A Class of Multi-Component Non-Isospectral TD Hierarchies and Their Bi-Hamiltonian Structures

: By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity.


Introduction
Nonlinear integrable evolution equations are important in many application fields, such as mathematics and theoretical physics.Deriving the nonlinear integrable evolution equations is a key subject.Magri [1] proposed that integrable systems can be obtained by applying Lax representation and zero curvature representation.Tu [2] found another effective method to generate integrable hierarchies of equations and the corresponding Hamiltonian structures.Ma [3,4] called this method the Tu scheme.By applying the Tu scheme, many important isospectral integrable systems were obtained [5][6][7][8][9][10][11][12].It is obvious that non-isospectral equations play a significant role in mathematics and theoretical physics.Ma [13] offered a universally applicable method of constructing Lax representations for isospectral and non-isospectral hierarchies of evolution equations.In [14], he gave a systematic method to construct integrable systems from the zero curvature representation non-isospectral flows associated with a given isospectral hierarchy of integrable systems.By using the Lenard recursive scheme, Qiao [15] displayed the hierarchies of generalized nonlinear evolution equations associated with the Harry-Dym spectral problem and constructed the corresponding generalized Lax representations by solving a key operator equation.Based on the methods, a large number of non-isospectral integrable systems are considered [16][17][18].
As we all know, integrable couplings originated from an investigation on soliton equations and have received increasing attention.The authors in [19] constructed a suitable transformation of Lax pairs to obtain the integrable coupling of TD hierarchy, which is called the direct method (see [20] for reference).Ma once provided a method to construct integrable couplings associated with the non-semisimple Lie algebra, which has the triangular block matrix form where m 1 and m 2 are two arbitrary square matrices of the same order.By applying this non-semi-simple Lie algebra, many isospectral or non-isospectral integrable couplings have been deduced [21][22][23][24][25]. Wang [26] extended the above non-semi-simple Lie algebra to generalized non-semi-simple Lie algebra, which has the following form: where ε is a constant.The generalized non-semi-simple Lie algebra, which depends on epsilon in a non-trivial way, is applied to construct the integrable couplings of isospectral and non-isopectral problems, the generalized integrable coupling of the MKdV isospectral problem and the Ablowitz-Kaup-Newell-Segur non-isopectral problem were constructed.Based on generalized non-semi-simple Lie algebra, a series of integrable coupling systems were obtained, such as two non-isospectral hierarchies of the Ablowitz-Kaup-Newell-Segur type and Kaup-Newell type, non-isospectral generalized nonlinear Schrödinger hierarchies [27].
Motivated by the generalized non-semi-simple Lie algebra, a new non-semi-simple Lie algebra that can extend the classical spectral problems to the multi-component spectral problems was determined.Wang [28] extended the integrable coupling MKdV isospectral and Ablowitz-Kaup-Newell-Segur non-isospectral hierarchies to the N-dimensional isospectral MKdV and non-isospectral Ablowitz-Kaup-Newell-Segur hierarchies.N-dimensional nonisospectral KdV and non-isospectral generalized nonlinear Schrödinger hierarchies were generated, respectively, in [27,29].The rest of this papper is organized as follows.In Section 2, with the help of the simple Lie algebra and the non-isospectral problem, we derive a non-isospectral TD hierarchy from zero curvature equations, and the Bi-Hamiltonian structure is presented by making use of the trace identity.In Section 3, using generalized non-semi-simple Lie algebra, by considering the extended non-isospectral problem and solving the extended zero curvature equation, the integrable coupling of a non-isospectral TD hierarchy is obtained.By applying Z ε 2 trace identity, we obtained the corresponding Bi-Hamiltonian structures.In Section 4, we present a multi-component non-semi-simple Lie algebra that can be used to generate higherdimensional isospectral and non-isospectral integrable hierarchies.We consider the special case of ε = 0 and derive the multi-component non-isospectral TD integrable coupling hierarchy by solving the enlarged zero curvature equations, for which the Bi-Hamiltonian structures are obtained.Section 5 provides the conclusions and discussions.

A Nonisospectral TD Integrable Hierarchy
Denote a simple Lie algebra, A, as follows: The basis of Lie algebra can be constituted by the following matrices: and this satisfies the commutator relations We consider the 2 × 2 matrix spectral problem: where ψ = (ψ 1 , ψ 2 ) T , u = (q, r) T .
To derive a nonlinear integrable hierarchy, the following stationary zero curvature equation needs to be solved By substituting M and N in (2) into (3), we obtain By , inserting them into (4), and comparing the same powers of λ, we have which leads to the following recurrence relations: By taking the initial conditions and imposing the integrable conditions {a i , b i , c i |i > 0} can be gained from the relations in (6).First, we list three sets as follows: Then, (3) can be decomposed as − .
The gradations of the left-hand side of (7) are obtained as follows: One can find that the minimum gradation of the left-hand side of ( 7) is zero.The maximum gradation of the right-hand side of (7) has no choice but to be zero.Thus, we obtain the following equation by taking the gradations of these terms to be zero: Then, we take the modified term + + △ n .By solving the non-isospectral zero curvature equation we obtain the well-known non-isospectral TD hierarchy: Here, we have used the first equation of (5).The first nontrivial system in the hierarchy ( 8) is as follows: Set k 1 (t) = k 2 (t) = 0, t = 2 t, and (9) reduces to which is the TD equation [2].

Bi-Hamiltonian Structures
In this subsection, we apply the trace identity discussed in [2] to consider the Bi-Hamiltonian structure of the nonlinear integrable hierarchy (8).
We calculate the following quantities: Bearing in mind that the trace identity , we arrive at By comparing the coefficients of λ −m−1 in the above equation, we obtain We figured out γ = 0 via inserting m = 1 into the above equation.Thus, we have δ δu where H m+1 = a m+2 m+1 dx.A recurrence relationship is generated by a direct calculation, as follows where According to ( 12) and ( 13), the non-isospectral TD hierarchy (8) can be represented as the following: with R 1 and the Hamiltonian operators J and Q as follows:

An Integrable Coupling of the TD Non-Isospectral Hierarchy
Similar to [26], in terms of the definition of Lie algebra, the Lie algebra A can be enlarged into a higher-dimensional Lie algebra.

The enlarged Lie algebra
i=1 has the following basis: where the commutator is given by We introduce an enlarged non-isospectral problem: where and u = (q 1 , r 1 , , and solving the stationary zero curvature equation we have A straightforward computation yields the following recursion equations: Assuming 18) gives rise to [q 2x − q 1x − r 2 (q 1 − εq 2 ) − r 1 (q 2 − q 1 )], [q 1x − q 2x − r 2 (q 1 − εq 2 ) − r 1 (q 2 − q 1 )], Letting and the modified term the non-isospectral zero curvature equation gives the integrable coupling of non-isospectral TD hierarchy When n = 2, (19) leads to Let ε = 0; the first and second equations in (20) coincide with (9).Therefore, the system (20) is an integrable coupling of the TD non-isospectral equations.Thus, ( 19) is an integrable coupling of the TD non-isospectral hierarchy.Let n = 3, and choose a suitable transformation; then, (19) becomes a coupled mKdV equation.

Bi-Hamiltonian Structures
With the aid of the following Z ε 2 -trace identities, which were discussed in [26], we shall construct the Bi-Hamiltonian structures of the integrable coupling of the non-isospectral hierarchy (19): From ( 15), we obtain Substituting the above equations into (21) and collecting the terms with the same powers of λ produces Set m = 1; we have γ = 0. Therefore, we conclude that where From (18), we obtain two kind of recursive relationships: where the recursion operator L 1 becomes and Then, the hierarchy in (19) determines the following two kinds of Bi-Hamiltonian structures.The first one is with the Hamiltonian operators J 1 , Q 1 , and R, as follows: Similarly, for the second component, we have with the Hamiltonian operators J 1 and Q 1 , as follows: L 12 ,

A Muti-Components Integrable TD Nonisospectral Hierarchy
In order to obtain the muti-components non-isospectral problem, we introduce the Lie algebra A 1N associated with the N-component non-isospectral problem.
First of all, a brief introduction to the Lie algebra A 1N related to the Lie algebra A will be given.Let us define a N × N square matrix: where m n (1 ≤ n ≤ N) represents n arbitrary square matrices of the same order.
For the arbitrary matrices T , one has the Lie product: where is subject to Similarly, the matrix product PQ can be easily written as and Secondly, we denote A 1N = span{ h i } 4N i=1 , with commutators and using the recursion relations (30), we can obtain r k (q j − q j−1 )], r k (q j − q j−1 )], We choose the modification term as By taking leads to the following N-component non-isospectral TD hierarchy: Letting N = n = 2, system (31) is reduced to system (19), in which ε = 0.Under the case of n = 2, we can infer that the system is a new N-component non-isospectral TD system.Therefore, (31) is a N-component non-isospectral TD integrable hierarchy.

Bi-Hamiltonian Structures
In order to furnish the Hamiltonian structures of the N-component non-isospectral TD hierarchy (31), we will use the Z ε N -trace identity ).With the aid of ( 30) and (37), the Bi-Hamiltonian structures of hierarchy (31) are obtained as follows: . .
where J 1 is the Hamiltonian operator: