Abstract
Based on the six-dimensional real special orthogonal Lie algebra , a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.
1. Introduction
As is well known, integrable equations are a remarkable class of nonlinear equations. To enrich multicomponent integrable equations, it has been an important task to explore more integrable properties for multi-integrable couplings. For example, one can find work on the integrable couplings [1, 2]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing nonsemisimple Lie algebras.
The trace identity provides a simple and effective way to search for Hamiltonian structures of the hierarchies, which is proposed by Tu [3, 4]. However the trace identity is based on the semisimple Lie algebras. It is no longer applicable, when the trace identity comes to the nonsemisimple Lie algebras. So the variational identity is proposed, which can be used to obtain the Hamiltonian structures in the case of nonsemisimple Lie algebras [5–7].
Searching for integrable couplings of systems will become more and more meaningful in the area of research in mathematical physics. In [8], Ma et al. propose a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices. Recently, bi-integrable couplings and tri-integrable couplings for the KDV hierarchy and the AKNS hierarchy have been studied considerably [9, 10]. Bi-integrable couplings of a new soliton hierarchy associated with have been constructed [11].
An integrable coupling of a given system is a triangular integrable system of the following form [12]:where is a function of variables and , , and . If is nonlinear with respect to the second dependent variable , the integrable coupling is called nonlinear.
A bi-integrable coupling of given integrable system (1) is an enlarged triangular integrable system of the following form [8]:
Integrable couplings correspond to nonsemisimple Lie algebras , and such Lie algebras can be written as semidirect sums [13]: 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 .
In this paper, we will construct bi-integrable couplings associated with for a new hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [11, 14, 15].
2. A New Soliton Hierarchy
Firstly, we will use the six-dimensional real special orthogonal Lie algebra [16] to construct a new soliton hierarchy and consider corresponding Hamiltonian structure by means of the trace identity [3, 4]. This Lie algebra has the following basis:Set . Then the circular commutator relations are The corresponding loop algebra is given by along with the commutators .
Consider an isospectral problem where are defined as
The stationary zero curvature representation gives
Take the initial values From (11), we have
Then the zero curvature equation leads to the following Lax integrable hierarchy: wherewhere and .
From recurrence relations (11), we have where is a recurrence operator and
In what follows, we will construct the Hamiltonian structure for soliton hierarchy (14) by the trace identity [3, 4]:
In this case, we can easily obtain
Substituting the above equations into trace identity (19) yields
Balancing coefficients of each power of in the above equality gives rise to
Checking the case with gives . Thus, we get the Hamiltonian structure for soliton hierarchy (14):
It is said that hierarchy (14) has the Hamiltonian structure, and it is easy to verity that . Therefore, hierarchy (14) is Liouville integrable.
3. Bi-Integrable Couplings and Hamiltonian Structures
3.1. Bi-Integrable Couplings Associated with
We take the class of block matrices in the form ofwhere is an arbitrary nonzero constant and , , and are square matrices of the same order. In the following, we define the corresponding nonsemisimple Lie algebra by a semidirect sum withwhere loop algebra is defined by
The corresponding matrix product reads with , , and being defined by
Based on this special nonsemisimple Lie algebra , we choose the corresponding enlarged spectral matrix , where is defined as in (9) and are defined as follows:
In order to solve the enlarged stationary zero curvature equations , we take , where is defined by (9): and are defined similarly:
It now follows from the enlarged stationary zero curvature equations that
The above equation system equivalently leads to (11) and the following equations:
Now, let us define the enlarged Lax matrices , where is defined as , and .
Solving the enlarged zero curvature equations , we get bi-integrable couplings for soliton hierarchy (14): with , , and being defined by (14) and (15), with , , , , , and being defined as follows:where where
3.2. Hamiltonian Structures
In this section, in order to generate the Hamiltonian structures of hierarchy (35), we use the corresponding variational identity [17] where is a required bilinear form, which is symmetric, nondegenerate, and invariant under the loop algebra .
First, we need to define a mapping [8] whereThis mapping induces a Lie algebraic structure on . The Lie bracket on can be computed as follows: where Mapping (41) is a Lie algebra isomorphism between the two Lie algebras.
Then we define a bilinear form on as follows: Following the properties of the matrix : and , we have where , and are arbitrary constants. We choose ; then the nondegenerate condition requires
So, the bilinear form on the semidirect sum is given by
As mentioned above, now we can compute that and furthermore we use the formula [17] to obtain that . Applying corresponding variational identity (40), we obtain the following Hamiltonian structures for the hierarchy of bi-integrable couplings (35): with the Hamiltonian operator where is defined by (15) and the Hamiltonian functionals
Based on (11) and (34), a direct computation yields a recursion relation where with being defined by (17) and are defined as follows:
4. Conclusion
In this paper, new Lax integrable hierarchy (14) is obtained by constructing an isospectral problem associated with . Furthermore, we have obtained a new class of bi-integrable couplings (35) for soliton hierarchy (14) based on nonsemisimple Lie algebra (25).
There are many soliton hierarchies obtained by using matrix spectral problems based on certain real Lie algebras such as and . Recently seeking for integrable couplings of systems has become more and more meaningful in the area of mathematics and physics. Certainly, there are many interesting aspects of integrable couplings we have not solved, for example, the hierarchies associated with and , the relations between the hierarchy of tri-integrable couplings associated with , and the hierarchy of tri-integrable couplings associated with . These problems are very interesting.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referee for valuable comments and suggestions on this paper. This work was supported by NNSF of China (nos. 11171055 and 11471090) and Scientific Research Fund of Heilongjiang Provincial Education Department (no. 12541184).