Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO ( 3 )

Copyright © 2017 Jian Zhang et al. 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. Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) 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.


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][7][8].
An integrable coupling equation   =  () =  (, , ,   ,   ,   ,   ,   , . ..) is 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 V, the integrable coupling is called nonlinear.
Integrable couplings correspond to non-semi-simple Lie algebras , and such Lie algebras can be written as semidirect sums [11]: (5)

Bi-Integrable Couplings and Hamiltonian Structures
2.1.Bi-Integrable Couplings Associated with SO(3).So as to generate bi-integrable couplings, we introduce a kind of block matrices: where  is an arbitrary nonzero constant and ,  1 , and  2 are square matrices of the same order.In the following, we define the corresponding non-semi-simple Lie algebra  by a semidirect sum: with where the loop algebra SO(3) is defined by Obviously, we have the matrix commutator relation: with ,  1 , and  2 being defined by Let us consider the Lie algebra SO (3).It has a basis with which the structure equations of SO(3) 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  1 =  2 (,  1 ,  2 ) and let supplementary spectral matrices be For purpose of solving the enlarged stationary zero curvature equation where  is defined as in [18]  =  (, ) = ( and the supplementary spectral matrices  1 and  2 read The enlarged stationary zero curvature equation The above equation system equivalently leads to Now, we define the enlarged Lax matrices 2 ),  ≥ 0, where  [] is defined as  [] = (  ) + , and 1 ] = 0,  ≥ 0, we get bi-integrable couplings of the soliton hierarchy in [18]    = ( ( ( ( (

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. 9; we define the Lie bracket [⋅, ⋅] on  9 as follows: where Following the properties of the matrix  1 :  1 ( 1 ())  = − 1 () 1 and  1 =   1 , we get where  1 ,  2 ,  3 ,  0 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  9 of the following form: Now we can compute that and furthermore, we use the following formular [20]: to obtain that  = 0. 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 ) , where  = / and  −1 = ∫(/).

Tri-Integrable Couplings Associated with SO(3)
. So as to generate bi-integrable couplings, we introduce a kind of block matrices: where , , and ] are arbitrary nonzero constants and ,  1 ,  2 , and  3 are square matrices of the same order.In the following, we define the corresponding non-semi-simple Lie algebra () by a semidirect sum: with where the loop algebra SO( 3) is defined by − Laurent series in } . (40) Obviously, we have the matrix commutator relation: with ,  1 ,  2 , and  3 being defined by We introduce the following enlarged spectral matrix to construct tri-integrable couplings for SO(3) hierarchy: with  = (, ) being defined as in (14), where  1 and  2 are defined by (15), and also the supplementary spectral matrix  3 reads As usual, we take a solution of the following form: where with (),  1 (), and  2 () being defined by (23), and Following the properties of the matrix  2 ,  2 ( 2 ())  =  2 () 2 and  2 =   2 , we have where ⊗ is the Kronecker product and  1 ,   In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form {, } on  12 of the following form: Now, we further compute that We use formula (28) and find that  = 0. 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  2 is given by with  1 and   (63)

Conclusion
In this paper, we take advantage of the non-semi-simple Lie algebras consisting of 3 × 3, 4 × 4 block matrices and apply them to the construction of bi-integrable couplings and tri-integrable couplings associated with SO(3), 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 SO(3) and SO(4) are used to form matrix loop algebras.In addition, we can also consider the relations between the hierarchy of triintegrable couplings associated with SO(3) and the hierarchy of tri-integrable couplings associated with SO(4).