AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 507540 10.1155/2014/507540 507540 Research Article Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System http://orcid.org/0000-0002-2660-9813 Yu Jing 1 http://orcid.org/0000-0001-7573-8446 Han Jingwei 2 Rui Weiguo 1 School of Science Hangzhou Dianzi University Hangzhou, Zhejiang 310018 China hdu.edu.cn 2 School of Information Engineering Hangzhou Dianzi University Hangzhou, Zhejiang 310018 China hdu.edu.cn 2014 3132014 2014 29 12 2013 04 03 2014 31 3 2014 2014 Copyright © 2014 Jing Yu and Jingwei Han. 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.

Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation and r -matrix are also given in this paper.

1. Introduction

The inverse scattering method provides us with a powerful tool to generate multicomponent soliton equations. In , they have constructed many multicomponent soliton equations, which are much more important for physicists and mathematicians than one-component ones, owing to the fact that they possess rich structure and have more extensive prospect.

Monononlinearization of Lax pair is a method to obtain finite-dimensional integrable Hamiltonian system, which was firstly proposed by Cao in . The main idea of monononlinearization includes the following three aspects. Firstly, they find a symmetry constraint between potential and eigenfunctions. Secondly, substituting the symmetry constraint into the spectral problem, they obtain constrained finite-dimensional system. Lastly, they show that obtained constrained system is Hamiltonian system and completely integrable in the Liouville sense. Many finite-dimensional integrable Hamiltonian systems are constructed in . This method was generalized by Ma and Strampp in . The main difference of binary-nonlinearization and monononlinearization lies in the following two aspects. One is that binary-nonlinearization needs to introduce adjoint spectral problem, and, thus, both spectral problem and adjoint spectral problem constitute even-dimensional system. The other is that symmetry constraint proposed in the procedure of binary-nonlinearization is not only associated with eigenfunctions but also associated with adjoint eigenfunctions. According to the Liouville integrable theorem which states that a 2 n -dimensional Hamiltonian system over some region Ω R 2 n with n independent integrals of motion in involution may be integrated by quadratures, we cannot apply monononlinearization to odd-dimensional spectral problem. However, we can apply both monononlinearization and binary-nonlinearization to the even-dimensional spectral problem. Many finite-dimensional integrable Hamiltonian systems are constructed in . Owing to one-component super integrable system which is associated with a 3 × 3 spectral matrix, we just consider binary-nonlinearization in our previous papers . From the above analysis, we propose the following questions. Does even-dimensional spectral problem of super integrable system exist? If it exists, can we apply monononlinearization to the super integrable system? All of these questions will be answered in this paper.

The paper is organized as follows. In Section 2, we derive two-component super AKNS system and write this new system as the super Hamiltonian form. In Section 3, we find a symmetry constraint which is only associated with eigenfunctions, and, after substituting the symmetry constraint into the N copies spectral problem, we obtained 6 N -dimensional constrained system. Furthermore, we rewrite the 6 N -dimensional system as the super Hamiltonian form. Lax representation and r -matrix of the constrained system are given in Section 4. In the last section, some conclusions and discussions are given.

2. Two-Component Super AKNS System

Let us start with the following linear space G = { e 1 , e 2 , e 3 , e 4 , e 5 } : (1) e 1 = ( E 0 0 0 - E 0 0 0 0 ) , e 2 = ( 0 E 0 0 0 0 0 0 0 ) , e 3 = ( 0 0 0 E 0 0 0 0 0 ) , e 4 = ( 0 0 E 0 0 0 0 - E 0 ) , e 5 = ( 0 0 0 0 0 E E 0 0 ) , where E = diag ( 1,1 ) is a 2 × 2 unit matrix, 0 is a 2 × 2 zero matrix, G 0 = { e 1 , e 2 , e 3 } is even, and G 1 = { e 4 , e 5 } is odd. After a direct calculation, we obtain that (2) [ e 1 , e 2 } = 2 e 2 , [ e 1 , e 3 } = - 2 e 3 , [ e 2 , e 3 } = e 1 , [ e 1 , e 4 } = e 4 , [ e 1 , e 5 } = - e 5 , [ e 2 , e 5 } = e 4 , [ e 3 , e 4 } = e 5 , [ e 4 , e 4 } = - 2 e 2 , [ e 4 , e 5 } = e 1 , [ e 5 , e 5 } = 2 e 3 , and the others are zeros, where (3) [ 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 matrix Lie superalgebra. The corresponding loop superalgebra G ~ is presented as (4) e i ( n ) = e i λ n , [ e 1 ( m ) , e 2 ( n ) } = 2 e 2 ( m + n ) , [ e 1 ( m ) , e 3 ( n ) } = - 2 e 3 ( m + n ) , [ e 2 ( m ) , e 3 ( n ) } = e 1 ( m + n ) , [ e 1 ( m ) , e 4 ( n ) } = e 4 ( m + n ) , [ e 1 ( m ) , e 5 ( n ) } = - e 5 ( m + n ) , [ e 2 ( m ) , e 4 ( n ) } = [ e 3 ( m ) , e 5 ( n ) } = 0 , [ e 2 ( m ) , e 5 ( n ) } = e 4 ( m + n ) , [ e 3 ( m ) , e 4 ( n ) } = e 5 ( m + n ) , [ e 4 ( m ) , e 4 ( n ) } = - 2 e 2 ( m + n ) , [ e 4 ( m ) , e 5 ( n ) } = e 1 ( m + n ) , [ e 5 ( m ) , e 5 ( n ) } = 2 e 3 ( m + n ) , deg e i ( n ) = n , ( i = 1,2 , 3,4 , 5 ) .

In what follows, we will construct multicomponent super integrable equations via the matrix Lie superalgebra G . Firstly, let us consider the following superisospectral problem: (5) ϕ x = U ( u , λ ) ϕ , where (6) U ( u , λ ) = - e 1 ( 1 ) + q e 2 ( 0 ) + r e 3 ( 0 ) + α e 4 ( 0 ) + β e 5 ( 0 ) , which can be written as the following matrix form: (7) ϕ x = ( - λ E q α r λ E β β - α 0 ) ϕ , where q = diag ( q 1 , q 2 ) , r = diag ( r 1 , r 2 ) , α = diag ( α 1 , α 2 ) , β = diag ( β 1 , β 2 ) , u = ( q , r , α , β ) T is a potential, λ is a spectral parameter which satisfies λ t n = 0 , and ϕ = ( ϕ 1 , , ϕ 6 ) T is an eigenfunction.

Taking (8) V = ( a b ρ c - a δ δ - ρ 0 ) , where a = diag ( a 1 , a 2 ) , b = diag ( b 1 , b 2 ) , c = diag ( c 1 , c 2 ) , ρ = diag ( ρ 1 , ρ 2 ) , and δ = diag ( δ 1 , δ 2 ) , the adjoint representation equation (9) V x = [ U , V ] leads to (10) a k , x = q k c k - r k b k + α k δ k + β k ρ k , k = 1,2 , b k , x = - 2 λ b k - 2 q k a k - 2 α k ρ k , k = 1,2 , c k , x = 2 λ c k + 2 r k a k + 2 β k δ k , k = 1,2 , ρ k , x = - λ ρ k + q k δ k - α k a k - β k b k , k = 1,2 , δ k , x = λ δ k + r k ρ k - α k c k + β k a k , k = 1,2 . On setting a k = m 0 a k ( m ) λ - m , b k = m 0 b k ( m ) λ - m , c k = m 0 c k ( m ) λ - m , ρ k = m 0 ρ k ( m ) λ - m , and δ k = m 0 δ k ( m ) λ - m ( k = 1,2 ) , (10) engenders equivalently (11) b k ( 0 ) = c k ( 0 ) = ρ k ( 0 ) = δ k ( 0 ) = 0 , k = 1,2 , a k , x ( m ) = q k c k ( m ) - r k b k ( m ) + α k δ k ( m ) + β k ρ k ( m ) , k = 1,2 , m 0 , b k , x ( m ) = - 2 b k ( m + 1 ) - 2 q k a k ( m ) - 2 α k ρ k ( m ) , k = 1,2 , m 0 , c k , x ( m ) = 2 c k ( m + 1 ) + 2 r k a k ( m ) + 2 β k δ k ( m ) , k = 1,2 , m 0 , ρ k , x ( m ) = - ρ k ( m + 1 ) + q k δ k ( m ) - α k a k ( m ) - β k b k ( m ) , hhhhhhhhhhhhlhhhhhhh k = 1,2 , m 0 , δ k , x ( m ) = δ k ( m + 1 ) + r k ρ k ( m ) - α k c k ( m ) + β k a k ( m ) , k = 1,2 , m 0 , which results in a recursion relation to determine a k ( m ) , b k ( m ) , c k ( m ) , ρ k ( m ) , δ k ( m ) ( k = 1,2 ) : (12) ( c 1 ( m + 1 ) c 2 ( m + 1 ) b 1 ( m + 1 ) b 2 ( m + 1 ) 2 δ 1 ( m + 1 ) 2 δ 2 ( m + 1 ) - 2 ρ 1 ( m + 1 ) - 2 ρ 2 ( m + 1 ) ) = L ( c 1 ( m ) c 2 ( m ) b 1 ( m ) b 2 ( m ) 2 δ 1 ( m ) 2 δ 2 ( m ) - 2 ρ 1 ( m ) - 2 ρ 2 ( m ) ) , where (13) L = ( L 11 L 12 L 13 L 14 L 21 L 22 L 23 L 24 L 31 L 32 L 33 L 34 L 41 L 42 L 43 L 44 ) , with (14) L 11 = diag ( 1 2 - r 1 - 1 q 1 , 1 2 - r 2 - 1 q 2 ) , L 12 = diag ( r 1 - 1 r 1 , r 2 - 1 r 2 ) , L 13 = diag ( - 1 2 r 1 - 1 α 1 - 1 2 β 1 , - 1 2 r 2 - 1 α 2 - 1 2 β 2 ) , L 14 = diag ( 1 2 r 1 - 1 β 1 , 1 2 r 2 - 1 β 2 ) , L 21 = diag ( - q 1 - 1 q 1 , - q 2 - 1 q 2 ) , L 22 = diag ( - 1 2 + q 1 - 1 r 1 , - 1 2 + q 2 - 1 r 2 ) , L 23 = diag ( - 1 2 q 1 - 1 α 1 , - 1 2 q 2 - 1 α 2 ) , L 24 = diag ( 1 2 α 1 + 1 2 q 1 - 1 β 1 , 1 2 α 2 + 1 2 q 2 - 1 β 2 ) , L 31 = diag ( 2 α 1 - 2 β 1 - 1 q 1 , 2 α 2 - 2 β 2 - 1 q 2 ) , L 32 = diag ( 2 β 1 - 1 r 1 , 2 β 2 - 1 r 2 ) , L 33 = diag ( - β 1 - 1 α 1 , - β 2 - 1 α 2 ) , L 34 = diag ( r 1 + β 1 - 1 β 1 , r 2 + β 2 - 1 β 2 ) , L 41 = diag ( 2 α 1 - 1 q 1 , 2 α 2 - 1 q 2 ) , L 42 = diag ( 2 β 1 - 2 α 1 - 1 r 1 , 2 β 2 - 2 α 2 - 1 r 2 ) , L 43 = diag ( - q 1 + α 1 - 1 α 1 , - q 2 + α 2 - 1 α 2 ) , L 44 = diag ( - - α 1 - 1 β 1 , - - α 2 - 1 β 2 ) . It is easy to obtain that a k , x ( 0 ) = 0 ( k = 1,2 ) . Therefore, we choose a k ( 0 ) = - 1 ( k = 1,2 ) and select constants of integral to be zero, which lead to the first few terms being worked out as follows: (15) a k ( 1 ) = 0 , b k ( 1 ) = q k , c k ( 1 ) = r k , ρ k ( 1 ) = α k , δ k ( 1 ) = β k , a k ( 2 ) = 1 2 q k r k + α k β k , b k ( 2 ) = - 1 2 q k , x , c k ( 2 ) = 1 2 r k , x , ρ k ( 2 ) = - α k , x , δ k ( 2 ) = β k , x , where k = 1,2 .

Secondly, let us associate (5) with the following auxiliary spectral problem: (16) ϕ t n = V ( n ) ϕ , where (17) V ( n ) = m = 0 n ( a 1 ( m ) 0 b 1 ( m ) 0 ρ 1 ( m ) 0 0 a 2 ( m ) 0 b 2 ( m ) 0 ρ 2 ( m ) c 1 ( m ) 0 - a 1 ( m ) 0 δ 1 ( m ) 0 0 c 2 ( m ) 0 - a 2 ( m ) 0 δ 2 ( m ) δ 1 ( m ) 0 - ρ 1 ( m ) 0 0 0 0 δ 2 ( m ) 0 - ρ 2 ( m ) 0 0 ) λ n - m . The compatibility condition of (5) and (16) leads to the famous zero curvature equations (18) U t n - V x ( n ) + [ U , V ( n ) ] = 0 , n 1 , which lead to isospectral super integrable equations (19) ( q 1 q 2 r 1 r 2 α 1 α 2 β 1 β 2 ) t n = ( - 2 b 1 ( n + 1 ) - 2 b 2 ( n + 1 ) 2 c 1 ( n + 1 ) 2 c 2 ( n + 1 ) - ρ 1 ( n + 1 ) - ρ 2 ( n + 1 ) δ 1 ( n + 1 ) δ 2 ( n + 1 ) ) . Under the special reduction of q 2 = r 2 = α 2 = δ 2 = 0 , (19) is equivalent to the super AKNS equations [15, 18, 19], and thus (19) is called a two-component super AKNS equation.

Lastly, super Hamiltonian structures of the two-component super AKNS equations (19) may be established by applying a powerful tool, that is, the so-called supertrace identity [20, 21] (20) δ δ u str ( V U λ ) d x = λ - γ λ λ γ str ( U u V ) . As is usual, we need the following equalities which are easy to calculate: (21) str ( V U λ ) = - 2 ( a 1 + a 2 ) , str ( U q k V ) = c k , str ( U r k V ) = b k , str ( U α k V ) = 2 δ k , str ( U β k V ) = - 2 ρ k , where k = 1,2 . Substituting the above equality (21) into the supertrace identity (20) and comparing the coefficients of λ - m - 2 on two sides, we arrive at (22) ( δ δ q 1 δ δ q 2 δ δ r 1 δ δ r 2 δ δ α 1 δ δ α 2 δ δ β 1 δ δ β 2 ) - 2 ( a 1 ( m + 2 ) + a 2 ( m + 2 ) ) d x = ( γ - m - 1 ) ( c 1 ( m + 1 ) c 2 ( m + 1 ) b 1 ( m + 1 ) b 2 ( m + 1 ) 2 δ 1 ( m + 1 ) 2 δ 2 ( m + 1 ) - 2 ρ 1 ( m + 1 ) - 2 ρ 2 ( m + 1 ) ) , which leads to the constant γ = 0 with m = 0 . Thus, we have (23) ( c 1 ( m + 1 ) c 2 ( m + 1 ) b 1 ( m + 1 ) b 2 ( m + 1 ) 2 δ 1 ( m + 1 ) 2 δ 2 ( m + 1 ) - 2 ρ 1 ( m + 1 ) - 2 ρ 2 ( m + 1 ) ) = ( δ δ q 1 δ δ q 2 δ δ r 1 δ δ r 2 δ δ α 1 δ δ α 2 δ δ β 1 δ δ β 2 ) H ¯ m , H ¯ m = 2 m + 2 ( a 1 ( m + 2 ) + a 2 ( m + 2 ) ) d x . Now, it follows from (23) that the two-component super AKNS system (19) has the following super bi-Hamiltonian structure: (24) u t n = ( q 1 q 2 r 1 r 2 α 1 α 2 β 1 β 2 ) t n = J ( c 1 ( n + 1 ) c 2 ( n + 1 ) b 1 ( n + 1 ) b 2 ( n + 1 ) 2 δ 1 ( n + 1 ) 2 δ 2 ( n + 1 ) - 2 ρ 1 ( n + 1 ) - 2 ρ 2 ( n + 1 ) ) = J δ H ¯ n δ u = M δ H ¯ n - 1 δ u , where the super Hamiltonian pair ( J , M = J L ) reads as (25) J = ( 0 - 2 E 0 0 2 E 0 0 0 0 0 0 1 2 E 0 0 1 2 E 0 ) , M = J L = ( - 2 L 21 - 2 L 22 - 2 L 23 - 2 L 24 2 L 11 2 L 12 2 L 13 2 L 14 1 2 L 41 1 2 L 42 1 2 L 43 1 2 L 44 1 2 L 31 1 2 L 32 1 2 L 33 1 2 L 34 ) , where L m n ( 1 m , n 4 ) are given by (13).

The first nonlinear equations in two-component AKNS system (24) are given by (26) q 1 , t 2 = - 1 2 q 1 , x x + q 1 2 r 1 + 2 q 1 α 1 β 1 - 2 α 1 α 1 , x , q 2 , t 2 = - 1 2 q 2 , x x + q 2 2 r 2 + 2 q 2 α 2 β 2 - 2 α 2 α 2 , x , r 1 , t 2 = 1 2 r 1 , x x - q 1 r 1 2 - 2 r 1 α 1 β 1 - 2 β 1 β 1 , x , r 2 , t 2 = 1 2 r 2 , x x - q 2 r 2 2 - 2 r 2 α 2 β 2 - 2 β 2 β 2 , x , α 1 , t 2 = - α 1 , x x - q 1 β 1 , x + 1 2 q 1 r 1 α 1 - 1 2 q 1 , x β 1 , α 2 , t 2 = - α 2 , x x - q 2 β 2 , x + 1 2 q 2 r 2 α 2 - 1 2 q 2 , x β 2 , β 1 , t 2 = β 1 , x x + r 1 α 1 , x + 1 2 r 1 , x α 1 - 1 2 q 1 r 1 β 1 , β 2 , t 2 = β 2 , x x + r 2 α 2 , x + 1 2 r 2 , x α 2 - 1 2 q 2 r 2 β 2 , and the corresponding temporal part of the Lax system is (27) ϕ t 2 = V ( 2 ) ϕ , where (28) V ( 2 ) = ( - λ 2 + v 1 0 q 1 λ - 1 2 q 1 , x 0 α 1 λ - α 1 , x 0 0 - λ 2 + v 2 0 q 2 λ - 1 2 q 2 , x 0 α 2 λ - α 2 , x r 1 λ + 1 2 r 1 , x 0 λ 2 - v 1 0 β 1 λ + β 1 , x 0 0 r 2 λ + 1 2 r 2 , x 0 λ 2 - v 2 0 β 2 λ + β 2 , x β 1 λ + β 1 , x 0 - α 1 λ + α 1 , x 0 0 0 0 β 2 λ + β 2 , x 0 - α 2 λ + α 2 , x 0 0 ) , with v k = ( 1 / 2 ) q k r k + α k β k ( k = 1,2 ) .

3. Finite-Dimensional Super Hamiltonian System

In this section, we will apply monononlinearization to the two-component super AKNS system (24). It is easy to find that, for the super spectral problem (1), variational derivative of λ with respect to the potential u reads (up to a constant factor) (29) δ λ δ u = ( δ λ δ q 1 δ λ δ q 2 δ λ δ r 1 δ λ δ r 2 δ λ δ α 1 δ λ δ α 2 δ λ δ β 1 δ λ δ β 2 ) = ( ϕ 3 2 ϕ 4 2 - ϕ 1 2 - ϕ 2 2 2 ϕ 3 ϕ 5 2 ϕ 4 ϕ 6 - 2 ϕ 1 ϕ 5 - 2 ϕ 2 ϕ 6 ) . When zero boundary conditions ( lim | x | + ϕ j = 0 , 1 j 6 ) are imposed, we can verify a simple characteristic property of the above variational derivative: (30) L δ λ δ u = λ δ λ δ u , where L is given by (13).

To carry out nonlinearization of N copies of systems (7) and (27) with N distinct parameters λ j ( 1 j N ) , we consider the following Bargmann symmetry constraint: (31) q 1 = - Φ 1 , Φ 1 , q 2 = - Φ 2 , Φ 2 , r 1 = Φ 3 , Φ 3 , r 2 = Φ 4 , Φ 4 , α 1 = Φ 1 , Φ 5 , α 2 = Φ 2 , Φ 6 , β 1 = Φ 3 , Φ 5 , β 2 = Φ 4 , Φ 6 , where Φ j = ( ϕ j 1 , , ϕ j N ) T ( 1 j 6 ) and · , · refers to the standard inner product of the Euclidian space N . Now, substituting the constraint (31) into N copies of system (7), we obtain the following constrained vector system: (32) Φ 1 , x = - Λ Φ 1 - Φ 1 , Φ 1 Φ 3 + Φ 1 , Φ 5 Φ 5 = H 1 Φ 3 , Φ 2 , x = - Λ Φ 2 - Φ 2 , Φ 2 Φ 4 + Φ 2 , Φ 6 Φ 6 = H 1 Φ 4 , Φ 3 , x = Φ 3 , Φ 3 Φ 1 + Λ Φ 3 + Φ 3 , Φ 5 Φ 5 = - H 1 Φ 1 , Φ 4 , x = Φ 4 , Φ 4 Φ 2 + Λ Φ 4 + Φ 4 , Φ 6 Φ 6 = - H 1 Φ 2 , Φ 5 , x = Φ 3 , Φ 5 Φ 1 - Φ 1 , Φ 5 Φ 3 = H 1 Φ 5 , Φ 6 , x = Φ 4 , Φ 6 Φ 2 - Φ 2 , Φ 6 Φ 4 = H 1 Φ 6 , where (33) H 1 = - Λ Φ 1 , Φ 3 - Λ Φ 2 , Φ 4 - 1 2 Φ 1 , Φ 1 Φ 3 , Φ 3 - 1 2 Φ 2 , Φ 2 Φ 4 , Φ 4 + Φ 1 , Φ 5 Φ 3 , Φ 5 + Φ 2 , Φ 6 Φ 4 , Φ 6 , with Λ = diag ( λ 1 , , λ N ) and the Poisson bracket is defined by (34) { f , g } = j = 1 N ( f ϕ 1 j g ϕ 3 j + f ϕ 2 j g ϕ 4 j - f ϕ 3 j g ϕ 1 j - f ϕ 4 j g ϕ 2 j + f ϕ 5 j g ϕ 5 j + f ϕ 6 j g ϕ 6 j ) .

Analogously, making use of (31) and (32), the t 2 -part (27) of the two-component AKNS equations (24) is constrained as the following 6 N -dimensional system: (35) ϕ 1 j , t 2 = ( - λ j 2 + 1 2 q ~ 1 r ~ 1 + α ~ 1 β ~ 1 ) ϕ 1 j + ( q ~ 1 λ j - 1 2 q ~ 1 , x ) ϕ 3 j + ( α ~ 1 λ j - α ~ 1 , x ) ϕ 5 j , 1 j N , ϕ 2 j , t 2 = ( - λ j 2 + 1 2 q ~ 2 r ~ 2 + α ~ 2 β ~ 2 ) ϕ 2 j + ( q ~ 2 λ j - 1 2 q ~ 2 , x ) ϕ 4 j + ( α ~ 2 λ j - α ~ 2 , x ) ϕ 6 j , 1 j N , ϕ 3 j , t 2 = ( r ~ 1 λ j + 1 2 r ~ 1 , x ) ϕ 1 j + ( λ j 2 - 1 2 q ~ 1 r ~ 1 - α ~ 1 β ~ 1 ) ϕ 3 j + ( β ~ 1 λ j + β ~ 1 , x ) ϕ 5 j , 1 j N , ϕ 4 j , t 2 = ( r ~ 2 λ j + 1 2 r ~ 2 , x ) ϕ 2 j + ( λ j 2 - 1 2 q ~ 2 r ~ 2 - α ~ 2 β ~ 2 ) ϕ 4 j + ( β ~ 2 λ j + β ~ 2 , x ) ϕ 6 j , 1 j N , ϕ 5 j , t 2 = ( β ~ 1 λ j + β ~ 1 , x ) ϕ 1 j + ( - α ~ 1 λ j + α ~ 1 , x ) ϕ 3 j , mmnmmmmmmmmmnm 1 j N , ϕ 6 j , t 2 = ( β ~ 2 λ j + β ~ 2 , x ) ϕ 2 j + ( - α ~ 2 λ j + α ~ 2 , x ) ϕ 4 j , mnmmmmmmmmmmmm 1 j N , where q ~ k , r ~ k , α ~ k , β ~ k denote the constrained q k , r k , α k , β k and q ~ k , x , r ~ k , x , α ~ k , x , β ~ k , x are given as follows: (36) q ~ 1 , x = 2 Λ Φ 1 , Φ 1 + 2 Φ 1 , Φ 1 Φ 1 , Φ 3 , q ~ 2 , x = 2 Λ Φ 2 , Φ 2 + 2 Φ 2 , Φ 2 Φ 2 , Φ 4 , r ~ 1 , x = 2 Λ Φ 3 , Φ 3 + 2 Φ 3 , Φ 3 Φ 1 , Φ 3 , r ~ 2 , x = 2 Λ Φ 4 , Φ 4 + 2 Φ 4 , Φ 4 Φ 2 , Φ 4 , α ~ 1 , x = - Λ Φ 1 , Φ 5 - Φ 1 , Φ 3 Φ 1 , Φ 5 , α ~ 2 , x = - Λ Φ 2 , Φ 6 - Φ 2 , Φ 4 Φ 2 , Φ 6 , β ~ 1 , x = Λ Φ 3 , Φ 5 + Φ 1 , Φ 3 Φ 3 , Φ 5 , β ~ 2 , x = Λ Φ 4 , Φ 6 + Φ 2 , Φ 4 Φ 4 , Φ 6 . After a direct calculation, constrained system (35) can be written as the vector form (37) Φ 1 , t 2 = H 2 Φ 3 , Φ 2 , t 2 = H 2 Φ 4 , Φ 3 , t 2 = - H 2 Φ 1 , Φ 4 , t 2 = - H 2 Φ 2 , Φ 5 , t 2 = H 2 Φ 5 , Φ 6 , t 2 = H 2 Φ 6 , where (38) H 2 = - Λ 2 Φ 1 , Φ 3 - Λ 2 Φ 2 , Φ 4 - 1 2 Φ 1 , Φ 1 Λ Φ 3 , Φ 3 - 1 2 Φ 2 , Φ 2 Λ Φ 4 , Φ 4 - 1 2 Λ Φ 1 , Φ 1 Φ 3 , Φ 3 - 1 2 Λ Φ 2 , Φ 2 Φ 4 , Φ 4 - 1 2 Φ 1 , Φ 1 Φ 1 , Φ 3 Φ 3 , Φ 3 - 1 2 Φ 2 , Φ 2 Φ 2 , Φ 4 Φ 4 , Φ 4 + Φ 1 , Φ 5 Λ Φ 3 , Φ 5 + Φ 2 , Φ 6 Λ Φ 4 , Φ 6 + Λ Φ 1 , Φ 5 Φ 3 , Φ 5 + Λ Φ 2 , Φ 6 Φ 4 , Φ 6 + Φ 1 , Φ 3 Φ 1 , Φ 5 Φ 3 , Φ 5 + Φ 2 , Φ 4 Φ 2 , Φ 6 Φ 4 , Φ 6 .

To this end, we show that the constrained N copies of systems (7) and (27) are super Hamiltonian systems (32) and (37).

4. <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M118"> <mml:mrow> <mml:mi>r</mml:mi></mml:mrow> </mml:math></inline-formula>-Matrix and Lax Representation

In what follows, we will show that super Hamiltonian systems (32) and (37) are both completely integrable in the Liouville sense. First of all, through a straightforward and tedious calculation, we arrive at the following proposition.

Proposition 1.

Super Hamiltonian systems (32) and (37) have the following Lax representations, respectively: (39) L x ( λ ) = [ U ~ , L ( λ ) ] , (40) L t 2 ( λ ) = [ V ~ ( 2 ) , L ( λ ) ] , where U ~ and V ~ ( 2 ) are, respectively, U and V ( 2 ) under symmetry constraint (31), and (41) L ( λ ) = ( A 1 ( λ ) 0 B 1 ( λ ) 0 ϱ 1 ( λ ) 0 0 A 2 ( λ ) 0 B 2 ( λ ) 0 ϱ 2 ( λ ) C 1 ( λ ) 0 - A 1 ( λ ) 0 ϖ 1 ( λ ) 0 0 C 2 ( λ ) 0 - A 2 ( λ ) 0 ϖ 2 ( λ ) ϖ 1 ( λ ) 0 - ϱ 1 ( λ ) 0 0 0 0 ϖ 2 ( λ ) 0 - ϱ 2 ( λ ) 0 0 ) , with (42) A 1 ( λ ) = - 1 + j = 1 N 1 λ - λ j ϕ 1 j ϕ 3 j , A 2 ( λ ) = - 1 + j = 1 N 1 λ - λ j ϕ 2 j ϕ 4 j , B 1 ( λ ) = - j = 1 N 1 λ - λ j ϕ 1 j 2 , B 2 ( λ ) = - j = 1 N 1 λ - λ j ϕ 2 j 2 , C 1 ( λ ) = j = 1 N 1 λ - λ j ϕ 3 j 2 , C 2 ( λ ) = j = 1 N 1 λ - λ j ϕ 4 j 2 , ϱ 1 ( λ ) = j = 1 N 1 λ - λ j ϕ 1 j ϕ 5 j , ϱ 2 ( λ ) = j = 1 N 1 λ - λ j ϕ 2 j ϕ 6 j , ϖ 1 ( λ ) = j = 1 N 1 λ - λ j ϕ 3 j ϕ 5 j , ϖ 2 ( λ ) = j = 1 N 1 λ - λ j ϕ 4 j ϕ 6 j .

Under the Poisson bracket (34), we have (43) { A k ( λ ) , B k ( λ ) } = 2 μ - λ ( - B k ( λ ) + B k ( μ ) ) = - { B k ( λ ) , A k ( λ ) } , k = 1,2 , { A k ( λ ) , C k ( λ ) } = 2 μ - λ ( C k ( λ ) - C k ( μ ) ) = - { C k ( λ ) , A k ( λ ) } , k = 1,2 , { A k ( λ ) , ϱ k ( λ ) } = 1 μ - λ ( - ϱ k ( λ ) + ϱ k ( μ ) ) = - { ϱ k ( λ ) , A k ( λ ) } , k = 1,2 , { A k ( λ ) , ϖ k ( λ ) } = 1 μ - λ ( ϖ k ( λ ) - ϖ k ( μ ) ) = - { ϖ k ( λ ) , A k ( λ ) } , k = 1,2 , { B k ( λ ) , C k ( λ ) } = 4 μ - λ ( - A k ( λ ) + A k ( μ ) ) = - { C k ( λ ) , B k ( λ ) } , k = 1,2 , { B k ( λ ) , ϖ k ( λ ) } = 2 μ - λ ( - ϱ k ( λ ) + ϱ k ( μ ) ) = - { ϖ k ( λ ) , B k ( λ ) } , k = 1,2 , { C k ( λ ) , ϱ k ( λ ) } = 2 μ - λ ( - ϖ k ( λ ) + ϖ k ( μ ) ) = - { ϱ k ( λ ) , C k ( λ ) } , k = 1,2 , { ϱ k ( λ ) , ϱ k ( λ ) } = 1 μ - λ ( - B k ( λ ) + B k ( μ ) ) , k = 1,2 , { ϖ k ( λ ) , ϖ k ( λ ) } = 1 μ - λ ( C k ( λ ) - C k ( μ ) ) , k = 1,2 , { ϱ k ( λ ) , ϖ k ( λ ) } = 1 μ - λ ( A k ( λ ) - A k ( μ ) ) = { ϖ k ( λ ) , ϱ k ( λ ) } , k = 1,2 , and the others are zero.

These relations imply the following proposition immediately.

Proposition 2.

The Lax matrix L ( λ ) satisfied the following r-matrix relation: (44) { L 1 ( λ ) , L 2 ( μ ) } = 1 μ - λ [ P , L 1 ( λ ) + L 2 ( μ ) ] , where (45) P = σ 1 σ 1 + σ 2 σ 2 + 2 ( σ 3 σ 5 + σ 5 σ 3 + σ 4 σ 6 + σ 6 σ 4 ) + σ 7 σ 9 + σ 9 σ 7 + σ 8 σ 10 + σ 10 σ 8 , with (46) σ 1 = E 11 - E 33 , σ 2 = E 22 - E 44 , σ 3 = E 13 , σ 4 = E 24 , σ 5 = E 31 , σ 6 = E 42 , σ 7 = E 15 - E 53 , σ 8 = E 26 - E 64 , σ 9 = E 35 + E 51 , σ 10 = E 46 + E 62 , E i j is the 6 × 6 matrix having 1 in the ( i , j ) th position and zeros elsewhere, and L 1 ( λ ) = L ( λ ) I and L 2 ( μ ) = I L ( μ ) .

Therefore, according to the general theory of the r -matrix , we know that ( 1 / 2 ) Str L 2 ( λ ) is a generating function of conserved integrals of motions. Explicitly, we can expand ( 1 / 2 ) Str L 2 ( λ ) as follows: (47) 1 2 Str L 2 ( λ ) = n 0 F n λ - n , where (48) F 1 = - 2 Φ 1 , Φ 3 - 2 Φ 2 , Φ 4 , F n = j = 1 n - 1 [ Λ j - 1 Φ 1 , Φ 3 Λ n - j - 1 Φ 1 , Φ 3 - Λ j - 1 Φ 1 , Φ 1 Λ n - j - 1 Φ 3 , Φ 3 + 2 Λ j - 1 Φ 1 , Φ 5 Λ n - j - 1 Φ 3 , Φ 5 + Λ j - 1 Φ 2 , Φ 4 Λ n - j - 1 Φ 2 , Φ 4 - Λ j - 1 Φ 2 , Φ 2 Λ n - j - 1 Φ 4 , Φ 4 + 2 Λ j - 1 Φ 2 , Φ 6 Λ n - j - 1 Φ 4 , Φ 6 ] - 2 Λ n - 1 Φ 1 , Φ 3 - 2 Λ n - 1 Φ 2 , Φ 4 , n 2 .

Therefore, from (44), we have (49) { F m , F n } = 0 , 1 m , n 3 N , which means that { F n } n = 1 3 N are in involution. Moreover, referring to proof of functional independence in , it is easy to find that { F n } n = 1 3 N are functionally independent over some region of 6 N .

To sum up the above results, we have the following theorem.

Theorem 3.

The super Hamiltonian systems given by (32) and (37) are both completely integrable in the sense of Liouville. That is to say, Hamiltonian systems (32) and (37) constitute an integrable decomposition of two-component super AKNS equations (26).

5. Conclusions and Discussions

Starting from a matrix Lie superalgebra, we constructed a two-component super AKNS system (19). For this new system, we considered a Bargmann symmetry constraint (31). Then, substituting the constraint (31) into N copies of systems (7) and (27), we obtained the constrained super Hamiltonian systems (32) and (37), and we showed that systems (32) and (37) are completely integrable in the Liouville sense. Accordingly, Lax matrix L ( λ ) and r -matrix representation were, respectively, given in Propositions 1 and 2. The difference between  and this paper will be listed in the following.

In , we applied binary-nonlinearization to one-component AKNS system which was associated with a 3 × 3 spectral matrix, while, in this paper, we applied monononlinearization to two-component AKNS system which was associated with a 6 × 6 spectral matrix.

In , symmetry constraint was associated with both eigenfunctions and adjoint eigenfunctions, while, in this paper, constraint was just associated with eigenfunctions.

Construction of integrals of motion is also different. In , we made use of constrained stationary equation ( N ~ 2 ) x = [ M ~ , N ~ 2 ] . And, in this paper, we made use of general theory of r -matrix.

According to the above conclusions, some future work is listed as follows.

Is this method applied to the other multicomponent super integrable system?

If potentials q and r are chosen as (2) in , is nonlinearization of Lax pairs applied to this multicomponent integrable system?

Is nonlinearization (including monononlinearization and binary-nonlinearization) extended to supersymmetry integrable system, such as supersymmetric Kadomtsev-Petviashvili system [24, 25]?

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant nos. 11001069 and 61273077 and Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ12A01002.

Ma W. Zhou R. Adjoint symmetry constraints of multicomponent AKNS equations Chinese Annals of Mathematics B 2002 23 3 373 384 2-s2.0-0036033067 10.1142/S0252959902000341 MR1930190 ZBL1183.37109 Zhang Y. A multi-component matrix loop algebra and a unified expression of the multi-component AKNS hierarchy and the multi-component BPT hierarchy Physics Letters A: General, Atomic and Solid State Physics 2005 342 1-2 82 89 2-s2.0-20444481030 10.1016/j.physleta.2005.01.098 MR2152838 ZBL1222.37072 Fordy A. P. Kulish P. P. Nonlinear Schrödinger equations and simple Lie algebras Communications in Mathematical Physics 1983 89 3 427 443 2-s2.0-0001565837 10.1007/BF01214664 MR709476 ZBL0563.35062 Tsuchida T. Wadati M. The coupled modified Korteweg-de vries equations Journal of the Physical Society of Japan 1998 67 4 1175 1187 2-s2.0-0032365143 Cao C. W. A cubic system which generates Bargmann potential and N-gap potential Chinese Quarterly Journal of Mathematics 1988 3 1 90 96 MR1012853 Zeng Y. Li Y. The constraints of potentials and the finite-dimensional integrable systems Journal of Mathematical Physics 1989 30 8 1679 1689 2-s2.0-0001238196 MR1006124 ZBL0695.58016 Cao C. W. Nonlinearization of the Lax system for AKNS hierarchy Science in China A: Mathematics, Physics, Astronomy 1990 33 5 528 536 MR1070539 ZBL0714.58026 Cewen C. Xianguo G. C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy Journal of Physics A: Mathematical and General 1990 23 18 4117 4125 2-s2.0-36149029754 MR1076724 10.1088/0305-4470/23/18/017 ZBL0719.35082 Zhou R. Dynamical r-matrices for the constrained Harry-Dym flows Physics Letters A: General, Atomic and Solid State Physics 1996 220 6 320 330 2-s2.0-0000853123 10.1016/0375-9601(96)00487-2 MR1431477 Zeng Y. Hietarinta J. Classical poisson structures and r-matrices from constrained flows Journal of Physics A: Mathematical and General 1996 29 16 5241 5252 2-s2.0-22244487348 10.1088/0305-4470/29/16/038 MR1418803 Ma W.-X. Strampp W. An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems Physics Letters A 1994 185 3 277 286 2-s2.0-0002796975 MR1261401 ZBL0941.37530 Ma W. X. Symmetry constraint of MKdV equations by binary nonlinearization Physica A: Statistical Mechanics and its Applications 1995 219 3-4 467 481 2-s2.0-0001190113 MR1353176 Ma W.-X. Zhou R. Adjoint symmetry constraints leading to binary nonlinearization Journal of Nonlinear Mathematical Physics 2002 9 supplement 1 106 126 10.2991/jnmp.2002.9.s1.10 MR1900189 Ji J. Zhou R. Two types of new integrable decompositions of the Kaup-Newell equation Chaos, Solitons & Fractals 2006 30 4 993 1003 2-s2.0-33745215400 10.1016/j.chaos.2005.09.009 MR2250269 ZBL1142.37359 He J. Yu J. Cheng Y. Zhou R. Binary nonlinearization of the super AKNS system Modern Physics Letters B 2008 22 4 275 288 2-s2.0-43949120750 10.1142/S0217984908014778 MR2407750 ZBL1154.37364 Yu J. He J. Ma W. Cheng Y. The Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems Chinese Annals of Mathematics B 2010 31 3 361 372 2-s2.0-77953083129 10.1007/s11401-009-0032-6 MR2652931 ZBL1200.35244 Yu J. He J. Cheng Y. Han J. A novel symmetry constraint of the super cKdV system Journal of Physics A: Mathematical and Theoretical 2010 43 44 12 2-s2.0-78649642966 10.1088/1751-8113/43/44/445201 445201 MR2733820 ZBL1203.35253 Yi-Shen L. Li-ning Z. Super AKNS scheme and its infinite conserved currents Il Nuovo Cimento A 1986 93 2 175 183 2-s2.0-51249173722 10.1007/BF02819989 MR862254 Yi-shen L. Li-ning Z. A note on the super AKNS equations Journal of Physics A: Mathematical and General 1988 21 7 1549 1552 10.1088/0305-4470/21/7/017 MR0951044 ZBL0695.35160 Hu X.-B. An approach to generate superextensions of integrable systems Journal of Physics A: Mathematical and General 1997 30 2 619 632 2-s2.0-0031581639 10.1088/0305-4470/30/2/023 MR1442879 ZBL0947.37039 Ma W.-X. He J.-S. Qin Z.-Y. A supertrace identity and its applications to superintegrable systems Journal of Mathematical Physics 2008 49 3 13 2-s2.0-41549127572 10.1063/1.2897036 033511 MR2406808 ZBL1153.81398 Babelon O. Viallet C.-M. Hamiltonian structures and Lax equations Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics 1990 237 3-4 411 416 2-s2.0-0001530153 MR1063458 Wang D.-S. Zhang D.-J. Yang J. Integrable properties of the general coupled nonlinear Schrödinger equations Journal of Mathematical Physics 2010 51 2 023510 10.1063/1.3290736 MR2605060 Manin Yu. I. Radul A. O. A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy Communications in Mathematical Physics 1985 98 1 65 77 2-s2.0-0002587975 MR785261 10.1007/BF01211044 ZBL0607.35075 He J. Cheng Y. The KW theorem for the SKP hierarchy Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics 1999 449 3-4 194 200 2-s2.0-10444242666 10.1016/S0370-2693(99)00041-6 MR1676120 ZBL0966.37033