Abstract
Based on the three-dimensional real special orthogonal Lie algebra , by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.
1. Introduction
Among the well-known soliton hierarchies are the KdV hierarchy, the AKNS hierarchy, and the Kaup-Newell hierarchy [1]. The trace identity is used for constructing Hamiltonian structures of soliton equations, which is proposed by Tu [2, 3]. In the case of non-semi-simple Lie algebras, integrable couplings of soliton equations are generated by zero curvature equations [4, 5] and the corresponding Hamiltonian structures are obtained by the variational identity [6–8].
An integrable coupling equationis a triangular integrable system of the following form [9]: where is a function of variables and , , and . If is nonlinear with respect to the second dependent variable , the integrable coupling is called nonlinear.
An integrable system of the following form [10] is called a bi-integrable of (1).
Similarly, an integrable system of the following form [10] is called a tri-integrable of (1).
Integrable couplings correspond to non-semi-simple Lie algebras , and such Lie algebras can be written as semidirect sums [11]: The notion of semidirect sums means that and satisfy , where , with denoting the Lie bracket of . Obviously, is an ideal of . The subscript indicates a contribution to the construction of coupling systems. We also require the closure property between and under the matrix multiplication: , where .
Integrable couplings are useful tools for describing and explaining nonlinear phenomena of new evaluation equations. There are very rich mathematical structures behind integrable couplings. In particular, integrable couplings generalize the symmetry problem and describe other integrable properties of integrable equations. In order to enrich multicomponent integrable equations, it has been an important task to explore more integrable properties from multi-integrable couplings. For example, one can find work on the integrable couplings [12, 13]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing non-semi-simple Lie algebras.
Recently, seeking new integrable systems including soliton hierarchies and integrable couplings forms a pretty important and interesting area of research in mathematical physics. To generate integrable couplings, bi-integrable couplings and tri-integrable couplings of soliton hierarchies, Ma proposed a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices [10]. Recently, bi-integrable couplings and tri-integrable couplings for the KdV hierarchy and the AKNS hierarchy have been studied considerably [14, 15]. From [16, 17], bi-integrable couplings of a new soliton hierarchy associated with and bi-integrable couplings of a new soliton hierarchy associated with have been studied.
In this paper, we will construct bi-integrable and tri-integrable couplings associated with for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [17–19].
2. Bi-Integrable Couplings and Hamiltonian Structures
2.1. Bi-Integrable Couplings Associated with
So as to generate bi-integrable couplings, we introduce a kind of block matrices:where is an arbitrary nonzero constant and , , and are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra by a semidirect sum: withwhere the loop algebra is defined by Obviously, we have the matrix commutator relation: with , , and being defined by
Let us consider the Lie algebra . It has a basis with which the structure equations of are , , .
The soliton hierarchy introduced in [18] has a spectral problem with the spectral matrix being chosen as
Based on this special non-semi-simple Lie algebra , we begin with the corresponding enlarged spectral matrix and let supplementary spectral matrices be
For purpose of solving the enlarged stationary zero curvature equation , we take , where is defined as in [18]and the supplementary spectral matrices and read
The enlarged stationary zero curvature equation is equivalent to
The above equation system equivalently leads to
Now, we define the enlarged Lax matrices , , where is defined as , and , .
Solving the enlarged zero curvature equations , , we get bi-integrable couplings of the soliton hierarchy in [18]
2.2. Hamiltonian Structures
In this section, for purpose of generating the Hamiltonian structure of hierarchy (20), we will use the corresponding variational identity [20]:where is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.
, ; we define the Lie bracket on as follows: where
Following the properties of the matrix : and , we get where are arbitrary constants. We are easy to have
In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form on of the following form:
Now we can compute that and furthermore, we use the following formular [20]:to obtain that . Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of bi-integrable coupling (20): where the Hamiltonian operator is and the Hamiltonian functions read
Based on (19), a direct computation yields a recursion relation: where with , , and being defined bywhere and .
3. Tri-Integrable Couplings and Hamiltonian Structures
3.1. Tri-Integrable Couplings Associated with
So as to generate bi-integrable couplings, we introduce a kind of block matrices:where , , and are arbitrary nonzero constants and , , , and are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra by a semidirect sum: withwhere the loop algebra is defined by Obviously, we have the matrix commutator relation: with , , , and being defined by
We introduce the following enlarged spectral matrix to construct tri-integrable couplings for hierarchy: with being defined as in (14), where and are defined by (15), and also the supplementary spectral matrix reads
As usual, we take a solution of the following form: where , , and are defined by (16) and (17); also reads
It now follows from the enlarged stationary zero curvature equation that
The above equation system is equivalent to
Now, we define the enlarged Lax matrices , , where is defined as , and , .
Solving the enlarged zero curvature equations , , we get tri-integrable couplings of the soliton hierarchy in [18]
3.2. Hamiltonian Structures
In this section, for the purpose of generating the Hamiltonian structure of the hierarchy (49), we will use the corresponding variational identity [20]: where is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.
, ; we define the Lie bracket on as follows:where with , and being defined by (23), and
Following the properties of the matrix , and , we have where is the Kronecker product and are arbitrary constants. It is easy to have
In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form on of the following form:
Now, we further compute that
We use formula (28) and find that . Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of tri-integrable couplings (49): where the Hamiltonian operator is and the Hamiltonian functions read
Based on (48), a direct computation shows a recursion relation: where the recursion operator is given by with and being given as in (34), and
4. Conclusion
In this paper, we take advantage of the non-semi-simple Lie algebras consisting of , block matrices and apply them to the construction of bi-integrable couplings and tri-integrable couplings associated with , based on the enlarged zero curvature equations. According to the associated variational identities, their Hamiltonian structures can be generated.
We can think about other related issues, for example, how we can get integrable couplings and their Hamiltonian structures when irreducible representations of and are used to form matrix loop algebras. In addition, we can also consider the relations between the hierarchy of tri-integrable couplings associated with and the hierarchy of tri-integrable couplings associated with .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by NNSF of China (nos. 11171055 and 11471090).