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

Abstract

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.

1. 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

$$M_1(m_1,m_2)=\left(\begin{array}{cc}{m}_1& 0\\ m_2& m_1\end{array}\right),$$

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:

$$M_2(m_1,m_2)=\left(\begin{array}{cc}{m}_1& \epsilon m_2\\ m_2& m_1\end{array}\right),$$

where $\epsilon$ 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 non-isospectral 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 $Z_2^{\epsilon }$ 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 higher-dimensional isospectral and non-isospectral integrable hierarchies. We consider the special case of $\epsilon =0$ 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.

2. A Nonisospectral TD Integrable Hierarchy

Denote a simple Lie algebra, A, as follows:

$$\begin{array}{c}A=\mathrm{span}\{h_1,h_2,h_3,h_4\}.\end{array}$$

The basis of Lie algebra can be constituted by the following matrices:

$$h_1=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),h_2=\left(\begin{array}{cc}0\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right),h_3=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),h_4=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),$$

(1)

and this satisfies the commutator relations

$$[h_1,h_2]=0,[h_1,h_3]=-[h_2,h_3]=h_3,[h_1,h_4]=-[h_2,h_4]=-h_4,[h_3,h_4]=h_1-h_2,$$

where $[a,b]=ab-ba,a,b\in A$.

We consider the $2\times 2$ matrix spectral problem:

$$\left\{\begin{array}{c}{\psi }_x=M\psi ,M=\left(\begin{array}{cc}-\lambda +r& q\\ q& \lambda \end{array}\right)=-\lambda h_1+\lambda h_2+r{h}_1+q(h_3+h_4),\hfill \\ {\psi }_t=N\psi ,N=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)=-a{h}_1+a{h}_2+b{h}_3+c{h}_4,\hfill \\ {\lambda }_t={\displaystyle \sum _{m\ge 0}}{k}_m\left(t\right){\lambda }^{-m},\hfill \end{array}\right.$$

(2)

where $\psi ={({\psi }_1,{\psi }_2)}^T,u={(q,r)}^T$.

To derive a nonlinear integrable hierarchy, the following stationary zero curvature equation needs to be solved

$$N_x=\frac{\partial M}{\partial \lambda }{\lambda }_t+[M,N].$$

(3)

By substituting M and N in (2) into (3), we obtain

$$\left\{\begin{array}{c}{a}_x=q(c-b)-{\lambda }_t,\hfill \\ b_x=-2b\lambda +rb-2qa,\hfill \\ c_x=2c\lambda -rc+2qa.\hfill \end{array}\right.$$

(4)

By assuming $a={\displaystyle \sum _{m\ge 0}}{a}_m{\lambda }^{-m},b={\displaystyle \sum _{m\ge 0}}{b}_m{\lambda }^{-m},c={\displaystyle \sum _{m\ge 0}}{c}_m{\lambda }^{-m}$, inserting them into (4), and comparing the same powers of $\lambda$, we have

$$\left\{\begin{array}{c}{a}_{mx}=q(c_m-b_m)-k_m\left(t\right),\hfill \\ b_{mx}=-2b_{m+1}+r{b}_m-2q{a}_m,\hfill \\ c_{mx}=2c_{m+1}-r{c}_m+2q{a}_m,\hfill \end{array}\right.$$

(5)

which leads to the following recurrence relations:

$$\left\{\begin{array}{c}{a}_m={\partial }_x^{-1}(q{c}_m-q{b}_m)-k_m\left(t\right)x,\hfill \\ b_{m+1}={\displaystyle \frac{1}{2}}(r{b}_m-b_{mx}-2q{a}_m),\hfill \\ c_{m+1}={\displaystyle \frac{1}{2}}(r{c}_m+c_{mx}-2q{a}_m).\hfill \end{array}\right.$$

(6)

By taking the initial conditions

$$\begin{array}{c}\hfill a_0=-1,b_0=0,c_0=0k_0=0,\end{array}$$

and imposing the integrable conditions

$$\begin{array}{c}\hfill a_i{|}_{q=r=0}=0,b_i{{|}_{q=r=0}=0,c_i|}_{q=r=0}=0,i\ge 0,\end{array}$$

$\{a_i,b_i,c_i|i>0\}$ can be gained from the relations in (6). First, we list three sets as follows:

$$\begin{array}{cc}\hfill & b_1=q,c_1=q,a_1=-k_1\left(t\right)x,\hfill \\ \hfill & b_2=q{k}_1\left(t\right)x+\frac{1}{2}(rq-q_x),c_2=q{k}_1\left(t\right)x+\frac{1}{2}(rq+q_x),a_2=\frac{1}{2}{q}^2-k_2\left(t\right)x,\hfill \\ \hfill & b_3=q{k}_2\left(t\right)x+\frac{1}{2}(rqx-x{q}_x-q)k_1\left(t\right)+\frac{1}{4}(r^{2}q-r{q}_x-{\left(rq\right)}_x+q_{xx})-\frac{1}{2}{q}^3,\hfill \\ \hfill & c_3=q{k}_2\left(t\right)x+\frac{1}{2}(rqx+x{q}_x+q)k_1\left(t\right)+\frac{1}{4}(r^{2}q+r{q}_x+{\left(rq\right)}_x+q_{xx})-\frac{1}{2}{q}^3,\hfill \\ \hfill & a_3={\partial }^{-1}q(x{q}_x+q)k_1\left(t\right)+\frac{1}{2}r{q}^2-k_3\left(t\right)x,\hfill \\ \hfill & \cdots .\hfill \end{array}$$

Denoting

$$\begin{array}{c}\hfill N_{+}^{\left(n\right)}=\sum _{m=0}^n(a_m(-h_1+h_2){\lambda }^{n-m}+b_m{h}_3{\lambda }^{n-m}+c_m{h}_4{\lambda }^{n-m}),{\lambda }_{t,+}^{\left(n\right)}=\sum _{m=0}^n{k}_m\left(t\right){\lambda }^{n-m},\end{array}$$

Then, (3) can be decomposed as

$$\begin{array}{c}\hfill \frac{\partial M}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}-N_{x,+}^{\left(n\right)}+\left[M,N_{+}^{\left(n\right)}\right]=-\frac{\partial M}{\partial \lambda }{\lambda }_{t,-}^{\left(n\right)}+N_{x,-}^{\left(n\right)}-\left[M,N_{-}^{\left(n\right)}\right].\end{array}$$

(7)

The gradations of the left-hand side of (7) are obtained as follows:

$$\begin{array}{c}\hfill \mathrm{deg}\frac{\partial M}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}\ge 0,\mathrm{deg}{N}_{+}^{\left(n\right)}\ge 0,\mathrm{deg}\left(\left[M,N_{+}^{\left(n\right)}\right]\right)\ge 0.\end{array}$$

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:

$$\begin{array}{c}\hfill \frac{\partial M}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}-N_{+,x}^{\left(n\right)}+\left[M,N_{+}^{\left(n\right)}\right]=2b_{n+1}{h}_3-2c_{n+1}{h}_4.\end{array}$$

Then, we take the modified term ${\Delta }_n=q^{-1}(b_{n+1}+c_{n+1})h_1$ so that for $N^{\left(n\right)}=N_{+}^{\left(n\right)}+{\Delta }_n$. By solving the non-isospectral zero curvature equation

$$\begin{array}{c}\hfill \frac{\partial M}{\partial u}{u}_t+\frac{\partial M}{\partial \lambda }{\lambda }_t^{\left(n\right)}-N_x^{\left(n\right)}+\left[M,N^{\left(n\right)}\right]=0,\end{array}$$

we obtain the well-known non-isospectral TD hierarchy:

$$\left\{\begin{array}{c}{q}_{t_n}=c_{n+1}-b_{n+1}=q^{-1}(\partial a_{n+1}+k_{n+1}\left(t\right)),\hfill \\ r_{t_n}=\partial \left[q^{-1}(c_{n+1}+b_{n+1})\right].\hfill \end{array}\right.$$

(8)

Here, we have used the first equation of (5).

The first nontrivial system in the hierarchy (8) is as follows:

$$\left\{\begin{array}{c}{q}_{t_2}=(x{q}_x+q)k_1\left(t\right)+{\displaystyle \frac{1}{2}}(r{q}_x+{\left(rq\right)}_x),\hfill \\ r_{t_2}=2k_2\left(t\right)+(x{r}_x+r)k_1\left(t\right)+{\displaystyle \frac{1}{2}}{(r^2+q^{-1}{q}_{xx}-2q^2)}_x.\hfill \end{array}\right.$$

(9)

Set $k_1\left(t\right)=k_2\left(t\right)=0,t=2\tilde{t}$, and (9) reduces to

$$\left\{\begin{array}{c}{q}_{{\tilde{t}}_2}=r{q}_x+{\left(rq\right)}_x,\hfill \\ r_{{\tilde{t}}_2}={(r^2+q^{-1}{q}_{xx}-2q^2)}_x,\hfill \end{array}\right.$$

(10)

which is the TD equation [2].

When $n=3$, (8) becomes

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill q_{t_3}& =(\frac{1}{2}xq{r}_x+rq+xr{q}_x)k_1\left(t\right)+(x{q}_x+q)k_2\left(t\right)+{\displaystyle \frac{1}{4}}(q_{xxx}-6q^2{q}_x+3r{\left(rq\right)}_x),\hfill \\ \hfill r_{t_3}& =2k_3\left(t\right)+\partial (x{r}_x+r)k_2\left(t\right)+\partial (\frac{1}{2}x{q}^{-1}{q}_{xx}+\frac{1}{2}x{r}^2+q^{-1}{q}_x-2{\partial }^{-1}xq{q}_x-2{\partial }^{-1}{q}^2)k_1\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}{(r^3+3q^{-1}{r}_x{q}_x+3r{q}^{-1}{q}_{xx}+r_{xx}-6q^{2}r)}_x.\hfill \end{array}\hfill \end{array}\right.$$

(11)

Let $k_1\left(t\right)=k_2\left(t\right)=k_3\left(t\right)=0,r=0,t=-4\tilde{t},q=i\tilde{q}$, (11) reduces to

$$\begin{array}{c}\hfill {\tilde{q}}_{{\tilde{t}}_3}+{\tilde{q}}_{xxx}+6{\tilde{q}}^2{\tilde{q}}_x=0,\end{array}$$

which is the MKdV equation.

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:

$$\begin{array}{c}\hfill \frac{\partial M}{\partial \lambda }=-h_1+h_2,\frac{\partial M}{\partial q}=h_3+h_4,\frac{\partial M}{\partial r}=h_1,\end{array}$$

$$\begin{array}{ccc}\hfill & \mathrm{tr}\left(N\frac{\partial M}{\partial \lambda }\right)=-2a,\mathrm{tr}\left(N\frac{\partial M}{\partial q}\right)=b+c,\hfill & \hfill \mathrm{tr}\left(N\frac{\partial M}{\partial r}\right)=a.\end{array}$$

Bearing in mind that the trace identity

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}\int \mathrm{tr}\left(N\frac{\partial M}{\partial \lambda }\right)dx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }{\left(\mathrm{tr}\left(N\frac{\partial M}{\partial q}\right),\mathrm{tr}\left(N\frac{\partial M}{\partial r}\right)\right)}^T,\end{array}$$

we arrive at

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}\int -2adx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}b+c\\ a\end{array}\right).\end{array}$$

By comparing the coefficients of ${\lambda }^{-m-1}$ in the above equation, we obtain

$$\frac{\delta }{\delta u}\int -2a_{m+1}dx=(\gamma -m)\left(\begin{array}{c}{b}_m+c_m\\ a_m\end{array}\right).$$

We figured out $\gamma =0$ via inserting $m=1$ into the above equation. Thus, we have

$$\begin{array}{cc}\hfill & \frac{\delta }{\delta u}\int \frac{a_{m+2}}{m+1}dx=\frac{1}{2}\left(\begin{array}{c}{b}_{m+1}+c_{m+1}\\ a_{m+1}\end{array}\right)=\frac{\delta H_{m+1}}{\delta u},\hfill \end{array}$$

(12)

where $H_{m+1}=\int \frac{a_{m+2}}{m+1}dx$. A recurrence relationship is generated by a direct calculation, as follows

$$\begin{array}{c}\hfill \left(\begin{array}{c}{b}_{m+1}+c_{m+1}\\ a_{m+1}\end{array}\right)=L\left(\begin{array}{c}{b}_m+c_m\\ a_m\end{array}\right)+R{k}_m\left(t\right)+R_0{k}_{m+1}\left(t\right),m\ge 0,\end{array}$$

(13)

where

$$\begin{array}{c}\hfill L=\frac{1}{2}\left(\begin{array}{cc}r& \partial q^{-1}\partial -4q\\ {\partial }^{-1}q\partial & {\partial }^{-1}r\partial \end{array}\right),R=\frac{1}{2}\left(\begin{array}{c}\partial q^{-1}\\ {\partial }^{-1}r\end{array}\right),R_0=\left(\begin{array}{c}0\\ x\end{array}\right).\end{array}$$

According to (12) and (13), the non-isospectral TD hierarchy (8) can be represented as the following:

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}& =\left(\begin{array}{c}{q}^{-1}(\partial a_{n+1}+k_{n+1}\left(t\right))\\ \partial \left[q^{-1}(b_{n+1}+c_{n+1})\right]\end{array}\right)=J\left(\begin{array}{c}{b}_{n+1}+c_{n+1}\\ a_{n+1}\end{array}\right)+R_1{k}_{n+1}\left(t\right)\hfill \\ & =:2J\frac{\delta H_{n+1}}{\delta u}+R_1{k}_{n+1}\left(t\right)\hfill \\ & =J\left(2L\frac{\delta H_n}{\delta u}+R{k}_n\left(t\right)+R_0{k}_{n+1}\left(t\right)\right)+R_1{k}_{n+1}\left(t\right)\hfill \\ & =:Q\frac{\delta H_n}{\delta u}+JR{k}_n\left(t\right)+(J{R}_0+R_1)k_{n+1}\left(t\right)\hfill \\ & =J{L}^n\left(\begin{array}{c}{b}_1+c_1\\ a_1\end{array}\right)+\sum _{i=0}^{n}J{L}^{n-i}(R{k}_i\left(t\right)+R_0{k}_{i+1}\left(t\right))+R_1{k}_{n+1}\left(t\right),\hfill \end{array}$$

with $R_1$ and the Hamiltonian operators J and Q as follows:

$$\begin{array}{c}\hfill R_1=\left(\begin{array}{c}{q}^{-1}\\ 0\end{array}\right),J=\left(\begin{array}{cc}0& q^{-1}\partial \\ \partial q^{-1}& 0\end{array}\right),Q=\left(\begin{array}{cc}\partial & q^{-1}r\partial \\ \partial q^{-1}r& {(\partial q^{-1})}^2\partial -4\partial \end{array}\right).\end{array}$$

3. An Integrable Coupling of the TD Non-Isospectral Hierarchy

Similar to [26], in terms of the definition of Lie algebra, the Lie algebra $\mathrm{A}$ can be enlarged into a higher-dimensional Lie algebra.

The enlarged Lie algebra $A_{12}=\mathrm{span}{\left\{h_i\right\}}_{i=1}^8$ has the following basis:

$$\begin{array}{c}\hfill {\tilde{h}}_1=\left(\begin{array}{cc}{h}_1& 0\\ 0& h_1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right),{\tilde{h}}_2=\left(\begin{array}{cc}{h}_2& 0\\ 0& h_2\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{h}}_3=\left(\begin{array}{cc}{h}_3& 0\\ 0& h_3\end{array}\right)=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right),{\tilde{h}}_4=\left(\begin{array}{cc}{h}_4& 0\\ 0& h_4\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right),\\ \hfill {\tilde{h}}_5=\left(\begin{array}{cc}0& \epsilon h_1\\ h_1& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& \epsilon & 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),{\tilde{h}}_6=\left(\begin{array}{cc}0& \epsilon h_2\\ h_2& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& \epsilon \\ 0& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right),\\ \hfill {\tilde{h}}_7=\left(\begin{array}{cc}0& \epsilon h_3\\ h_3& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& \epsilon \\ 0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right),{\tilde{h}}_8=\left(\begin{array}{cc}0& \epsilon h_4\\ h_4& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& \epsilon & 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right),\end{array}$$

where the commutator is given by

$$\begin{array}{cc}\hfill & [{\tilde{h}}_1,{\tilde{h}}_2]=0,[{\tilde{h}}_1,{\tilde{h}}_3]={\tilde{h}}_3,[{\tilde{h}}_1,{\tilde{h}}_4]=-{\tilde{h}}_4,[{\tilde{h}}_1,{\tilde{h}}_5]=0,[{\tilde{h}}_1,{\tilde{h}}_6]=0,\hfill \\ \hfill & [{\tilde{h}}_1,{\tilde{h}}_7]={\tilde{h}}_7,[{\tilde{h}}_1,{\tilde{h}}_8]=-{\tilde{h}}_8,[{\tilde{h}}_2,{\tilde{h}}_3]=-{\tilde{h}}_3,[{\tilde{h}}_2,{\tilde{h}}_4]={\tilde{h}}_4,[{\tilde{h}}_2,{\tilde{h}}_5]=0,\hfill \\ \hfill & [{\tilde{h}}_2,{\tilde{h}}_6]=0,[{\tilde{h}}_2,{\tilde{h}}_7]=-{\tilde{h}}_7,[{\tilde{h}}_2,{\tilde{h}}_8]={\tilde{h}}_8,[{\tilde{h}}_3,{\tilde{h}}_4]={\tilde{h}}_1-{\tilde{h}}_2,[{\tilde{h}}_3,{\tilde{h}}_5]=-{\tilde{h}}_7,\hfill \\ \hfill & [{\tilde{h}}_3,{\tilde{h}}_6]={\tilde{h}}_7,[{\tilde{h}}_3,{\tilde{h}}_7]=0,[{\tilde{h}}_3,{\tilde{h}}_8]={\tilde{h}}_5-{\tilde{h}}_6,[{\tilde{h}}_4,{\tilde{h}}_5]={\tilde{h}}_8,[{\tilde{h}}_4,{\tilde{h}}_6]=-{\tilde{h}}_8,\hfill \\ & [{\tilde{h}}_4,{\tilde{h}}_7]=-{\tilde{h}}_5+{\tilde{h}}_6,[{\tilde{h}}_4,{\tilde{h}}_8]=0,[{\tilde{h}}_5,{\tilde{h}}_6]=0,[{\tilde{h}}_5,{\tilde{h}}_7]=\epsilon {\tilde{h}}_3,\hfill \\ \hfill & [{\tilde{h}}_5,{\tilde{h}}_8]=-\epsilon {\tilde{h}}_4,[{\tilde{h}}_6,{\tilde{h}}_7]=-\epsilon {\tilde{h}}_3,[{\tilde{h}}_6,{\tilde{h}}_8]=\epsilon {\tilde{h}}_4,[{\tilde{h}}_7,{\tilde{h}}_8]=\epsilon ({\tilde{h}}_1-{\tilde{h}}_2).\hfill \end{array}$$

Assume $G_1=\mathrm{span}\{{\tilde{h}}_1,{\tilde{h}}_2,{\tilde{h}}_3,{\tilde{h}}_4\}$, $G_2=\mathrm{span}\{{\tilde{h}}_5,{\tilde{h}}_6,{\tilde{h}}_7,{\tilde{h}}_8\}$, then $A_{12}=G_1\oplus G_2$. It is obvious that

$$[G_1,G_1]\subseteq G_1,[G_1,G_2]\subseteq G_2,[G_2,G_2]\subseteq G_1.$$

We introduce an enlarged non-isospectral problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\psi }_x=U\psi ,U=-\lambda {\tilde{h}}_1+\lambda {\tilde{h}}_2+r_1{\tilde{h}}_1+q_1{\tilde{h}}_3+q_1{\tilde{h}}_4-\lambda {\tilde{h}}_5+\lambda {\tilde{h}}_6+r_2{\tilde{h}}_5+q_2{\tilde{h}}_7+q_2{\tilde{h}}_8,\hfill \\ {\psi }_t=V\psi ,V=a{\tilde{h}}_1-a{\tilde{h}}_2+b{\tilde{h}}_3+c{\tilde{h}}_4+d{\tilde{h}}_5-d{\tilde{h}}_6+e{\tilde{h}}_7+f{\tilde{h}}_8,\hfill \\ {\lambda }_t={\displaystyle \sum _{m\ge 0}}{k}_m\left(t\right){\lambda }^{-m},\hfill \end{array}\right.\end{array}$$

(14)

where

$$\begin{array}{cc}\hfill & U=\left(\begin{array}{cc}{U}_1& \epsilon U_2\\ U_2& U_1\end{array}\right),V=\left(\begin{array}{cc}{V}_1& \epsilon V_2\\ V_2& V_1\end{array}\right),\hfill \\ \hfill & U_1=-\lambda h_1+\lambda h_2+r_1{h}_1+q_1{h}_3+q_1{h}_4,U_2=-\lambda h_1+\lambda h_2+r_2{h}_1+q_2{h}_3+q_2{h}_4,\hfill \\ \hfill & V_1=a{h}_1-a{h}_2+b{h}_3+c{h}_4,V_2=\widehat{a}{h}_1-\widehat{a}{h}_2+\widehat{b}{h}_3+\widehat{c}{h}_4.\hfill \end{array}$$

(15)

and $\tilde{u}={(q_1,r_1,q_2,r_2)}^T$. Choosing $a={\displaystyle \sum _{m\ge 0}}{a}_m{\lambda }^{-m},b={\displaystyle \sum _{m\ge 0}}{b}_m{\lambda }^{-m},c={\displaystyle \sum _{m\ge 0}}{c}_m{\lambda }^{-m},\widehat{a}={\displaystyle \sum _{m\ge 0}}{\widehat{a}}_m{\lambda }^{-m},\widehat{b}={\displaystyle \sum _{m\ge 0}}{\widehat{b}}_m{\lambda }^{-m},\widehat{c}={\displaystyle \sum _{m\ge 0}}{\widehat{c}}_m{\lambda }^{-m}$, and solving the stationary zero curvature equation

$$V_x=\frac{\partial U}{\partial \lambda }{\lambda }_t+[U,V],$$

(16)

we have

$$\left\{\begin{array}{c}{a}_{mx}=q_1(c_m-b_m)+\epsilon q_2({\widehat{c}}_m-{\widehat{b}}_m)-k_m\left(t\right),\hfill \\ b_{mx}=-2b_{m+1}-2\epsilon {\widehat{b}}_{m+1}+r_1{b}_m+\epsilon r_2{\widehat{b}}_m-2q_1{a}_m-2\epsilon q_2{\widehat{a}}_m,\hfill \\ c_{mx}=2c_{m+1}+2\epsilon {\widehat{c}}_{m+1}-r_1{c}_m-\epsilon r_2{\widehat{c}}_m+2q_1{a}_m+2\epsilon q_2{\widehat{a}}_m,\hfill \\ {\widehat{a}}_{mx}=q_1({\widehat{c}}_m-{\widehat{b}}_m)+q_2(c_m-b_m)-k_m\left(t\right),\hfill \\ {\widehat{b}}_{mx}=-2b_{m+1}-2{\widehat{b}}_{m+1}+r_1{\widehat{b}}_m+r_2{b}_m-2q_1{\widehat{a}}_m-2q_2{a}_m,\hfill \\ {\widehat{c}}_{mx}=2c_{m+1}+2{\widehat{c}}_{m+1}-r_1{\widehat{c}}_m-r_2{c}_m+2q_1{\widehat{a}}_m+2q_2{a}_m.\hfill \end{array}\right.$$

(17)

A straightforward computation yields the following recursion equations:

$$\left\{\begin{array}{c}{a}_m={\partial }^{-1}[q_1(c_m-b_m)+\epsilon q_2({\widehat{c}}_m-{\widehat{b}}_m)]-k_m\left(t\right)x,\hfill \\ b_{m+1}=\frac{1}{2(\epsilon -1)}[b_{mx}-\epsilon {\widehat{b}}_{mx}+2q_1(a_m-\epsilon {\widehat{a}}_m)-r_1(b_m-\epsilon {\widehat{b}}_m)+\epsilon (2q_2({\widehat{a}}_m-a_m)-r_2({\widehat{b}}_m-b_m))],\hfill \\ c_{m+1}=\frac{1}{2(\epsilon -1)}[\epsilon {\widehat{c}}_{mx}-c_{mx}+2q_1(a_m-\epsilon {\widehat{a}}_m)-r_1(c_m-\epsilon {\widehat{c}}_m)+\epsilon (2q_2({\widehat{a}}_m-a_m)-r_2({\widehat{c}}_m-c_m))],\hfill \\ {\widehat{a}}_m={\partial }^{-1}[q_1({\widehat{c}}_m-{\widehat{b}}_m)+q_2(c_m-b_m)]-k_m\left(t\right)x,\hfill \\ {\widehat{b}}_{m+1}=\frac{1}{2(\epsilon -1)}[{\widehat{b}}_{mx}-b_{mx}+2q_2(a_m-\epsilon {\widehat{a}}_m)-r_2(b_m-\epsilon {\widehat{b}}_m)+2q_1({\widehat{a}}_m-a_m)-r_1({\widehat{b}}_m-b_m)],\hfill \\ {\widehat{c}}_{m+1}=\frac{1}{2(\epsilon -1)}[c_{mx}-{\widehat{c}}_{mx}+2q_2(a_m-\epsilon {\widehat{a}}_m)-r_2(c_m-\epsilon {\widehat{c}}_m)+2q_1({\widehat{a}}_m-a_m)-r_1({\widehat{c}}_m-c_m)].\hfill \end{array}\right.$$

(18)

Assuming

$$\begin{array}{c}\hfill a_0=-1,b_0=0,c_0=0,k_0=0,{\widehat{a}}_0=-1,{\widehat{b}}_0=0,{\widehat{c}}_0=0,\end{array}$$

then (18) gives rise to

$$\begin{array}{cc}\hfill & b_1=q_1,c_1=q_1,{\widehat{b}}_1=q_2,{\widehat{c}}_1=q_2,a_1=-k_1\left(t\right)x,{\widehat{a}}_1=-k_1\left(t\right)x,\hfill \\ \hfill & b_2=q_1{k}_1\left(t\right)x+\frac{1}{2(\epsilon -1)}[q_{1x}-\epsilon q_{2x}-r_1(q_1-\epsilon q_2)-\epsilon r_2(q_2-q_1)],\hfill \\ \hfill & c_2=q_1{k}_1\left(t\right)x+\frac{1}{2(\epsilon -1)}[\epsilon q_{2x}-q_{1x}-r_1(q_1-\epsilon q_2)-\epsilon r_2(q_2-q_1)],\hfill \\ \hfill & {\widehat{b}}_2=q_2{k}_1\left(t\right)x+\frac{1}{2(\epsilon -1)}[q_{2x}-q_{1x}-r_2(q_1-\epsilon q_2)-r_1(q_2-q_1)],\hfill \\ \hfill & {\widehat{c}}_2=q_2{k}_1\left(t\right)x+\frac{1}{2(\epsilon -1)}[q_{1x}-q_{2x}-r_2(q_1-\epsilon q_2)-r_1(q_2-q_1)],\hfill \\ \hfill & a_2=\frac{1}{2(\epsilon -1)}[-q_1^2+\epsilon q_2^2+2\epsilon q_2(q_1-q_2)]-k_2\left(t\right)x,\hfill \\ \hfill & {\widehat{a}}_2=\frac{1}{2(\epsilon -1)}[q_1^2-\epsilon q_2^2+2q_2(\epsilon q_2-q_1)]-k_2\left(t\right)x,\hfill \\ \hfill & \cdots .\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill V_{+}^{\left(n\right)}=& a_m{\tilde{h}}_1{\lambda }^{n-m}-a_m{\tilde{h}}_2{\lambda }^{n-m}+b_m{\tilde{h}}_3{\lambda }^{n-m}+c_m{\tilde{h}}_4{\lambda }^{n-m}\hfill \\ \hfill & +{\widehat{a}}_m{\tilde{h}}_5{\lambda }^{n-m}-{\widehat{a}}_m{\tilde{h}}_6{\lambda }^{n-m}+{\widehat{b}}_m{\tilde{h}}_7{\lambda }^{n-m}+{\widehat{c}}_m{\tilde{h}}_8{\lambda }^{n-m},\hfill \end{array}$$

then, we have

$$\begin{array}{cc}\hfill -V_{+,x}^{\left(n\right)}+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}+\left[U,V_{+}^{\left(n\right)}\right]& =V_{-,x}^{\left(n\right)}-\frac{\partial U}{\partial \lambda }{\lambda }_{t,-}^{\left(n\right)}-\left[U,V_{-}^{\left(n\right)}\right]\hfill \\ \hfill & =2(b_{n+1}+\epsilon {\widehat{b}}_{n+1}){\tilde{h}}_3-2(c_{n+1}+\epsilon {\widehat{c}}_{n+1}){\tilde{h}}_4\hfill \\ \hfill & +2(b_{n+1}+{\widehat{b}}_{n+1}){\tilde{h}}_7-2(c_{n+1}+{\widehat{c}}_{n+1}){\tilde{h}}_8.\hfill \end{array}$$

Letting

$$V^{\left(n\right)}=V_{+}^{\left(n\right)}+{\tilde{\Delta }}_n$$

and the modified term

$$\begin{array}{cc}\hfill {\tilde{\Delta }}_n=& \frac{[q_1(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})-\epsilon q_2(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})]{\tilde{h}}_1}{q_1^2-\epsilon q_2^2}\hfill \\ \hfill & +\frac{[q_1(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})-q_2(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})]{\tilde{h}}_5}{q_1^2-\epsilon q_2^2},\hfill \end{array}$$

the non-isospectral zero curvature equation

$$\begin{array}{c}\hfill U_{t_n}+\frac{\partial U}{\partial \lambda }{\lambda }_t^{\left(n\right)}-V_x^{\left(n\right)}+\left[U,V^{\left(n\right)}\right]=0,\end{array}$$

gives the integrable coupling of non-isospectral TD hierarchy

$$\left\{\begin{array}{c}\begin{array}{c}{q}_{1t_n}=c_{n+1}+\epsilon {\widehat{c}}_{n+1}-b_{n+1}-\epsilon {\widehat{b}}_{n+1}\hfill \\ ={\displaystyle \frac{q_1(a_{n,x}+\epsilon {\widehat{a}}_{n,x})-\epsilon q_2(a_{n,x}+{\widehat{a}}_{n,x})+(q_1-\epsilon q_2)k_n\left(t\right)+\epsilon (q_1-q_2)k_n\left(t\right)}{q_1^2-\epsilon q_2^2}},\hfill \\ r_{1t_n}=\partial \left({\displaystyle \frac{q_1(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})-\epsilon q_2(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right),\hfill \\ q_{2t_n}=c_{n+1}+{\widehat{c}}_{n+1}-b_{n+1}-{\widehat{b}}_{n+1}\hfill \\ ={\displaystyle \frac{q_1(a_{n,x}+{\widehat{a}}_{n,x})-q_2(a_{n,x}+\epsilon {\widehat{a}}_{n,x})+(q_1-q_2)k_n\left(t\right)+(q_1-\epsilon q_2)k_n\left(t\right)}{q_1^2-\epsilon q_2^2}},\hfill \\ r_{2t_n}=\partial \left({\displaystyle \frac{q_1(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})-q_2(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right).\hfill \end{array}\hfill \end{array}\right.$$

(19)

When $n=2$, (19) leads to

$$\left\{\begin{array}{c}\begin{array}{c}{q}_{1,t_2}=(x{q}_{1,x}+q_1)k_1\left(t\right)\hfill \\ +{\displaystyle \frac{1}{2(\epsilon -1)}}\left[2r_1(\epsilon q_{2,x}-q_{1,x})+2\epsilon r_2(q_{1,x}-q_{2,x})+q_1(\epsilon r_{2,x}-r_{1,x})+\epsilon q_2(r_{1,x}-r_{2,x})\right],\hfill \\ r_{1,t_2}=2k_2\left(t\right)+(r_1+x{r}_{1,x})k_1\left(t\right)\hfill \\ +{\displaystyle \frac{1}{2(\epsilon -1)}}\partial \left[2\epsilon r_1{r}_2-r_1^2-\epsilon r_2^2-4\epsilon q_1{q}_2-\frac{q_1(q_{1,xx}-\epsilon q_{2,xx})+\epsilon q_2(q_{1,xx}-q_{2,xx})+2{\epsilon }^2{q}_2^4-2q_1^4}{q_1^2-\epsilon q_2^2}\right],\hfill \\ q_{2,t_2}=(x{q}_{2,x}+q_2)k_1\left(t\right)\hfill \\ +{\displaystyle \frac{1}{2(\epsilon -1)}}\left[2r_1(q_{1,x}-q_{2,x})+2r_2(\epsilon q_{2,x}-q_{1,x})+q_1(r_{1,x}-r_{2,x})+q_2(\epsilon r_{2,x}-r_{1,x})\right],\hfill \\ r_{2,t_2}=2k_2\left(t\right)+(r_2+x{r}_{2,x})k_1\left(t\right)\hfill \\ +{\displaystyle \frac{1}{2(\epsilon -1)}}\partial \left[r_1^2+\epsilon r_2^2-2r_1{r}_2+4q_1{q}_2+\frac{q_1(q_{1,xx}-q_{2,xx})+q_2(q_{1,xx}-\epsilon q_{2,xx})+2{\epsilon }^2{q}_2^4-2q_1^4}{q_1^2-\epsilon q_2^2}\right].\hfill \end{array}\hfill \end{array}\right.$$

(20)

Let $\epsilon =0$; 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 $Z_2^{\epsilon }$-trace identities, which were discussed in [26], we shall construct the Bi-Hamiltonian structures of the integrable coupling of the non-isospectral hierarchy (19):

$$\begin{array}{c}\hfill \frac{\delta }{\delta \tilde{u}}\int \left(tr\left(V_1\frac{\partial U_2}{\partial \lambda }\right)+tr\left(V_2\frac{\partial U_1}{\partial \lambda }\right)\right)dx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(tr\left(V_1\frac{\partial U_2}{\partial \tilde{u}}\right)+tr\left(V_2\frac{\partial U_1}{\partial \tilde{u}}\right)\right),\\ \hfill \frac{\delta }{\delta \tilde{u}}\int \left(tr\left(V_1\frac{\partial U_1}{\partial \lambda }\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial \lambda }\right)\right)dx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(tr\left(V_1\frac{\partial U_1}{\partial \tilde{u}}\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial \tilde{u}}\right)\right).\end{array}$$

(21)

From (15), we obtain

$$\begin{array}{c}tr\left(V_1\frac{\partial U_2}{\partial q_1}\right)+tr\left(V_2\frac{\partial U_1}{\partial q_1}\right)=\widehat{b}+\widehat{c},tr\left(V_1\frac{\partial U_2}{\partial q_2}\right)+tr\left(V_2\frac{\partial U_1}{\partial q_2}\right)=b+c,\hfill \\ tr\left(V_1\frac{\partial U_2}{\partial r_1}\right)+tr\left(V_2\frac{\partial U_1}{\partial r_1}\right)=\widehat{a},tr\left(V_1\frac{\partial U_2}{\partial r_2}\right)+tr\left(V_2\frac{\partial U_1}{\partial r_2}\right)=a,\hfill \end{array}$$

$$\begin{array}{c}tr\left(V_1\frac{\partial U_1}{\partial q_1}\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial q_1}\right)=b+c,tr\left(V_1\frac{\partial U_1}{\partial q_2}\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial q_2}\right)=\epsilon (\widehat{b}+\widehat{c}),\hfill \\ tr\left(V_1\frac{\partial U_1}{\partial r_1}\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial r_1}\right)=a,tr\left(V_1\frac{\partial U_1}{\partial r_2}\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial r_2}\right)=\epsilon \widehat{a},\hfill \\ tr\left(V_1\frac{\partial U_2}{\partial \lambda }\right)+tr\left(V_2\frac{\partial U_1}{\partial \lambda }\right)=-2(a+\widehat{a}),tr\left(V_1\frac{\partial U_1}{\partial \lambda }\right)+\epsilon tr\left(V_2\frac{\partial U_2}{\partial \lambda }\right)=-2(a+\epsilon \widehat{a}).\hfill \end{array}$$

Substituting the above equations into (21) and collecting the terms with the same powers of $\lambda$ produces

$$\begin{array}{c}\hfill \frac{\delta }{\delta \tilde{u}}\int -2(a_{m+1}+{\widehat{a}}_{m+1})dx=(\gamma -m)\left(\begin{array}{c}{\widehat{b}}_m+{\widehat{c}}_m\\ {\widehat{a}}_m\\ b_m+c_m\\ a_m\end{array}\right),\\ \hfill \frac{\delta }{\delta \tilde{u}}\int -2(a_{m+1}+\epsilon {\widehat{a}}_{m+1})dx=(\gamma -m)\left(\begin{array}{c}{b}_m+c_m\\ a_m\\ \epsilon ({\widehat{b}}_m+{\widehat{c}}_m)\\ \epsilon {\widehat{a}}_m\end{array}\right).\end{array}$$

Set $m=1$; we have $\gamma =0$. Therefore, we conclude that

$$\begin{array}{cc}\hfill & \frac{\delta }{\delta \tilde{u}}\int \frac{a_{m+2}+{\widehat{a}}_{m+2}}{m+1}dx=\frac{1}{2}\left(\begin{array}{c}{\widehat{b}}_{m+1}+{\widehat{c}}_{m+1}\\ {\widehat{a}}_{m+1}\\ b_{m+1}+c_{m+1}\\ a_{m+1}\end{array}\right)=\frac{\delta H_{m+1}}{\delta \tilde{u}},\hfill \\ \hfill & \frac{\delta }{\delta \tilde{u}}\int \frac{a_{m+2}+\epsilon {\widehat{a}}_{m+2}}{m+1}dx=\frac{1}{2}\left(\begin{array}{c}{b}_{m+1}+c_{m+1}\\ a_{m+1}\\ \epsilon ({\widehat{b}}_{m+1}+{\widehat{c}}_{m+1})\\ \epsilon {\widehat{a}}_{m+1}\end{array}\right)=\frac{\delta {\widehat{H}}_{m+1}}{\delta \tilde{u}},\hfill \end{array}$$

(22)

where

$$\begin{array}{c}\hfill H_{m+1}=\int \frac{a_{m+2}+{\widehat{a}}_{m+2}}{m+1}dx,{\widehat{H}}_{m+1}=\int \frac{a_{m+2}+\epsilon {\widehat{a}}_{m+2}}{m+1}dx.\end{array}$$

From (18), we obtain two kind of recursive relationships:

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\widehat{b}}_{m+1}+{\widehat{c}}_{m+1}\\ {\widehat{a}}_{m+1}\\ b_{m+1}+c_{m+1}\\ a_{m+1}\end{array}\right)={\tilde{L}}_1\left(\begin{array}{c}{\widehat{b}}_m+{\widehat{c}}_m\\ {\widehat{a}}_m\\ b_m+c_m\\ a_m\end{array}\right)+{\tilde{R}}_1{k}_m\left(t\right)+{\tilde{R}}_0{k}_{m+1}\left(t\right),m\ge 0,\end{array}$$

(23)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{b}_{m+1}+c_{m+1}\\ a_{m+1}\\ \epsilon ({\widehat{b}}_{m+1}+{\widehat{c}}_{m+1})\\ \epsilon {\widehat{a}}_{m+1}\end{array}\right)={\tilde{L}}_1\left(\begin{array}{c}{b}_m+c_m\\ a_m\\ \epsilon ({\widehat{b}}_m+{\widehat{c}}_m)\\ \epsilon {\widehat{a}}_m\end{array}\right)+{\widehat{R}}_1{k}_m\left(t\right)+{\widehat{R}}_0{k}_{m+1}\left(t\right),m\ge 0,\end{array}$$

(24)

where the recursion operator ${\tilde{L}}_1$ becomes

$$\begin{array}{c}\hfill {\tilde{L}}_1=\frac{1}{2(\epsilon -1)}\left(\begin{array}{cccc}{\tilde{L}}_{11}& {\tilde{L}}_{12}& {\tilde{L}}_{13}& {\tilde{L}}_{14}\\ {\tilde{L}}_{21}& {\tilde{L}}_{22}& {\tilde{L}}_{23}& {\tilde{L}}_{24}\\ \epsilon {\tilde{L}}_{13}& \epsilon {\tilde{L}}_{14}& {\tilde{L}}_{11}& {\tilde{L}}_{12}\\ \epsilon {\tilde{L}}_{23}& \epsilon {\tilde{L}}_{24}& {\tilde{L}}_{21}& {\tilde{L}}_{22}\end{array}\right),\end{array}$$

with

$$\begin{array}{cc}\hfill & {\tilde{L}}_{11}=-r_1+\epsilon r_2,{\tilde{L}}_{12}=-\partial \left(\frac{q_1+\epsilon q_2}{q_1^2-\epsilon q_2^2}\right)\partial +4q_1-4\epsilon q_2,{\tilde{L}}_{13}=r_1-r_2,\hfill \\ \hfill & {\tilde{L}}_{14}=\partial \left(\frac{q_1+q_2}{q_1^2-\epsilon q_2^2}\right)\partial -4q_1+4q_2,{\tilde{L}}_{21}={\partial }^{-1}(\epsilon q_2-q_1)\partial ,\hfill \\ \hfill & {\tilde{L}}_{22}={\partial }^{-1}(\epsilon r_2-r_1)\partial ,{\tilde{L}}_{23}={\partial }^{-1}(q_1-q_2)\partial ,{\tilde{L}}_{24}={\partial }^{-1}(r_1-r_2)\partial ,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\tilde{R}}_1=\frac{1}{2}\left(\begin{array}{c}-\partial \left({\displaystyle \frac{q_2}{q_1^2-\epsilon q_2^2}}\right)\\ {\partial }^{-1}{r}_2\\ \partial \left({\displaystyle \frac{q_1}{q_1^2-\epsilon q_2^2}}\right)\\ {\partial }^{-1}{r}_1\end{array}\right),{\widehat{R}}_1=\frac{1}{2}\left(\begin{array}{c}\partial \left({\displaystyle \frac{q_1}{q_1^2-\epsilon q_2^2}}\right)\\ {\partial }_x^{-1}{r}_1\\ -\epsilon \partial \left({\displaystyle \frac{q_2}{q_1^2-\epsilon q_2^2}}\right)\\ \epsilon {\partial }^{-1}{r}_2\end{array}\right),{\tilde{R}}_0=\left(\begin{array}{c}0\\ x\\ 0\\ x\end{array}\right),{\widehat{R}}_0=\left(\begin{array}{c}0\\ x\\ 0\\ \epsilon x\end{array}\right).\end{array}$$

Then, the hierarchy in (19) determines the following two kinds of Bi-Hamiltonian structures. The first one is

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}{q}_1\\ r_1\\ q_2\\ r_2\end{array}\right)}_{t_n}& =\left(\begin{array}{c}{\displaystyle \frac{q_1(a_{n+1,x}+\epsilon {\widehat{a}}_{n+1,x})-\epsilon q_2(a_{n+1,x}+{\widehat{a}}_{n+1,x})+(q_1-\epsilon q_2)k_{n+1}+\epsilon (q_1-q_2)k_{n+1}}{q_1^2-\epsilon q_2^2}}\\ \partial \left({\displaystyle \frac{q_1(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})-\epsilon q_2(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right)\\ {\displaystyle \frac{q_1(a_{n+1,x}+{\widehat{a}}_{n+1,x})-q_2(a_{n+1,x}+\epsilon {\widehat{a}}_{n+1,x})+(q_1-q_2)k_{n+1}+(q_1-\epsilon q_2)k_{n+1}}{q_1^2-\epsilon q_2^2}}\\ \partial \left({\displaystyle \frac{q_1(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})-q_2(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right)\end{array}\right)\hfill \\ & ={\tilde{J}}_1\left(\begin{array}{c}{\widehat{b}}_{n+1}+{\widehat{c}}_{m+1}\\ {\widehat{a}}_{n+1}\\ b_{n+1}+c_{n+1}\\ a_{n+1}\end{array}\right)+\tilde{R}{k}_{n+1}=:2{\tilde{J}}_1\frac{\delta H_{n+1}}{\delta \tilde{u}}+\tilde{R}{k}_{n+1}\hfill \\ & ={\tilde{J}}_1(2{\tilde{L}}_1\frac{\delta H_n}{\delta \tilde{u}}+{\tilde{R}}_1{k}_n\left(t\right)+{\tilde{R}}_0{k}_{n+1}\left(t\right))+\tilde{R}{k}_{n+1}\left(t\right)=:{\tilde{Q}}_1\frac{\delta H_n}{\delta \tilde{u}}+{\tilde{J}}_1{\tilde{R}}_1{k}_n\left(t\right)+({\tilde{J}}_1{\tilde{R}}_0+\tilde{R})k_{n+1}\hfill \\ & ={\tilde{J}}_1{\tilde{L}}_1^n\left(\begin{array}{c}{\widehat{b}}_1+{\widehat{c}}_1\\ {\widehat{a}}_1\\ b_1+c_1\\ a_1\end{array}\right)+\sum _{i=0}^n{\tilde{J}}_1{\tilde{L}}_1^{n-i}({\tilde{R}}_1{k}_i\left(t\right)+{\tilde{R}}_0{k}_{i+1}\left(t\right))+\tilde{R}{k}_{n+1},\hfill \end{array}$$

with the Hamiltonian operators ${\tilde{J}}_1$, ${\tilde{Q}}_1$, and $\tilde{R}$, as follows:

$$\begin{array}{cc}\hfill & {\tilde{J}}_1=\left(\begin{array}{cccc}0& {\displaystyle \frac{\epsilon (q_1-q_2)}{q_1^2-\epsilon q_2^2}}\partial & 0& {\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial \\ \epsilon \partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right)& 0& \partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right)& 0\\ 0& {\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial & 0& {\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial \\ \partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right)& 0& \partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right)& 0\end{array}\right),\hfill \\ \hfill & {\tilde{Q}}_1=\frac{1}{\epsilon -1}\left(\begin{array}{cccc}\epsilon {\tilde{Q}}_{11}& \epsilon {\tilde{Q}}_{12}& {\tilde{Q}}_{13}& {\tilde{Q}}_{14}\\ \epsilon {\tilde{Q}}_{21}& \epsilon {\tilde{Q}}_{22}& {\tilde{Q}}_{23}& {\tilde{Q}}_{24}\\ {\tilde{Q}}_{13}& {\tilde{Q}}_{14}& {\tilde{Q}}_{11}& {\tilde{Q}}_{12}\\ {\tilde{Q}}_{23}& {\tilde{Q}}_{24}& {\tilde{Q}}_{21}& {\tilde{Q}}_{22}\end{array}\right),\tilde{R}=\frac{1}{q_1^2-\epsilon q_2^2}\left(\begin{array}{c}(1+\epsilon )q_1-2\epsilon q_2\\ 0\\ 2q_1-(1+\epsilon )q_2\\ 0\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}& {\tilde{Q}}_{11}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{21}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{23},{\tilde{Q}}_{12}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{22}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{24},\hfill \\ & {\tilde{Q}}_{13}={\displaystyle \frac{\epsilon (q_1-q_2)}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{23}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{21},{\tilde{Q}}_{14}={\displaystyle \frac{\epsilon (q_1-q_2)}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{24}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{22},\hfill \\ & {\tilde{Q}}_{21}=\partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{11}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{13},{\tilde{Q}}_{22}=\partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{12}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{14},\hfill \\ & {\tilde{Q}}_{23}=\epsilon \partial \left(\frac{q_1-q_2}{q_1^2-\epsilon q_2^2}\right){\tilde{L}}_{13}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{11},{\tilde{Q}}_{24}=\epsilon \partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{14}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{12}.\hfill \end{array}$$

Similarly, for the second component, we have

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}{q}_1\\ r_1\\ q_2\\ r_2\end{array}\right)}_{t_n}& =\left(\begin{array}{c}{\displaystyle \frac{q_1(a_{n+1,x}+\epsilon {\widehat{a}}_{n+1,x})-\epsilon q_2(a_{n+1,x}+{\widehat{a}}_{n+1,x})+(q_1-\epsilon q_2)k_{n+1}+\epsilon (q_1-q_2)k_{n+1}}{q_1^2-\epsilon q_2^2}}\\ \partial \left({\displaystyle \frac{q_1(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})-\epsilon q_2(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right)\\ {\displaystyle \frac{q_1(a_{n+1,x}+{\widehat{a}}_{n+1,x})-q_2(a_{n+1,x}+\epsilon {\widehat{a}}_{n+1,x})+(q_1-q_2)k_{n+1}+(q_1-\epsilon q_2)k_{n+1}}{q_1^2-\epsilon q_2^2}}\\ \partial \left({\displaystyle \frac{q_1(b_{n+1}+{\widehat{b}}_{n+1}+c_{n+1}+{\widehat{c}}_{n+1})-q_2(b_{n+1}+\epsilon {\widehat{b}}_{n+1}+c_{n+1}+\epsilon {\widehat{c}}_{n+1})}{q_1^2-\epsilon q_2^2}}\right)\end{array}\right)\hfill \\ & ={\widehat{J}}_1\left(\begin{array}{c}{b}_{n+1}+c_{n+1}\\ a_{n+1}\\ \epsilon ({\widehat{b}}_{n+1}+{\widehat{c}}_{n+1})\\ \epsilon {\widehat{a}}_{n+1}\end{array}\right)+\tilde{R}{k}_{n+1}=:2{\widehat{J}}_1\frac{\delta {\widehat{H}}_{n+1}}{\delta \tilde{u}}+\tilde{R}{k}_{n+1}\hfill \\ & ={\widehat{J}}_1(2{\tilde{L}}_1\frac{\delta {\widehat{H}}_n}{\delta \tilde{u}}+{\widehat{R}}_1{k}_n\left(t\right)+{\widehat{R}}_0{k}_{n+1}\left(t\right))+\tilde{R}{k}_{n+1}=:{\widehat{Q}}_1\frac{\delta {\widehat{H}}_n}{\delta \tilde{u}}+{\widehat{J}}_1({\widehat{R}}_1{k}_n\left(t\right)++{\widehat{R}}_0{k}_{n+1}\left(t\right))+\tilde{R}{k}_{n+1}\hfill \\ & ={\widehat{J}}_1{\tilde{L}}_1^n\left(\begin{array}{c}{b}_1+c_1\\ a_1\\ \epsilon ({\widehat{b}}_1+{\widehat{c}}_1)\\ \epsilon {\widehat{a}}_1\end{array}\right)+\sum _{i=0}^n{\widehat{J}}_1{\tilde{L}}^{n-i}({\widehat{R}}_1{k}_i\left(t\right)+{\widehat{R}}_0{k}_{i+1}\left(t\right))+\tilde{R}{k}_{n+1},\hfill \end{array}$$

with the Hamiltonian operators ${\widehat{J}}_1$ and ${\widehat{Q}}_1$, as follows:

$$\begin{array}{cc}\hfill & {\widehat{J}}_1=\left(\begin{array}{cccc}0& {\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial & 0& {\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial \\ \partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right)& 0& \partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right)& 0\\ 0& {\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial & 0& {\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\partial \\ \partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right)& 0& \partial \left({\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\right)& 0\end{array}\right),\hfill \\ \hfill & {\widehat{Q}}_1=\frac{1}{\epsilon -1}\left(\begin{array}{cccc}\epsilon {\widehat{Q}}_{11}& \epsilon {\widehat{Q}}_{12}& {\widehat{Q}}_{13}& {\widehat{Q}}_{14}\\ \epsilon {\widehat{Q}}_{21}& \epsilon {\widehat{Q}}_{22}& {\widehat{Q}}_{23}& {\widehat{Q}}_{24}\\ {\widehat{Q}}_{13}& {\widehat{Q}}_{14}& {\widehat{Q}}_{11}& {\widehat{Q}}_{12}\\ {\widehat{Q}}_{23}& {\widehat{Q}}_{24}& {\widehat{Q}}_{21}& {\widehat{Q}}_{22}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}& {\widehat{Q}}_{11}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{23}+{\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\partial {\tilde{L}}_{21},{\widehat{Q}}_{12}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{24}+{\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\partial {\tilde{L}}_{22},\hfill \\ & {\widehat{Q}}_{13}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{21}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{23},{\widehat{Q}}_{14}={\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{22}+{\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\partial {\tilde{L}}_{24},\hfill \\ & {\widehat{Q}}_{21}=\partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{13}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\right){\tilde{L}}_{11},{\widehat{Q}}_{22}=\partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{14}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{\epsilon (q_1^2-\epsilon q_2^2)}}\right){\tilde{L}}_{12},\hfill \\ & {\widehat{Q}}_{23}=\partial \left({\displaystyle \frac{q_1-q_2}{(q_1^2-\epsilon q_2^2)}}\right){\tilde{L}}_{11}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{13},{\widehat{Q}}_{24}=\partial \left({\displaystyle \frac{q_1-q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{12}+\partial \left({\displaystyle \frac{q_1-\epsilon q_2}{q_1^2-\epsilon q_2^2}}\right){\tilde{L}}_{14}.\hfill \end{array}$$

4. A Muti-Components Integrable TD Nonisospectral Hierarchy

In order to obtain the muti-components non-isospectral problem, we introduce the Lie algebra ${\mathrm{A}}_{1N}$ associated with the N-component non-isospectral problem.

First of all, a brief introduction to the Lie algebra ${\mathrm{A}}_{1N}$ related to the Lie algebra $\mathrm{A}$ will be given. Let us define a $N\times N$ square matrix:

$$\begin{array}{cc}\hfill M\left(m_1,m_2,\cdots ,m_N\right)& =\left(\begin{array}{ccccccc}{m}_1& \epsilon m_N& \epsilon m_{N-1}& \cdots & \epsilon m_4& \epsilon m_3& \epsilon m_2\\ m_2& m_1& \epsilon m_N& \cdots & \epsilon m_5& \epsilon m_4& \epsilon m_3\\ m_3& m_2& m_1& \cdots & \epsilon m_6& \epsilon m_5& \epsilon m_4\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ m_{N-2}& m_{N-3}& m_{N-4}& \cdots & m_1& \epsilon m_N& \epsilon m_{N-1}\\ m_{N-1}& m_{N-2}& m_{N-3}& \cdots & m_2& m_1& \epsilon m_N\\ m_N& m_{N-1}& m_{N-2}& \cdots & m_3& m_2& m_1\end{array}\right)\hfill \\ \hfill & =:{\left[m_1,m_2,\cdots ,m_N\right]}^T,\hfill \end{array}$$

(25)

where $m_n(1\le n\le N)$ represents n arbitrary square matrices of the same order.

For the arbitrary matrices $P={[P_1,P_2,\cdots ,P_N]}^T,Q={[Q_1,Q_2,\cdots ,Q_N]}^T,$ one has the Lie product:

$$\begin{array}{c}\hfill [P,Q]=\left(\begin{array}{ccccccc}{R}_1& \epsilon R_N& \epsilon R_{N-1}& \cdots & \epsilon R_4& \epsilon R_3& \epsilon R_2\\ R_2& R_1& \epsilon R_N& \cdots & \epsilon R_5& \epsilon R_4& \epsilon R_3\\ R_3& R_2& R_1& \cdots & \epsilon R_6& \epsilon R_5& \epsilon R_4\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ R_{N-2}& R_{N-3}& R_{N-4}& \cdots & R_1& \epsilon R_N& \epsilon R_{N-1}\\ R_{N-1}& R_{N-2}& R_{N-3}& \cdots & R_2& R_1& \epsilon R_N\\ R_N& R_{N-1}& R_{N-2}& \cdots & R_3& R_2& R_1\end{array}\right)=:{\left[R_1,R_2,\cdots ,R_N\right]}^T,\end{array}$$

where

$$R_l=\sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}[P_k,Q_j]+\sigma \epsilon \sum _{\begin{array}{c}s+t=l+1+N\\ l+1\le s,t\le N\end{array}}[P_s,Q_t],$$

is subject to

$$\begin{array}{c}\hfill \sigma =\left\{\begin{array}{cc}1,\hfill & 1\le l\le N-1,\hfill \\ 0,\hfill & l=N.\hfill \end{array}\right.\end{array}$$

Similarly, the matrix product $PQ$ can be easily written as

$$\begin{array}{cc}PQ=\hfill & [P_1{Q}_1+\sigma \epsilon \sum _{\begin{array}{c}s+t=2+N\\ 2\le s,t\le N\end{array}}{P}_s{Q}_t,\sum _{\begin{array}{c}k+j=3\\ 1\le k,j\le 2\end{array}}{P}_k{Q}_j+\sigma \epsilon \sum _{\begin{array}{c}s+t=3+N\\ 3\le s,t\le N\end{array}}{P}_s{Q}_t,\hfill \\ \hfill & \cdots ,\sum _{\begin{array}{c}k+j=l\\ 1\le k,j\le l\end{array}}{P}_k{Q}_j+\sigma \epsilon \sum _{\begin{array}{c}s+t=l+1+N\\ l+1\le s,t\le N\end{array}}{P}_s{Q}_t,\cdots ,\sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le N\end{array}}{P}_k{Q}_j{]}^T.\hfill \end{array}$$

When $\epsilon =0$, ${\left[R_1,R_2,\cdots ,R_N\right]}^T$ is a lower triangular matrix, $R_l={\displaystyle \sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}}[P_k,Q_j]$, and

$$PQ={[P_1{Q}_1,\sum _{\begin{array}{c}k+j=3\\ 1\le k,j\le 2\end{array}}{P}_k{Q}_j,\cdots ,\sum _{\begin{array}{c}k+j=l\\ 1\le k,j\le l\end{array}}{P}_k{Q}_j,\sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le N\end{array}}{P}_k{Q}_j]}^T.$$

Secondly, we denote ${\mathrm{A}}_{1N}=\mathrm{span}{\left\{{\widehat{h}}_i\right\}}_{i=1}^{4N}$, with commutators

$$\begin{array}{cc}\hfill & {\widehat{h}}_1=M(h_1,0,\cdots ,0),{\widehat{h}}_2=M(h_2,0,\cdots ,0),{\widehat{h}}_3=M(h_3,0,\cdots ,0),{\widehat{h}}_4=M(h_4,0,\cdots ,0),\hfill \\ \hfill & {\widehat{h}}_5=M(0,h_1,\cdots ,0),{\widehat{h}}_6=M(0,h_2,\cdots ,0),{\widehat{h}}_7=M(0,h_3,\cdots ,0),{\widehat{h}}_8=M(0,h_4,\cdots ,0),\hfill \\ \hfill & \cdots \hfill \\ \hfill & {\widehat{h}}_{4N-3}=M(0,0,\cdots ,h_1),{\widehat{h}}_{4N-2}=M(0,0,\cdots ,h_2),\hfill \\ \hfill & {\widehat{h}}_{4N-1}=M(0,0,\cdots ,h_3),{\widehat{h}}_{4N}=M(0,0,\cdots ,h_4),\hfill \\ \hfill & \left[{\widehat{h}}_{4l-3},{\widehat{h}}_{4k-2}\right]=\left[{\widehat{h}}_{4l-3},{\widehat{h}}_{4k-3}\right]=\left[{\widehat{h}}_{4l-2},{\widehat{h}}_{4k-2}\right]=\left[{\widehat{h}}_{4l-1},{\widehat{h}}_{4k-1}\right]=\left[{\widehat{h}}_{4l},{\widehat{h}}_{4k}\right]=0,l,k=1,2,\cdots ,N,\hfill \\ \hfill & \left[{\widehat{h}}_{4l-3},{\widehat{h}}_{4k-1}\right]=-\left[{\widehat{h}}_{4l-2},{\widehat{h}}_{4k-1}\right]=\left\{\begin{array}{c}{\widehat{h}}_{4(k+l-1)-1},1\le k\le N-l+1,\hfill \\ \epsilon {\widehat{h}}_{4(k+l-1-N)-1},N-l+2\le k\le N,\hfill \end{array}\right.\hfill \\ \hfill & \left[{\widehat{h}}_{4l-3},{\widehat{h}}_{4k}\right]=-\left[{\widehat{h}}_{4l-2},{\widehat{h}}_{4k}\right]=\left\{\begin{array}{c}-{\widehat{h}}_{4(k+l-1)},1\le k\le N-l+1,\hfill \\ -\epsilon {\widehat{h}}_{4(k+l-1-N)},N-l+2\le k\le N,\hfill \end{array}\right.\hfill \\ \hfill & \left[{\widehat{h}}_{4l-1},{\widehat{h}}_{4k}\right]=\left\{\begin{array}{c}{\widehat{h}}_{4(k+l-1)-3}-{\widehat{h}}_{4(k+l-1)-2},1\le k\le N-l+1,\hfill \\ \epsilon ({\widehat{h}}_{4(k+l-1-N)-3}-{\widehat{h}}_{4(k+l-1-N)-2}),N-l+2\le k\le N.\hfill \end{array}\right.\hfill \end{array}$$

Let ${\widehat{G}}_k=\mathrm{span}\left\{{\widehat{h}}_{4k-3},{\widehat{h}}_{4k-2},{\widehat{h}}_{4k-1},{\widehat{h}}_{4k}\right\},k=1,2,\cdots N$, then ${\mathrm{A}}_{1N}={\widehat{G}}_1\oplus {\widehat{G}}_2\oplus \cdots \oplus {\widehat{G}}_N$. Thus, we obtain

$$\begin{array}{c}\hfill \left[{\widehat{G}}_s,{\widehat{G}}_t\right]\subseteq {\widehat{G}}_{s+t-1-\delta N},\delta =\left\{\begin{array}{cc}0,\hfill & 2\le s+t\le N+1,\hfill \\ 1,\hfill & N+2\le s+t\le 2N,\hfill \end{array}s,t=1,2,\cdots ,N.\right.\end{array}$$

In terms of $A_{1N}$ under the case of $\epsilon =0$, we introduce the following N-component non-isospectral problem:

$$\left\{\begin{array}{c}{\psi }_x=\widehat{U}\psi ,\widehat{U}={\displaystyle \sum _{l=1}^N}[-\lambda {\widehat{h}}_{4l-3}+\lambda {\widehat{h}}_{4l-2}+r_l{\widehat{h}}_{4l-3}+q_l({\widehat{h}}_{4l-1}+{\widehat{h}}_{4l})],\hfill \\ {\psi }_t=\widehat{V}\psi ,\widehat{V}={\displaystyle \sum _{l=1}^N}[a_l{\widehat{h}}_{4l-3}-a_l{\widehat{h}}_{4l-2}+b_l{\widehat{h}}_{4l-1}+c_l{\widehat{h}}_{4l}],\hfill \\ {\lambda }_t={\displaystyle \sum _{m\ge 0}}{k}_m\left(t\right){\lambda }^{-m},\hfill \end{array}\right.$$

(26)

where $\widehat{u}={(q_1,r_1,q_2,r_2,\cdots ,q_N,r_N)}^T,\widehat{U}={\left[{\widehat{U}}_1,{\widehat{U}}_2,\cdots ,{\widehat{U}}_N\right]}^T,\widehat{V}={\left[{\widehat{V}}_1,{\widehat{V}}_2,\cdots ,{\widehat{V}}_N\right]}^T$ and

$$\begin{array}{c}\hfill {\widehat{U}}_l=-\lambda h_1+\lambda h_2+r_l{h}_1+q_l(h_3+h_4),{\widehat{V}}_l=a_l{h}_1-a_l{h}_2+b_l{h}_3+c_l{h}_4.\end{array}$$

By choosing

$$a_l={\displaystyle \sum _{m\ge 0}}{a}_{lm}{\lambda }^{-m},b_l={\displaystyle \sum _{m\ge 0}}{b}_{lm}{\lambda }^{-m},c_l={\displaystyle \sum _{m\ge 0}}{c}_{lm}{\lambda }^{-m},$$

(27)

and solving the stationary zero curvature equation

$${\widehat{V}}_x=\frac{\partial \widehat{U}}{\partial \lambda }{\lambda }_t+[\widehat{U},\widehat{V}],$$

(28)

we arrive at the following recursion equations:

$$\left\{\begin{array}{c}\begin{array}{c}{a}_{lm,x}={\displaystyle \sum _{\begin{array}{c}i+j=l+1\\ 1\le i,j\le l\end{array}}}\left[q_i(c_{jm}-b_{jm})\right]-k_m\left(t\right),\hfill \\ b_{lm,x}=-2{\displaystyle \sum _{1\le p\le l}}{b}_{p,m+1},\hfill \\ c_{lm,x}=2{\displaystyle \sum _{1\le p\le l}}{c}_{p,m+1},\hfill \end{array}\hfill \end{array}\right.$$

(29)

which are equal to

$$\left\{\begin{array}{c}\begin{array}{c}{a}_{lm}=\sum _{\begin{array}{c}i+j=l+1\\ 1\le i,j\le l\end{array}}{\partial }^{-1}\left[q_i(c_{jm}-b_{jm})\right]-k_m\left(t\right)x,\hfill \\ b_{1,m+1}=\frac{1}{2}(r_1{b}_{1m}-b_{1m,x}-2q_1{a}_{1m}),\hfill \\ b_{l,m+1}=\frac{1}{2}[b_{(l-1)m,x}-b_{lm,x}-2q_l{a}_{1m}+r_l{b}_{1m}-\sum _{\begin{array}{c}k+j=l+1\\ 1\le k<l\\ 1<j\le l\end{array}}(2q_k(a_{jm}-a_{j-1,m})-r_k(b_{jm}-b_{j-1,m}))],\hfill \\ c_{1,m+1}=\frac{1}{2}(c_{1m,x}-2q_1{a}_{1m}+r_1{c}_{1m}),\hfill \\ c_{l,m+1}=\frac{1}{2}[c_{lm,x}-c_{(l-1)m,x}-2q_l{a}_{1m}+r_l{c}_{1m}-\sum _{\begin{array}{c}k+j=l+1\\ 1\le k<l\\ 1<j\le l\end{array}}(2q_k(a_{jm}-a_{j-1,m})-r_k(c_{jm}-c_{j-1,m}))].\hfill \end{array}\hfill \end{array}\right.$$

(30)

By choosing the initial data

$$\begin{array}{c}\hfill a_{l0}=-1,b_{l0}=0,c_{l0}=0,k_0=0,l=1,2,\cdots ,N,\end{array}$$

letting

$$\begin{array}{c}\hfill a_{lk}{|}_{q_k=r_k=0}=0,b_{lk}{{|}_{q_k=r_k=0}=0,c_{lk}|}_{q_k=r_k=0}=0,l=1,2,\cdots ,N,\end{array}$$

and using the recursion relations (30), we can obtain

$$\begin{array}{cc}\hfill & b_{l1}=q_l,c_{l1}=q_l,a_{l1}=-k_1\left(t\right)x,\hfill \\ \hfill & b_{1,2}=q_1{k}_1\left(t\right)x+\frac{1}{2}(r_1{q}_1-q_{1,x}),\hfill \\ \hfill & b_{l,2}=q_l{k}_1\left(t\right)x-\frac{1}{2}[q_{l,x}-q_{l-1,x}-r_l{q}_1-\sum _{\begin{array}{c}k+j=l+1\\ 1\le k<l\\ 1<j\le l\end{array}}{r}_k(q_j-q_{j-1})],\hfill \\ \hfill & c_{1,2}=q_1{k}_1\left(t\right)x+\frac{1}{2}(r_1{q}_1+q_{1,x}),\hfill \\ \hfill & c_{l,2}=q_l{k}_1\left(t\right)x-\frac{1}{2}[q_{l-1,x}-q_{l,x}-r_l{q}_1-\sum _{\begin{array}{c}k+j=l+1\\ 1\le k<l\\ 1<j\le l\end{array}}{r}_k(q_j-q_{j-1})],\hfill \\ \hfill & a_{2l-1,2}=\frac{1}{2}[q_l^2-2\sum _{\begin{array}{c}i+j=2l-1\\ 1\le i\le \left[\frac{2l-1}{2}\right]\end{array}}{q}_i(q_j-q_{j+1})]-k_2\left(t\right)x,\hfill \\ \hfill & a_{2l,2}=\frac{1}{2}[-q_l^2+q_{2l}{q}_1-2\sum _{\begin{array}{c}i+j=2l\\ l<i<2l\end{array}}{q}_i(q_j-q_{j+1})]-k_2\left(t\right)x,\hfill \\ \hfill & \cdots .\hfill \end{array}$$

By denoting

$$\begin{array}{c}\hfill {\widehat{V}}_{+}^{\left(n\right)}=\sum _{k=1}^N\sum _{m=0}^n(a_{km}{\widehat{h}}_{4k-1}{\lambda }^{n-m}+a_{km}{\widehat{h}}_{4k-2}{\lambda }^{n-m}+b_{km}{\widehat{h}}_{4k-3}{\lambda }^{n-m}+c_{km}{\widehat{h}}_{4k}{\lambda }^{n-m}),\end{array}$$

one can find that

$$\begin{array}{c}\hfill -{\widehat{V}}_{+,x}^{\left(n\right)}+\frac{\partial \widehat{U}}{\partial \lambda }{\lambda }_{+,t}^{\left(n\right)}+\left[\widehat{U},{\widehat{V}}_{+}^{\left(n\right)}\right]=2\sum _{k=1}^N\left(\sum _{p=1}^k{b}_{p,n+1}{\widehat{h}}_{4k-1}-\sum _{p=1}^k{c}_{p,n+1}{\widehat{h}}_{4k}\right).\end{array}$$

We choose the modification term as

$$\begin{array}{cc}\hfill {\widehat{\Delta }}_n=& q_1^{-1}\sum _{i=1}^N\left[\sum _{p=1}^i(b_{p,n+1}+c_{p,n+1})\right]{\widehat{h}}_{4i-3}\hfill \\ \hfill & +\sum _{k=2}^N{(-1)}^{k-1}{q}_1^{-k}\sum _{i=2}^N\left[\sum _{\begin{array}{c}{s}_1+\cdots +s_k=i+k-1\\ s_l\ge 2,l=1,\cdots ,k-1\\ s_k>0\end{array}}[\sum _{p=1}^{s_k}(b_{p,n+1}+c_{p,n+1})q_{s_1}{q}_{s_2}\cdots q_{s_{k-1}}]\right]{\widehat{h}}_{4i-3}.\hfill \end{array}$$

By taking ${\widehat{V}}^{\left(n\right)}={\widehat{V}}_{+}^{\left(n\right)}+{\widehat{\Delta }}_n$, the non-isospectral zero curvature equation

$$\begin{array}{c}\hfill {\widehat{U}}_{t_n}+\frac{\partial \widehat{U}}{\partial \lambda }{\lambda }_t^{\left(n\right)}-{\widehat{N}}_x+\left[\widehat{U},{\widehat{V}}^{\left(n\right)}\right]=0\end{array}$$

leads to the following N-component non-isospectral TD hierarchy:

$$\begin{array}{cc}\hfill q_{j,t_n}=& \sum _{p=1}^j(c_{p,n+1}-b_{p,n+1})=q_1^{-1}(\sum _{i=1}^j{a}_{in,x}+j{k}_n\left(t\right))\hfill \\ \hfill & +\sum _{i=2}^j{(-1)}^{i-1}{q}_1^{-i}\sum _{\begin{array}{c}{s}_1+\cdots +s_i=j+i-1\\ s_l\ge 2,l=1,\cdots ,i-1\\ s_i>0\end{array}}[\sum _{p=1}^{s_i}{a}_{pn,x}{q}_{s_1}{q}_{s_2}\cdots q_{s_{i-1}}+s_i{q}_{s_1}{q}_{s_2}\cdots q_{s_{i-1}}{k}_n\left(t\right)],\hfill \\ \hfill r_{j,t_n}=& \sum _{i=2}^j{(-1)}^{i-1}\partial q_1^{-i}\sum _{\begin{array}{c}{s}_1+\cdots +s_i=j+i-1\\ s_l\ge 2,l=1,\cdots ,i-1\\ s_i>0\end{array}}[\sum _{p=1}^{s_i}(b_{p,n+1}+c_{p,n+1})q_{s_1}{q}_{s_2}\cdots q_{s_{i-1}}]\hfill \\ \hfill & +\partial q_1^{-1}\sum _{i=1}^j(c_{i,n+1}+b_{i,n+1}),j=1,\cdots ,N.\hfill \end{array}$$

(31)

Letting $N=n=2$, system (31) is reduced to system (19), in which $\epsilon =0$. 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 $Z_N^{\epsilon }$-trace identity

$$\begin{array}{c}\hfill \frac{\delta }{\partial \widehat{u}}\int \sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}tr\left({\widehat{V}}_k\frac{\partial {\widehat{U}}_j}{\partial \lambda }\right)dx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}tr\left({\widehat{V}}_k\frac{\partial {\widehat{U}}_j}{\partial \widehat{u}}\right).\end{array}$$

(32)

From (26), we have

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}tr\left({\widehat{V}}_k\frac{\partial {\widehat{U}}_j}{\partial \lambda }\right)=-2\sum _{1\le p\le l}{a}_p,\end{array}$$

(33)

and

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}k+j=l+1\\ 1\le k,j\le l\end{array}}tr\left({\widehat{V}}_k\frac{\partial {\widehat{U}}_j}{\partial \widehat{u}}\right)=\left(\begin{array}{c}{\displaystyle \sum _{1\le p\le l}}(b_p+c_p)\\ {\displaystyle \sum _{1\le p\le l}}{a}_p\end{array}\right).\end{array}$$

(34)

By substituting (33) and (34) into (32), we have

$$\begin{array}{cc}\hfill & \frac{\delta }{\partial \widehat{u}}\int -2\sum _{1\le p\le l}{a}_{p}dx={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}{\displaystyle \sum _{1\le p\le l}}(b_p+c_p)\\ {\displaystyle \sum _{1\le p\le l}}{a}_p\end{array}\right).\hfill \end{array}$$

(35)

By inserting (27) into (35) and comparing the coefficients of $\lambda$, we obtain

$$\begin{array}{c}\hfill \frac{\delta }{\partial \widehat{u}}\int -2{\displaystyle \sum _{1\le p\le l}}{a}_{p,m+1}dx=(\gamma -m)\left(\begin{array}{c}{\displaystyle \sum _{1\le p\le l}}(b_{p,m}+c_{p,m})\\ {\displaystyle \sum _{1\le p\le l}}{a}_{p,m}\end{array}\right).\end{array}$$

(36)

By fixing the constant $m=1$ in the above equation, we conclude that $\gamma =0$. Then, we obtain

$$\begin{array}{cc}\hfill & \frac{\delta }{\partial \widehat{u}}\int \frac{{\displaystyle \sum _{1\le p\le l}}{a}_{p,m+2}}{m+1}dx=\frac{1}{2}\left(\begin{array}{c}{\displaystyle \sum _{1\le p\le l}}(b_{p,m+1}+c_{p,m+1})\\ {\displaystyle \sum _{1\le p\le l}}{a}_{p,m+1}\end{array}\right)=\frac{\delta {\widehat{H}}_{l,m+1}}{\partial \widehat{u}}.\hfill \end{array}$$

(37)

Therefore,

$$\begin{array}{c}\hfill {\widehat{H}}_{l,m+1}=\int \frac{{\displaystyle \sum _{1\le p\le l}}{a}_{p,m+2}}{m+1}dx,l=1,2,\cdots ,N.\end{array}$$

With the aid of (30) and (37), the Bi-Hamiltonian structures of hierarchy (31) are obtained as follows:

$$\begin{array}{cc}\hfill {\widehat{u}}_{t_n}& ={\left(\begin{array}{c}{q}_1\hfill \\ r_1\hfill \\ q_2\hfill \\ r_2\hfill \\ ⋮\hfill \\ q_N\hfill \\ r_N\hfill \end{array}\right)}_{t_n}=\left(\begin{array}{c}{q}_1^{-1}(a_{1n+1,x}+k_{n+1}\left(t\right))\\ \partial [q_1^{-1}(c_{1,n+1}+b_{1,n+1})]\\ q_1^{-1}(a_{1n+1,x}+a_{2n+1,x}+2k_{n+1}\left(t\right))-{\displaystyle \frac{q_2(a_{1n+1,x}+k_{n+1}\left(t\right))}{q_1^2}}\\ \partial [q_1^{-1}{\displaystyle \sum _{i=1}^2}(c_{i,n+1}+b_{i,n+1})-{\displaystyle \frac{{\displaystyle \sum _{\begin{array}{c}{s}_1+s_2=3\\ s_1\ge 2,s_2>0\end{array}}}{\displaystyle \sum _{p=1}^{s_2}}(c_{p,n+1}+b_{p,n+1})q_{s_1}}{q_1^2}}]\\ ⋮\\ A\\ B\end{array}\right)\hfill \\ \hfill & ={\overline{\mathrm{\Phi }}}_1^n{\overline{J}}_1\frac{\delta }{\delta \widehat{u}}\left(\begin{array}{c}{\overline{H}}_{1,1}\hfill \\ {\overline{H}}_{2,1}\hfill \\ {\overline{H}}_{3,1}\hfill \\ {\overline{H}}_{4,1}\hfill \\ ⋮\hfill \\ {\overline{H}}_{N-1,1}\hfill \\ {\overline{H}}_{N,1}\hfill \end{array}\right)+{\displaystyle \sum _{m=0}^n}{\overline{\mathrm{\Phi }}}_1^{n-m}{\overline{J}}_1({\overline{R}}_1{k}_n\left(t\right)+{\overline{R}}_0{k}_{n+1}\left(t\right))+\overline{R}{k}_{n+1}\left(t\right),\hfill \end{array}$$

(38)

where ${\overline{J}}_1$ is the Hamiltonian operator:

$$\begin{array}{c}\hfill {\overline{J}}_1=\left(\begin{array}{ccccc}{J}_{11}& 0& \cdots & 0& 0\\ J_{12}& J_{11}& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ J_{1,N-1}& J_{1,N-2}& \cdots & J_{13}& J_{12}\\ J_{1N}& J_{1,N-1}& \cdots & J_{12}& J_{11}\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill & J_{11}=\left(\begin{array}{cc}0& q_1^{-1}\partial \\ \partial q_1^{-1}& 0\end{array}\right),J_{12}=\left(\begin{array}{cc}0& -{\displaystyle \frac{q_2\partial }{q_1^2}}\\ -{\displaystyle \frac{\partial q_2}{q_1^2}}& 0\end{array}\right),J_{1l}=\left(\begin{array}{cc}0& C\partial \\ \partial C\end{array}\right),l=3,\cdots ,N,\hfill \\ \hfill & C=-{\displaystyle \frac{q_l}{q_1^2}}+{\displaystyle \sum _{i=3}^l}{(-1)}^{i-1}{q}_1^{-i}{\displaystyle \sum _{\begin{array}{c}{s}_1+\cdots +s_{i-1}=l+i-2\\ s_j\ge 2,j=1,\cdots ,i-1\end{array}}}{q}_{s_1}{q}_{s_2}\cdots q_{s_{i-1}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \overline{R}=\left(\begin{array}{c}{q}_1^{-1}\\ 0\\ 2q_1^{-1}-{\displaystyle \frac{q_2}{q_1^2}}\\ 0\\ ⋮\\ N{q}_1^{-1}+{\displaystyle \sum _{i=2}^N}{(-1)}^{i-1}{q}_1^{-i}{\displaystyle \sum _{\begin{array}{c}{s}_1+\cdots +s_i=N+i-1\\ s_l\ge 2,l=1,\cdots ,i-1\\ s_i>0\end{array}}}{s}_i{q}_{s_1}{q}_{s_2}\cdots q_{s_{i-1}}\\ 0\end{array}\right),\hfill \end{array}$$

${\overline{Q}}_1={\overline{J}}_1{\overline{L}}_1$ and ${\overline{\mathrm{\Phi }}}_1={\overline{J}}_1{\overline{L}}_1{\overline{J}}_1^{-1}$.

5. Conclusions and Discussion

In this paper, we consider three kinds of Lie algebras: classical Lie algebras, A, the enlarged Lie algebras, $A_{12}$, and Lie algebras, $A_{1N}$, which are associated with the N-component non-isospectral problem. Based on these Lie algebras, we derive three classes of non-isospectral TD hierarchies, and the corresponding Bi-Hamiltonian structures are established by using trace identity. It should be noted that the first nontrivial system in the first kind of TD hierarchy can be reduced to the TD equation, and the second nontrivial member can be reduced to the mKdV equation. We find that the second kind of integrable coupling of the TD hierarchy can be reduced to the integrable coupling of TD equations and the coupled mKdV equations. The method we used in this paper can be applied to generate other multi-component non-isospectral integrable hierarchies in the future.