag ( _{C 1} , ... , _{C m} ) , ρ = diag ( _{ρ 1} , ... , _{ρ m} ) , and δ = diag ( _{δ 1} , ... , _{δ m} ) , we have [figure omitted; refer to PDF] where 1 ...4; k ...4; m . Let _{A k} = _{∑ j ...5; 0} ^{A k ( j ) λ - j} , _{B k} = _{∑ j ...5; 0} ^{B k ( j ) λ - j} , _{C k} = _{∑ j ...5; 0} ^{C k ( j ) λ - j} , _{ρ k} = _{∑ j ...5; 0} ^{ρ k ( j ) λ - j} , and _{δ k} = _{∑ j ...5; 0} ^{δ k ( j ) λ - j} , and then we have [figure omitted; refer to PDF] where j ...5; 0,1 ...4; k ...4; m , which can be written as a recursive form: [figure omitted; refer to PDF] where the recursive operator is given by [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

It is easy to verify that ^{B k , x ( 0 )} = 0 ( 1 ...4; k ...4; m ) . Choosing ^{B k ( 0 )} = 1 and constant of integration to be zero, the first few terms can be worked out as follows: [figure omitted; refer to PDF]

In what follows, the spectral problem (29) is associated with the auxiliary spectral problem: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The compatibility condition between (29) and (37) gives the following super nonlinear soliton hierarchy: [figure omitted; refer to PDF] where 1 ...4; k ...4; m . By denoting ^{A ( j )} = diag ( ^{A 1 ( j )} , ... , ^{A m ( j )} ) , ^{B ( j )} = diag ( ^{B 1 ( j )} , ... , ^{B m ( j )} ) , ^{C ( j )} = diag ( ^{C 1 ( j )} , ... , ^{C m ( j )} ) , ^{ρ ( j )} = diag ( ^{ρ 1 ( j )} , ... , ^{ρ m ( j )} ) , and ^{δ ( j )} = diag ( ^{δ 1 ( j )} , ... , ^{δ m ( j )} ) , with j ...5; 0 , and (39) can be written as follows: [figure omitted; refer to PDF] Similarly, when m = 1 , (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: [figure omitted; refer to PDF] Thus, the supertrace identity (7) becomes the following equality: [figure omitted; refer to PDF] Equating the coefficients of ^{λ - j - 2} in the above equality, we have [figure omitted; refer to PDF] Let j = 0 , and we have γ = 0 . Thus, we obtain [figure omitted; refer to PDF] Therefore, (39) or (40) can be written in the following super bi-Hamiltonian structure: [figure omitted; refer to PDF] where J ~ = ( 0 2 E 0 0 - 2 E 0 0 0 0 0 E 0 0 0 0 E ) is the supersymplectic operator.

Example 2.

Let m = 2 , n = 2 in (45), and we obtain two-component super Dirac equations: [figure omitted; refer to PDF]

5. Conclusions and Discussions

Starting from the matrix Lie superalgebra A ( 2 m - 1 , m - 1 ) , 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.

Acknowledgments

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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