The Bi-Integrable Couplings of Two-Component Casimir-Qiao-Liu Type Hierarchy and Their Hamiltonian Structures

A new type of two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is produced from a new spectral problem and their biHamiltonian structures are constructed. Particularly, a new completely integrable two-component Casimir-Qiao-Liu type equation (2-CQLTE) is presented. Furthermore, based on the semidirect sums of matrix Lie algebras consisting of 3 × 3 block matrix Lie algebra, the bi-integrable couplings of the 2-CQLTH are constructed and their bi-Hamiltonian structures are furnished.


Introduction
Solitons and integrable systems play an important role in nonlinear wave and dynamics systems.It has been significant in soliton theory to find more new integrable systems.In 1996, Olver and Rosenau obtained a Casimir equation [1]: which is an integrable case of the general class of equations   = (1 ±  2 )  .In 2009, Qiao and Liu proposed a thirdorder integrable peakon equation [2,3]: which possesses Lax representation and bi-Hamiltonian structures.Here we can find that (2) can be obtained from the Casimir equation ( 1) by setting  = 3  √ 2.We will use Casimir-Qiao-Liu equation (CQLE) to denote (2) in this paper.The CQLE (2) represents a third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids.Moreover, it may be reduced from the two-dimensional Euler equation by an approximation procedure, and its solutions may be useful to construct new solvable potentials in Newtonian dynamics and to model electrophysiological phenomena in neuroscience.It has attracted many scholars' attention in recent years [4][5][6].
In recent years, the construction of soliton hierarchies and integrable couplings have become important research fields in soliton theory [7][8][9][10].The soliton hierarchies are generated from the zero curvature equations [11,12] which are based on semisimple Lie algebras, while the integrable couplings are generated from the zero curvature equations based on semidirect sums of Lie algebras [13][14][15].
The trace identity proposed by Tu is a useful tool for constructing the Hamiltonian structures for both continuous and discrete integrable systems [11,12].Many integrable Hamiltonian systems of infinite dimensions with various physics and mathematical backgrounds have been obtained [16][17][18].But when it comes to the integrable couplings, since they are based on semidirect sums of Lie algebras, the trace identity cannot be used properly.In order to solve this problem, Ma proposes the variational identity [19] while Guo and Zhang propose the quadratic-form identity [20].By using them, the Hamiltonian structures of many integrable couplings systems have been furnished [21][22][23].

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The notion of semidirect sums  =  ⊕   means that the two Lie subalgebras  and   satisfy where [,   ] = {[, ] |  ∈ ,  ∈   }, with [⋅, ⋅] denoting the Lie bracket of .We also require the closure property between  and   under the matrix multiplication: where Now we make the following assumptions: where ,  ∈ ,   ,   ∈   and ,  satisfying (3), (4), and (5).In this condition, we can construct the enlarged spectral problem as From the enlarged zero curvature equations we have The first equation of ( 9) is the original soliton hierarchy while the second equation is the integrable couplings.The bi-Hamiltonian structures of ( 9) can be obtained by using the following variational identity: where ⟨⋅, ⋅⟩ is a bilinear form [19,20,25].In this paper, starting from a new eigenvalue problem, the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is worked out.It is proved that the 2-CQLTH has Lax pairs and bi-Hamiltonian structures, so it is completely integrable.Particularly, the two-component Casimir-Qiao-Liu type equation (2-CQLTE) is given.Then, by constructing the enlarged spectral problem, we obtain the the bi-integrable couplings of the 2-CQLTH.Similarly, we have proved the integrability of the bi-integrable couplings by constructing their Lax pairs and bi-Hamiltonian structures.This paper is organized as follows.In Section 2, a new 2-CQLTH is given.In Section 3, the bi-Hamiltonian structures of the 2-CQLTH are constructed.In Section 4, we obtain the bi-integrable couplings of the 2-CQLTH.In Section 5, we generate the bi-Hamiltonian structures of the integrable couplings.The conclusion is given in Section 6.

The Two-Component Casimir-Qiao-Liu Type Hierarchy
Consider a spatial matrix isospectral problem: where ,  are potential functions and  is the spectral parameter.
Assuming that  has the form then the stationary zero curvature equation gives rise to Further, let , ,  possess the Laurent expansions: where the initial values are as follows: Advances in Mathematical Physics 3 In this case, system (15) becomes Imposing the following conditions on constants of integration: the sequence of {  ,   ,   |  ≥ 1} can be uniquely determined and the first two sets are as follows: Now we introduce where By considering the compatibility of ( 12) and (23) gives the zero curvature equation: From the zero curvature equation (24), we can obtain the 2-CQLTH: where Remark 1.The Lax pair for the 2-CQLTH (see (25)) is given by ( 12) and ( 23).This implies that the 2-CQLTH (see (25)) is integrable in Lax sense.

Bi-Hamiltonian Structures and Liouville Integrability
By direct computation, we can get so we have

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Now, the trace identity [11] gives Comparing coefficients of all powers of  in the equality (31), we have Checking a particular case with  = 1 in (32), we have  = 1.Thus, we obtain where the Hamiltonian functionals are defined by It now follows that the soliton hierarchy (25) has the following Hamiltonian structures: From the recursion relations (18), we can get where It is easy to verify that  =  * .Actually, we know that the hierarchy ( 25) is bi-Hamiltonian: where the second Hamiltonian operator is and  and  constitute the Hamiltonian pairs.Thus, the soliton hierarchy ( 25) is Liouville integrable.
Remark 3. Based on SO(3) [26,27], a 3 × 3 matrix isospectral problem is constructed: with the spectral matrix as By a standard procedure similar to Sections 2 and 3, another 2-QLTH can be worked out: with the Hamiltonian operators and the Hamiltonian functionals Particularly, when we take  = 1, a new 2-CQLTE is presented as Equation ( 45) can be transformed into (27) in complex field, but they are different equations in real field.
Considering the enlarged spectral matrix where and , ,  1 , When , , , , , , , ℎ,  are assumed to be and the initial values are taken as we can write (49) in the following forms: Furthermore, by imposing the sequence of {  ,   ,   ,   ,   ,   ,   , ℎ  ,   |  ≥ 1} has the unique expressions In order to get the integrable couplings systems from the enlarged zero curvature equations we define the Lax matrices and modification terms as where and  + denotes the polynomial part of .

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In this condition, the integrable couplings system can be obtained as which is equal to where Remark 4.  and  [] are the Lax pairs of the integrable couplings of the 2-CQLTH; this implies that the integrable couplings are integrable in Lax sense.
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Bi-Hamiltonian Structures and Liouville Integrability of the Bi-Integrable Couplings
Because the associated matrix Lie algebras are nonsemisimple, we should use the variational identity (11) to furnish the Hamiltonian structures of the integrable couplings system (58) (or (59)).Before doing this, we should find out the nondegenerate, symmetric, and adinvariant bilinear form corresponding to the nonsemisimple matrix loop algebras first [22].By defining a mapping from  to  9 as where we have the following results according to [22,28,30].The mapping (61) is a Lie algebra isomorphism from  to  9 and the Lie bracket [⋅, ⋅] can be calculated as where Assuming that the bilinear form on  9 possesses the following form: where  is a constant matrix, the symmetric property and adinvariance property of give rise to So the form of the constant matrix  can be written as where  1 ,  2 , and  3 are arbitrary constants.Thus, the bilinear form on the semidirect sums  of  and   can be defined as where In order to guarantee that the bilinear form is nondegenerate, we should require that is to say,  ̸ = 0 and  3 ̸ = 0.By calculation, we have ) . (71) Further, we have ) . ( Substituting the above expressions into the variational identity (11), we can obtain Comparing the coefficients of all powers of  in (73), we know that where Setting  = 1 in (74), we know that the constant  = 1, so we have Thus, the Hamiltonian functions can be expressed as and the Hamiltonian structures of (58) as Furthermore, by using (52), we can get where ) , ) , ) , where  and  are the Hamiltonian pairs of the integrable couplings of the 2-CQLTH.So the integrable couplings (58) are Liouville integrable.

Conclusion
In this paper, we construct a new eigenvalue problem and give the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH).The 2-CQLTH has Lax representation and possesses bi-Hamiltonian structures, which implies that it is integrable in both Lax and Liouville sense.Furthermore, we obtain the integrable couplings of the integrable hierarchy as well as its Lax pairs and bi-Hamiltonian structures and prove that the integrable couplings are also integrable in both Lax and Liouville senses.
2 ,  1 ,  2 are dependent variables, if we substitute ,  into the corresponding enlarged stationary zero curvature equation, we can get