Study of Integrability and Exact Solutions for Nonlinear Evolution EquationsView this Special Issue
Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System
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 -matrix are also given in this paper.
The inverse scattering method provides us with a powerful tool to generate multicomponent soliton equations. In [1–4], 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 [6–10]. 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 -dimensional Hamiltonian system over some region with 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 [12–14]. Owing to one-component super integrable system which is associated with a spectral matrix, we just consider binary-nonlinearization in our previous papers [15–17]. 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 copies spectral problem, we obtained -dimensional constrained system. Furthermore, we rewrite the -dimensional system as the super Hamiltonian form. Lax representation and -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 : where is a unit matrix, is a zero matrix, is even, and is odd. After a direct calculation, we obtain that and the others are zeros, where is the super Lie bracket and denotes the parity of the arbitrary element .
It is easy to prove that the linear space is matrix Lie superalgebra. The corresponding loop superalgebra is presented as
In what follows, we will construct multicomponent super integrable equations via the matrix Lie superalgebra . Firstly, let us consider the following superisospectral problem: where which can be written as the following matrix form: where , , , , is a potential, is a spectral parameter which satisfies , and is an eigenfunction.
Taking where , , , , and , the adjoint representation equation leads to On setting , , , , and , (10) engenders equivalently which results in a recursion relation to determine : where with It is easy to obtain that . Therefore, we choose and select constants of integral to be zero, which lead to the first few terms being worked out as follows: where .
Secondly, let us associate (5) with the following auxiliary spectral problem: where The compatibility condition of (5) and (16) leads to the famous zero curvature equations which lead to isospectral super integrable equations Under the special reduction of , (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] As is usual, we need the following equalities which are easy to calculate: where . Substituting the above equality (21) into the supertrace identity (20) and comparing the coefficients of on two sides, we arrive at which leads to the constant with . Thus, we have Now, it follows from (23) that the two-component super AKNS system (19) has the following super bi-Hamiltonian structure: where the super Hamiltonian pair () reads as where are given by (13).
The first nonlinear equations in two-component AKNS system (24) are given by and the corresponding temporal part of the Lax system is wherewith .
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 with respect to the potential reads (up to a constant factor) When zero boundary conditions (, ) are imposed, we can verify a simple characteristic property of the above variational derivative: where is given by (13).
To carry out nonlinearization of copies of systems (7) and (27) with distinct parameters , we consider the following Bargmann symmetry constraint: where and refers to the standard inner product of the Euclidian space . Now, substituting the constraint (31) into copies of system (7), we obtain the following constrained vector system: where with and the Poisson bracket is defined by
Analogously, making use of (31) and (32), the -part (27) of the two-component AKNS equations (24) is constrained as the following -dimensional system: where , , , denote the constrained , , , and , , , are given as follows: After a direct calculation, constrained system (35) can be written as the vector form where
4. -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.
Under the Poisson bracket (34), we have and the others are zero.
These relations imply the following proposition immediately.
Proposition 2. The Lax matrix satisfied the following r-matrix relation: where with is the matrix having 1 in the th position and zeros elsewhere, and and .
Therefore, according to the general theory of the -matrix , we know that is a generating function of conserved integrals of motions. Explicitly, we can expand as follows: where
Therefore, from (44), we have which means that are in involution. Moreover, referring to proof of functional independence in , it is easy to find that are functionally independent over some region of .
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 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 and -matrix representation were, respectively, given in Propositions 1 and 2. The difference between  and this paper will be listed in the following.(1)In , we applied binary-nonlinearization to one-component AKNS system which was associated with a spectral matrix, while, in this paper, we applied monononlinearization to two-component AKNS system which was associated with a spectral matrix.(2)In , symmetry constraint was associated with both eigenfunctions and adjoint eigenfunctions, while, in this paper, constraint was just associated with eigenfunctions.(3)Construction of integrals of motion is also different. In , we made use of constrained stationary equation . And, in this paper, we made use of general theory of -matrix.
According to the above conclusions, some future work is listed as follows.(1)Is this method applied to the other multicomponent super integrable system?(2)If potentials and are chosen as (2) in , is nonlinearization of Lax pairs applied to this multicomponent integrable system?(3)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.
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.
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