Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

Abstract

By introducing a $3\times 3$ matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the $3\times 3$ matrix Lie algebra into a $2\times 2$ matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

1. Introduction

It has been an important issue to generate new integrable hierarchies and further investigate their related properties, such as symmetries, Bäcklund transformations, algebraic–geometric solutions, covering, etc. [1,2,3,4,5,6,7]. Blaszak and Ma [8] started from AKNS $3\times 3$ matrix Lax pairs to discuss the Liouville integrable noncanonical systems with variable coefficient symplectic form by using binary symmetry constraints of the AKNS hierarchy, and they further provided a class of integrable factorization for every AKNS system in the hierarchy. Ma, Fuchssteiner and Oevel [9] adopted $3\times 3$ matrix spectral problems

$${\mathsf{\Phi }}_x=U\mathsf{\Phi },U=\left(\begin{array}{ccc}-2\lambda & \sqrt{2}q& 0\\ \sqrt{2}r& 0& \sqrt{2}q\\ 0& \sqrt{2}r& 2\lambda \end{array}\right),$$

(1)

$${\mathsf{\Phi }}_t=V\mathsf{\Phi },V=\sum _{i=0}^{\infty }\left(\begin{array}{ccc}2a_i& \sqrt{2}{b}_i& 0\\ \sqrt{2}{c}_i& 0& \sqrt{2}{b}_i\\ 0& \sqrt{2}{c}_i& -2a_i\end{array}\right){\lambda }^{-i},$$

(2)

and employed zero curvature equations for deriving the standard AKNS hierarchy. Furthermore, they exploited the binary nonlinearization theory to extend a case of $3\times 3$ matrix spectral problems for AKNS hierarchy. Fuchssteiner [10] proposed the notation on integrable couplings of some known integrable systems while investigating properties of Virasora algebras. Later, Ma and Fuchssteiner [11] employed the perturbation technique for generating the integrable couplings of the KdV equation. However, this method is too tedious and only obtains the integrable couplings of single integrable equations. In 2002, Zhang et al. [12,13,14] adopted finite dimensional Lie algebras to introduce spectral problems, then employed the Tu Scheme [15] to have generated integrable couplings of some known integrable hierarchy. In order to produce Hamiltonian structures of integrable couplings, Guo and Zhang [16] introduced a s-dimensional vector space V and defined a Lie bracket to make V a Lie algebra $\overline{V}$. On $\overline{V}$, a linear functional was again introduced, by employing the variational method, a formula called the quadratic-form identity was obtained. Later, Ma and Chen [17,18] further improved the formula to get a variational identity, which is a powerful method to generate Hamiltonian structures of some integrable couplings of the known integrable hierarchies. The quadratic-form identity is generalized form of the trace identity proposed by Tu Guizhang [15]. It is remarkable that not all integrable couplings of the known integrable hierarchies possess Hamiltonian structures deduced by the quadratic-form identity or the variational identity. For example, the Lie algebra

$$\begin{array}{cc}\hfill & A_{2,1}=\mathbf{span}\{e_1,e_2,e_3,e_4,e_5\},\hfill \\ \hfill & e_1=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right),e_2=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),e_3=\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),\hfill \\ \hfill & e_4=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),e_5=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),\hfill \end{array}$$

has a resulting loop algebra:

$$\widetilde{A_{2,1}}=\mathbf{span}\{e_i\left(n\right),i=1,\cdots ,5\},e_i\left(n\right)=e_i{\lambda }^n.$$

By applying the loop algebra $\widetilde{A_{2,1}}$, some integrable couplings of the AKNS hierarchy, the TD hierarchy, etc., could be obtained by the use of the Tu Scheme. However, the Hamiltonian structures of such the integrable couplings cannot be generated by the quadratic-form identity or the variational identity. Therefore, it is necessary to choose appropriate Lie algebras for deducing their Hamiltonian structures. In addition, the Lie algebras for generating integrable couplings are usually enlarged by the basis of the Lie algebra $A_1$. In the paper, we want to start from the $3\times 3$ AKNS spectral problems which are represented by $3\times 3$ matrix Lie algebras to derive integrable couplings of the known integrable hierarchies through turning the $3\times 3$ matrix Lie algebras to the $2\times 2$ Lie algebra by choosing proper matrix commutative transformations. Based on those, we choose two appropriate enlarged Lie algebras for which the integrable couplings of the AKNS hierarchy from the $3\times 3$ spectral problems are generated, respectively. Besides, their Hamiltonian structures are also obtained by employing the quadratic-form identity and the trace identity. Specifically, we obtain new nonlinear isospectral and nonisospectral integrable couplings and their initial symmetries of the nonlinear Schröger equation, the KdV equations. In particular, we obtain a nonisospectral nonlinear Schröger equation.

2. Isospectral and Nonisospectral $\mathbf{3}\times \mathbf{3}$ ANKS Hierarchies

Based on the $3\times 3$ spectral problem (1) and (2), we take the spatial spectral equation

$${\mathsf{\Psi }}_x=U\mathsf{\Psi },U=\left(\begin{array}{ccc}-\lambda & q& 0\\ r& 0& q\\ 0& r& \lambda \end{array}\right),$$

(3)

Set

$$\begin{array}{c}\hfill e_1=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right),e_2=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),e_3=\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),\end{array}$$

and define a commutative operation as follows:

$$[A,B]=AB-BA,$$

then we have

$$[e_1,e_2]=e_2,[e_1,e_3]=-e_3,[e_2,e_3]=e_1.$$

(4)

Making a matrix commutative transformation

$$e_1\sim \frac{1}{2}h,e_2\sim e,e_3\sim \frac{1}{2}f,$$

(5)

where

$$\begin{array}{c}\hfill h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),\end{array}$$

(6)

is a basis of the Lie algebra $A_1$.

By using (6), an enlarge Lie algebra is given by

$$H=:\mathbf{span}\{h_1,h_2,h_3,h_4,h_5,h_6\},$$

where

$$\begin{array}{c}\hfill h_1=\frac{1}{2}\left(\begin{array}{cc}h& 0\\ 0& h\end{array}\right),h_2=\left(\begin{array}{cc}e& 0\\ 0& e\end{array}\right),h_3=\frac{1}{2}\left(\begin{array}{cc}f& 0\\ 0& f\end{array}\right),\\ \hfill h_4=\frac{1}{2}\left(\begin{array}{cc}0& h\\ 0& h\end{array}\right),h_5=\left(\begin{array}{cc}0& e\\ 0& e\end{array}\right),h_6=\frac{1}{2}\left(\begin{array}{cc}0& f\\ 0& f\end{array}\right).\end{array}$$

It is easy to see that

$$\begin{array}{cc}\hfill & [h_1,h_2]=h_2,[h_1,h_3]=-h_3,[h_1,h_4]=0,[h_1,h_5]=h_5,\hfill \\ \hfill & [h_1,h_6]=-h_6,[h_2,h_3]=h_1,[h_2,h_4]=-h_5,[h_2,h_5]=0,\hfill \\ \hfill & [h_2,h_6]=h_4,[h_3,h_4]=h_6,[h_3,h_5]=-h_4,[h_3,h6]=0,\hfill \\ \hfill & [h_4,h_5]=h_5,[h_4,h_6]=-h_6,[h_5,h_6]=h_6.\hfill \end{array}$$

Denoting

$$H_1=\mathbf{span}\{h_1,h_2,h_3\},H_2=\mathbf{span}\{h_4,h_5,h_6\},$$

then

$$H=H_1\oplus H_2,[H_1,H_1]\subset H_1,[H_1,H_2]\subset H_2,[H_2,H_2]\subset H_2,$$

(7)

hence, H is a semi-simple Lie algebra. Specially, we find the Lie algebra $A_1=\mathbf{span}\{e_1,e_2,e_3\}$ has the same commutative relations with the Lie algebra $H_1$. Therefore, $\mathbf{span}\{e_1,e_2,e_3\}$ is isomorphic to $H_1$. Thus, if employing the loop algebras $\widetilde{A_1}$ and $\widetilde{H_1}$ with the same degrations and the Tu Scheme, we could generate the common integrable hierarchies. In what follows, we first apply the loop algebra $\widetilde{A_1}$ and the Tu Scheme to deduce the $3\times 3$ isospectral and nonisospectral hierarchy.

Set

$$\widetilde{A_1}=\mathbf{span}\{e_1\left(n\right),e_2\left(n\right),e_3\left(n\right)\},e_i\left(n\right)=e_i{\lambda }^n,i=1,2,3,n\in \mathbb{Z}.$$

Taking

$$\begin{array}{cc}\hfill V& =\sum _{i\ge 0}(a_i{e}_1(-i)+b_i{e}_2(-i)+c_i{e}_3(-i))+\sum _{j\ge 0}({\overline{a}}_j{e}_1(-j)+{\overline{b}}_j{e}_2{e}_2(-j)+{\overline{c}}_j{e}_3(-j))\hfill \\ \hfill & =:V_1+V_2,\hfill \end{array}$$

then the compatibility condition of the spectral problems

$${\phi }_x=U(u,\lambda )\phi ,{\phi }_t=V(u,\lambda )\phi ,{\lambda }_t\ne 0$$

(8)

reads that

$$\frac{\partial U}{\partial u}{u}_t+\frac{\partial U}{\partial \lambda }{\lambda }_t-V_x+[U,V]=0.$$

According to the scheme called the generalized Tu Scheme presented in [19,20,21], we first solve the equation for V

$$\frac{\partial U}{\partial \lambda }{\lambda }_t-V_x+[U,V]=0,{\lambda }_t=\sum _{j\ge 0}{k}_j\left(t\right){\lambda }^{-j},$$

(9)

which admits that

$$\begin{array}{cc}\hfill & a_i={\partial }^{-1}(q{c}_i-r{b}_i)-{\alpha }_i\left(t\right),\hfill \\ \hfill & {\overline{a}}_j={\partial }^{-1}(q{\overline{c}}_j-r{\overline{b}}_j)-k_j\left(t\right)x,\hfill \\ \hfill & b_{i+1}=(-\partial +q{\partial }^{-1}r)b_i-q{\partial }^{-1}q{c}_i+{\alpha }_i\left(t\right)q,\hfill \\ \hfill & {\overline{b}}_{j+1}=(-\partial +q{\partial }^{-1}r){\overline{b}}_j-q{\partial }^{-1}q{\overline{c}}_j+k_j\left(t\right)xq,\hfill \\ \hfill & c_{i+1}=(\partial -r{\partial }^{-1}q)c_i+r{\partial }^{-1}r{b}_i+{\alpha }_i\left(t\right)r,\hfill \\ \hfill & {\overline{c}}_{j+1}=(\partial -r{\partial }^{-1}q){\overline{c}}_j+r{\partial }^{-1}r{\overline{b}}_j+k_j\left(t\right)xr.\hfill \end{array}$$

Noting

$$\begin{array}{cc}\hfill & V_{+}^{(n,m)}=\sum _{i=0}^n(a_i{e}_1(n-i)+b_i{e}_2(n-i)+c_i{e}_3(n-i))\hfill \\ \hfill & +\sum _{j=0}^m({\overline{a}}_j{e}_1(m-j)+{\overline{b}}_j{e}_2(m-j)+{\overline{c}}_j{e}_3(m-j))\hfill \\ \hfill & ={\lambda }^n{V}_1+{\lambda }^m{V}_2-V_{-}^{(n,m)},\hfill \\ \hfill & {\lambda }_{t,+}^{\left(m\right)}={\lambda }^m{\lambda }_t-{\lambda }_{t,-}^{\left(m\right)},\hfill \end{array}$$

then (9) can be decomposed into

$$-V_{+,x}^{(n,m)}+[U,V_{+}^{(n,m)}]-\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}=V_{-,x}^{(n,m)}-[U,V_{-}^{(n,m)}]-\frac{\partial U}{\partial \lambda }{\lambda }_{t,-}^{\left(m\right)}.$$

(10)

The gradations of the left-hand side in (10) are more than 0, while the right-hand side less than 1. Thus, the gradations of (10) read $0,1$, which lead us to the following

$$-V_{+,x}^{(n,m)}+[U,V_{+}^{(n,m)}]+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}=(b_{n+1}+{\overline{b}}_{m+1})e_2\left(0\right)-(c_{n+1}+{\overline{c}}_{m+1})e_3\left(0\right).$$

The zero curvature equation

$$\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}+\frac{\partial U}{\partial u}{u}_t-V_{+,x}^{(n,m)}+[U,V_{+}^{(n,m)}]=0$$

gives rise to the following isospectra and nonisospectral AKNS hierarchy

$$\begin{array}{cc}\hfill u_t=:{\left(\begin{array}{c}q\\ r\end{array}\right)}_t& =\left(\begin{array}{c}-b_{n+1}-{\overline{b}}_{m+1}\\ c_{n+1}+{\overline{c}}_{m+1}\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}0& -\frac{1}{2}\\ \frac{1}{2}& 0\end{array}\right)\left(\begin{array}{c}2c_{n+1}+2{\overline{c}}_{m+1}\\ 2b_{n+1}+2{\overline{b}}_{m+1}\end{array}\right)=:J_1\left(\begin{array}{c}2c_{n+1}+2{\overline{c}}_{m+1}\\ 2b_{n+1}+2{\overline{b}}_{m+1}\end{array}\right)\hfill \end{array},$$

(11)

where $J_1=\left(\begin{array}{cc}0& -\frac{1}{2}\\ \frac{1}{2}& 0\end{array}\right)$ is a Hamiltonian operator. Equation (11) can be written again

$$\begin{array}{cc}\hfill u_t=& \frac{1}{2}\left(\begin{array}{cc}-q{\partial }^{-1}q& \partial -q{\partial }^{-1}q\\ \partial -r{\partial }^{-1}q& r{\partial }^{-1}r\end{array}\right)\left(\begin{array}{c}2c_n+2{\overline{c}}_m\\ 2b_n+2{\overline{b}}_m\end{array}\right)\hfill \\ \hfill & +{\alpha }_n\left(t\right)\left(\begin{array}{c}-q\\ r\end{array}\right)+k_m\left(t\right)\left(\begin{array}{c}-xq\\ xr\end{array}\right)\hfill \end{array}.$$

(12)

It is easy to compute that

$$\begin{array}{cc}\hfill \left(\begin{array}{c}{c}_{n+1}\\ b_{n+1}\end{array}\right)& =\left(\begin{array}{cc}\partial -r{\partial }^{-1}r& r{\partial }^{-1}r\\ -q{\partial }^{-1}q& -\partial +q{\partial }^{-1}r\end{array}\right)\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)+{\alpha }_n\left(\begin{array}{c}r\\ q\end{array}\right)\hfill \\ \hfill & =:L\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)+{\alpha }_n\left(\begin{array}{c}r\\ q\end{array}\right)=\cdots \hfill \\ \hfill & =L^n\left(\begin{array}{c}{c}_1\\ b_1\end{array}\right)+[{\alpha }_1{L}^{n-1}+{\alpha }_2{L}^{n-2}+\cdots +{\alpha }_{n-1}L+{\alpha }_n]\left(\begin{array}{c}r\\ q\end{array}\right).\hfill \end{array}$$

Hence, (11) can be written as

$$\begin{array}{cc}\hfill u_t& ={\left(\begin{array}{c}q\\ t\end{array}\right)}_t={\mathsf{\Phi }}^{n}J\left(\begin{array}{c}2{\alpha }_{0}r\\ 2{\alpha }_{0}q\end{array}\right)+(2{\alpha }_1{\mathsf{\Phi }}^{n-1}+2{\alpha }_2{\varphi }^{n-2}+\cdots +2{\alpha }_n)J\left(\begin{array}{c}r\\ q\end{array}\right)\hfill \\ \hfill & +{\mathsf{\Phi }}^{m}J\left(\begin{array}{c}2k_{0}xr\\ 2k_{0}xq\end{array}\right)+(2k_1{\mathsf{\Phi }}^{m-1}+2k_2{\mathsf{\Phi }}^{m-2}+\cdots +2k_m)J\left(\begin{array}{c}xr\\ xq\end{array}\right)\hfill \\ \hfill & =\sum _{i=0}^{n}2{\alpha }_{n-i}{\mathsf{\Phi }}^{i}J\left(\begin{array}{c}r\\ q\end{array}\right)+\sum _{j=0}^{m}2k_{m-j}{\mathsf{\Phi }}^{i}J\left(\begin{array}{c}xr\\ xq\end{array}\right).\hfill \end{array}$$

Due to

$$J\left(\begin{array}{c}r\\ q\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}-q\\ r\end{array}\right),$$

we have the $3\times 3$ AKNS isospectral–nonisospectral hierarchy:

$$u_t=\sum _{i=0}^n{\alpha }_{n-i}{\left(\frac{1}{4}\right)}^i{\overline{\mathsf{\Phi }}}^i\left(\begin{array}{c}-q\\ r\end{array}\right)+\sum _{j=0}^m{k}_{m-j}{\left(\frac{1}{4}\right)}^j{\overline{\mathsf{\Phi }}}^j\left(\begin{array}{c}-xq\\ xr\end{array}\right),$$

(13)

where

$$\overline{\mathsf{\Phi }}=\left(\begin{array}{cc}-\partial +q{\partial }^{-1}r& q{\partial }^{-1}q\\ -r{\partial }^{-1}r& \partial -r{\partial }^{-1}q\end{array}\right).$$

Let $j=0,i=n,{\alpha }_0\left(t\right)=4^n$, (13) reduces to the isospectral hierarchy:

$$u_t={\overline{\mathsf{\Phi }}}^n\left(\begin{array}{c}-q\\ r\end{array}\right)=:K_n.$$

(14)

It is easy to verify that

$$T_0^n=nt{K}_{n-1}+x\left(\begin{array}{c}-q\\ r\end{array}\right)$$

are nonlocal symmetries of (14). Since $\overline{\mathsf{\Phi }}$ is a strong symmetric operator, ${\tau }_n^m={\overline{\mathsf{\Phi }}}^m{\tau }_0^n=nt{K}_{m+n-1}+{\overline{\mathsf{\Phi }}}^{n}x\left(\begin{array}{c}-q\\ r\end{array}\right)$ are still symmetries of (14).

Remark 1.

Through the discussion as above, we declare that the integrable hierarchies derived from $3\times 3$ matrix Lie algebras can be worked out by employing $2\times 2$ matrix Lie algebras via such the transformation (5). Actually, some other $3\times 3$ spectral problems can also transform to $2\times 2$ cases by the transformation (5). For example, the following $3\times 3$ spectral problem [22]:

$${\mathsf{\Psi }}_x=\left(\begin{array}{ccc}2\lambda -2s& \sqrt{2}q& 0\\ -\sqrt{2}\lambda r& 0& \sqrt{2}q\\ 0& -\sqrt{2}\lambda r& -2\lambda -2s\end{array}\right)\mathsf{\Psi }.$$

The advantage for turning $3\times 3$ spectral problems to $2\times 2$ cases by using (5) lies in further investigating integrable couplings of the associating integrable hierarchies derived from $3\times 3$ spectral problems. In what follows, we still take the $3\times 3$ AKNS integrable hierarchy for example to illustrate the question.

3. Two Kinds of Integrable Models

The so-called integrable models in the paper imply integrable couplings of some known integrable hierarchies.

Case 3.1: The first kind of integrable model

Now, we shall employ the enlarged Lie algebra H (7) obtained by using the transformation (5) for deducing a kind of integrable coupling of (13) and discuss its Hamiltonian structure.

Set

$$\begin{array}{cc}& {\phi }_x=U\phi ,U=-h_1\left(1\right)+q{h}_2\left(0\right)+r{h}_3\left(0\right)+u_1{h}_5\left(0\right)+u_2{h}_6\left(0\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}& {\phi }_t=V\phi ,V=V_1+V_2,\hfill \\ & V_1=\sum _{i\ge 0}{a}_i{h}_1(-i)+b_i{h}_2(-i)+c_i{h}_3(-i)+d_i{h}_4(-i)+e_i{h}_5(-i)+g_i{h}_6(-i),\hfill \\ & V_2=\sum _{j\ge 0}{\overline{a}}_i{h}_1(-j)+{\overline{b}}_j{h}_2(-j)+{\overline{c}}_j{h}_3(-j)+d_j{h}_4(-j)+e_j{h}_5(-j)+g_j{h}_6(-j),\hfill \\ & {\lambda }_t=\sum _{j\ge 0}{k}_j\left(t\right){\lambda }^{-j}.\hfill \end{array}$$

(16)

By using (9) along with (15) and (16), we have

$$\left\{\begin{array}{c}{a}_i={\partial }^{-1}(q{c}_i-r{b}_i)-{\alpha }_i\left(t\right),\hfill \\ {\overline{a}}_j={\partial }^{-1}(q{\overline{c}}_j-r{\overline{b}}_j)-k_j\left(t\right)x,\hfill \\ b_{i+1}=-b_{i}x-q{a}_i,\hfill \\ {\overline{b}}_{j+1}=-{\overline{b}}_{j}x-q{\overline{a}}_j,\hfill \\ c_{i+1}=c_{i}x-r{a}_i,\hfill \\ {\overline{c}}_{j+1}={\overline{c}}_{j}x-r{\overline{a}}_j,\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{c}{d}_i={\partial }^{-1}[(q+u_1)g_i-(r+u_2)e_i+u_1{c}_i-u_2{b}_i]-{\beta }_i\left(t\right),\hfill \\ {\overline{d}}_j={\partial }^{-1}[(q+u_1){\overline{g}}_j-(r+u_2){\overline{e}}_j+u_1{\overline{c}}_j-u_2{\overline{b}}_j]-{\gamma }_j\left(t\right)x,\hfill \\ e_{i+1}=-e_{i}x-(q+u_1)d_i-u_1{a}_i,\hfill \\ {\overline{e}}_{j+1}=-{\overline{e}}_{j}x-(q+u_1){\overline{d}}_j-u_1{\overline{a}}_j,\hfill \\ g_{i+1}=g_{i}x-(r+u_2)d_i-u_2{a}_i,\hfill \\ {\overline{g}}_{j+1}={\overline{g}}_{j}x-(r+u_2){\overline{d}}_j-u_2{\overline{a}}_j.\hfill \end{array}\right.$$

(18)

Denoting

$$\begin{array}{cc}\hfill V_{+}^{(n,m)}=& \sum _{i=0}^n(a_i{h}_1(n-i)+b_i{h}_2(n-i)+c_i{h}_3(n-i)+d_i{h}_4(n-i)+e_i{h}_5(n-i)\hfill \\ \hfill & +g_i{h}_6(n-i))+\sum _{j=0}^m({\overline{a}}_j{h}_1(m-j)+{\overline{b}}_j{h}_2(m-j)+{\overline{c}}_j{h}_3(m-j)\hfill \\ \hfill & +{\overline{d}}_j{h}_4(m-j)+{\overline{e}}_j{h}_5(m-j)+{\overline{g}}_j{h}_6(m-j))=:V_{1,+}^{\left(n\right)}+V_{2,+}^{\left(m\right)},\hfill \\ \hfill {\lambda }_{t,+}^{(n,m)}=& \sum _{j=0}^m{k}_j\left(t\right){\lambda }^{m-j}={\lambda }^m{\lambda }_t-{\lambda }_{t,-}^{\left(m\right)}.\hfill \end{array}$$

A direct calculation gives rise to

$$\begin{array}{cc}\hfill -V_{+,x}^{(n,m)}+[U,V_{+}^{(n,m)}]+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}=& (b_{n+1}+{\overline{b}}_{m+1})h_2\left(0\right)-\left(c_{n+1}{\overline{c}}_{m+1}\right)h_3\left(0\right)\hfill \\ \hfill & +(e_{n+1}+{\overline{e}}_{m+1})h_5\left(0\right)-(g_{n+1}+{\overline{g}}_{m+1})h_6\left(0\right).\hfill \end{array}$$

Noting $V^{(n,m)}=V_{+}^{(n,m)}$, then the nonisospectral zero curvature equation

$$\frac{\partial U}{\partial u}{u}_t+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{(n,m)}-V_x^{n,m}+[U,V^{(n,m)}]=0$$

admits an integrable model

$$u_t=:{\left(\begin{array}{c}q\\ r\\ u_1\\ u_2\end{array}\right)}_t=\left(\begin{array}{c}-b_{n+1}-{\overline{b}}_{m+1}\\ c_{n+1}+{\overline{c}}_{m+1}\\ -e_{n+1}-{\overline{e}}_{m+1}\\ g_{n+1}+{\overline{g}}_{m+1}\end{array}\right).$$

(19)

Obviously, when $u_1=u_2=0$, (19) reduces to (13). Therefore, (19) is an integrable coupling of the isospectral–nonisospectral hierarchy (13). In the following, we consider some reductions of (19).

Set

$$b_0=c_0=g_0=e_0=0,a_0={\alpha }_0\left(t\right),d_0=-{\beta }_0\left(t\right),$$

then from (17) and (18) we get that

$$\begin{array}{cc}\hfill & b_1={\alpha }_{0}q,c_1={\alpha }_{0}r,a_1=-{\alpha }_1\left(t\right),\hfill \\ \hfill & b_2=-{\alpha }_0{q}_x+{\alpha }_{1}q,c_2={\alpha }_0{r}_x+{\alpha }_{1}r,a_2={\alpha }_{0}qr-{\alpha }_2\left(t\right),\hfill \\ \hfill & b_3={\alpha }_0{q}_{xx}-{\alpha }_1{q}_x-{\alpha }_0{q}^{2}r+{\alpha }_{2}q,\hfill \\ \hfill & c_3={\alpha }_0{r}_{xx}+{\alpha }_1{r}_x-{\alpha }_{0}q{r}^2+{\alpha }_{2}r,\hfill \\ \hfill & a_3={\alpha }_0(q{r}_x-q_{x}r)+{\alpha }_{1}qr-{\alpha }_3,\hfill \\ \hfill & b_4=-{\alpha }_0{q}_{xxx}+3{\alpha }_{0}q{q}_{x}r+{\alpha }_1{q}_{xx}-{\alpha }_1{q}^{2}r-{\alpha }_2{q}_x+{\alpha }_{3}q,\hfill \\ \hfill & c_4={\alpha }_0{r}_{xxx}-3{\alpha }_{0}qr{r}_x+{\alpha }_1{r}_{xx}-{\alpha }_{1}q{r}^2+{\alpha }_2{r}_x+{\alpha }_{3}r,\hfill \\ \hfill & \cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & g_1={\beta }_0(r+u_2)+{\alpha }_0{u}_2,\hfill \\ \hfill & e_1={\beta }_0(q+u_1)+{\alpha }_0{u}_1,\hfill \\ \hfill & d_1=-{\beta }_1\left(t\right),\hfill \\ \hfill & e_2=-{\beta }_0{(q+u_1)}_x-{\alpha }_0{u}_{1,x}+{\beta }_1(q+u_1)+{\alpha }_1{u}_1,\hfill \\ \hfill & g_2={\beta }_0{(r+u_2)}_x+{\alpha }_0{u}_{2,x}+{\beta }_1(r+u_2)+{\alpha }_1{u}_2,\hfill \\ \hfill & d_2={\beta }_0(q+u_1)(r+u_2)+{\alpha }_0{u}_{1}r+{\alpha }_0{u}_1{u}_2+{\alpha }_0{u}_{2}q-{\beta }_2,\hfill \\ \hfill & \cdots \hfill \end{array}.$$

When $n=1,m=0$, (19) reduces to

$$\left\{\begin{array}{c}{q}_t={\alpha }_0{q}_x-{\alpha }_{1}q,\hfill \\ r_t={\alpha }_0{r}_x+{\alpha }_{1}r,\hfill \\ u_{1t}={\beta }_0{(q+u_1)}_x+{\alpha }_0{u}_{1,x}-\beta (q+u_1)-{\alpha }_1{u}_1,\hfill \\ u_{2t}={\beta }_0{(r+u_2)}_x+{\alpha }_0{u}_{2,x}+{\beta }_1(r+u_2)+{\alpha }_1{u}_2.\hfill \end{array}\right.$$

When $n=2,m=0,$ (19) becomes

$$\left\{\begin{array}{cc}\hfill q_t=& -{\alpha }_0{q}_{xx}+{\alpha }_1{q}_x+{\alpha }_0{q}^{2}r-{\alpha }_{2}q,\hfill \\ \hfill r_t=& {\alpha }_0{r}_{xx}+{\alpha }_1{r}_x-{\alpha }_{0}q{r}^2+{\alpha }_{2}r,\hfill \\ \hfill u_{1t}=& -{\beta }_0{(q+u_1)}_{xx}-{\alpha }_0{u}_{1,xx}+{\beta }_1{(q+u_1)}_x+{\alpha }_1{u}_{1,x},\hfill \\ \hfill & +{\beta }_0{(q+u_1)}^2(r+u_2)+{\alpha }_0(q+u_1)(u_{1}r+u_{2}q+u_1{u}_2)\hfill \\ \hfill & -{\beta }_2(q+u_1)+{\alpha }_{0}qr{u}_1-{\alpha }_2{u}_1,\hfill \\ \hfill u_{2t}=& {\beta }_0{(r+u_2)}_{xx}+{\alpha }_0{u}_{2,xx}+{\beta }_1{(r+u_2)}_x+{\alpha }_1{u}_{2,x}\hfill \\ \hfill & -{\beta }_0(q+u_1){(r+u_2)}^2-{\alpha }_0(r+u_2)(u_{1}r+u_1{u}_2+u_{2}q)\hfill \\ \hfill & +{\beta }_0(r+u_2)-{\alpha }_{0}qr{u}_2+{\alpha }_2{u}_2.\hfill \end{array}\right.$$

(20)

Specially, set ${\alpha }_1={\alpha }_2=0,{\alpha }_0=1,{\beta }_0={\beta }_1={\beta }_2=0,u_1=u_2=0,$ (20) reduces to the well-known nonlinear Schrödinger system

$$\{\begin{array}{c}{q}_t=-q_{xx}+q^{2}r,\hfill \\ r_t=r_{xx}-q{r}^2.\hfill \end{array}$$

(21)

Remark 2.

We first time obtained such the nonlinear integrable coupling of the nonlinear Schrödinger equation. The so-called nonlinear integrable coupling means that if $u_t=K\left(u\right)$ is a known integrable hierarchy, $v_t=S(u,v)$ is also integrable and is nonlinear with respect to the new potential variable v, then the integrable system

$$\left\{\begin{array}{c}{u}_t=K\left(u\right),\hfill \\ v_t=S(u,v)\hfill \end{array}\right.$$

is called a nonlinear integrable coupling.

When $n=3,m=0$, (19) again reduces to

$$\left\{\begin{array}{c}{q}_t=-b_4={\alpha }_0{q}_{xxx}-3{\alpha }_{0}q{q}_{x}r-{\alpha }_1{q}_{xx}-{\alpha }_1{q}^{2}r+{\alpha }_2{q}_x-{\alpha }_{3}q,\hfill \\ r_t=c_4={\alpha }_0{r}_{xxx}+{\alpha }_1{r}_{xx}-3{\alpha }_{0}qr{r}_x-{\alpha }_{1}q{r}^2+{\alpha }_2{r}_x+{\alpha }_{3}r,\hfill \end{array}\right.$$

(22)

$$\begin{array}{cc}\hfill u_{1t}=-e_4=& {\beta }_0{(q+u_1)}_{xxx}+{\alpha }_0{u}_{1,xxx}-{\beta }_1{(q+u_1)}_{xx}-{\alpha }_1{u}_{1,xx}\hfill \\ \hfill & -{\alpha }_0{\left[(q+u_1)(u_{1}r+u_{2}q+u_1{u}_2)\right]}_x+{\beta }_2{(q+u_1)}_x-{\alpha }_0{\left(qr{u}_1\right)}_x\hfill \\ \hfill & +{\alpha }_2{u}_{1,x}-3{\beta }_0(q+u_1){(q+u_1)}_x(r+u_2)+{\alpha }_0{(q+u_1)}^2{u}_{2,x}\hfill \\ \hfill & -{\alpha }_0(q+u_1){(q+u_1)}_x{u}_2+{\alpha }_{0}q{u}_{2,x}(q+u_1)-{\alpha }_0{q}_x{u}_2(q+u_1)\hfill \\ \hfill & +{\alpha }_0(q+u_1)(u_1{u}_{2,x}-u_{1,x}{u}_2+u_1{r}_x-u_{1,x}r)+{\beta }_1{(q+u_1)}^2(q+u_2)\hfill \\ \hfill & +{\alpha }_1(q+u_1)(q+u_2+u_1{u}_2+u_{1}r)-{\beta }_3(q+u_1)+{\alpha }_0{u}_1(q{r}_x-q_{x}r)\hfill \\ \hfill & +{\alpha }_{1}qr{u}_1-{\alpha }_3{u}_1,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill u_{2t}=g_4=& {\beta }_0{(r+u_2)}_{xxx}+{\alpha }_0{u}_{2,xxx}+{\beta }_1{(r+u_2)}_{xx}+{\alpha }_1{u}_{2,xx}\hfill \\ & -3{\beta }_0(q+u_1)(r+u_2){(r+u_2)}_x-{\alpha }_0{\left[(r+u_2)(u_{1}r+u_1{u}_2+u_{2}q)\right]}_x\hfill \\ \hfill & +{\beta }_0{(r+u_2)}_x-{\alpha }_0{\left(qr{u}_2\right)}_x+{\alpha }_2{u}_{2,x}-{\alpha }_0(r+u_2)(q+u_1)u_{2,x}\hfill \\ \hfill & +{\alpha }_0(r+u_2){(q+u_1)}_x{u}_2-{\alpha }_0(r+u_2)q{u}_{2,x}+{\alpha }_0{q}_x{u}_2(r+u_2)\hfill \\ \hfill & -{\alpha }_0(r+u_2)(u_1{u}_{2,x}-u_{1,x}{u}_2+u_1{r}_x-u_{1,x}r)-{\beta }_1(r+u_2)(q+u_1)\hfill \\ \hfill & \times (q+u_2)-{\alpha }_1(r+u_2)(q{u}_2+u_1{u}_2+u_{1}r)+{\beta }_3(r+u_2)\hfill \\ \hfill & -{\alpha }_0{u}_2(q{r}_x-q_{x}r)-{\alpha }_{1}qr{u}_2+{\alpha }_3{u}_2.\hfill \end{array}$$

(24)

When ${\alpha }_1={\alpha }_2={\alpha }_3=0,{\alpha }_0=1$, (22) reduces to

$$\left\{\begin{array}{c}{q}_t=q_{xxx}+3q{q}_{x}r,\hfill \\ r_t=r_{xxx}-3qr{r}_x.\hfill \end{array}\right.$$

(25)

When $r=1$, (25) turns to the KdV equation

$$q_t=q_{xxx}+3q{q}_x.$$

When $r=-iq$, (25) becomes

$$q_t=q_{xxx}-3i{q}^2{q}_x,$$

(26)

which is a complex modified KdV equation.

When ${\alpha }_0={\alpha }_2={\alpha }_3=0,{\alpha }_1=1$, (22) reduces to

$$\left\{\begin{array}{c}{q}_t=-q_{xx}+q^{2}r,\hfill \\ r_t=r_{xx}-q{r}^2,\hfill \end{array}\right.$$

(27)

which represents the Schrödinger equation.

Therefore, we call (22) a generalized KdV-Schrödinger integrable system. Obviously, when ${\beta }_i=0(i=1,2,3),u_1=u_2=0$, Equations (23) and (24) identically hold. Hence, (22)–(24) consist of an nonlinear integrable coupling of the generalized KdV-Schrödinger integrable system (22).

Remark 3.

From the above discussion, we conclude that a simple and efficient approach for generating nonlinear integrable couplings just right take multiple parameter functions.

In what follows, we look for the Hamiltonian structure and the symmetries of the isospectral integrable hierarchy:

$$u_t={\left(\begin{array}{c}q\\ r\\ u_1\\ u_2\end{array}\right)}_t=\left(\begin{array}{c}-b_{n+1}\\ c_{n+1}\\ -e_{n+1}\\ g_{n+1}\end{array}\right),$$

(28)

which is the integrable coupling of the hierarchy (14). For arbitrary elements $a,b$ in the Lie algebra H, we represent them as

$$a=\sum _{i=1}^6{a}_i{h}_i,b=\sum _{i=1}^6{b}_i{h}_i$$

which can be used to define a commutative operation in the vector space $R^6$ as follows:

$$[a,b]=\left(\begin{array}{c}{[a,b]}_1\\ {[a,b]}_2\end{array}\right),$$

(29)

where

$$\begin{array}{cc}\hfill {[a,b]}_1^T=& (a_2{b}_3-a_3{b}_2,a_1{b}_2-a_2{b}_1,a_3{b}_1-a_1{b}_3),\hfill \\ \hfill {[a,b]}_2^T=& (a_2{b}_6-a_6{b}_2+a_5{b}_3-a_3{b}_5+a_5{b}_6-a_6{b}_5,\hfill \\ & a_1{b}_5-a_5{b}_1+a_4{b}_2-a_2{b}_4+a_4{b}_5-a_5{b}_4,\hfill \\ & a_3{b}_4-a_4{b}_3+a_6{b}_1-a_1{b}_6+a_6{b}_4-a_4{b}_6).\hfill \end{array}$$

It can be verified that $R^6$ becomes a vector Lie algebra if equipped with (29). Besides, (29) can be written as

$$[a,b]=a^{T}R\left(b\right)=a^T\left(\begin{array}{cc}{R}_1& R_2\\ 0& R_3\end{array}\right),$$

where

$$\begin{array}{cc}\hfill & R_1=\left(\begin{array}{ccc}0& b_2& -b_3\\ b_3& -b_1& 0\\ -b_2& 0& b_1\end{array}\right),R_2=\left(\begin{array}{ccc}0& b_5& -b_6\\ b_6& -b_4& 0\\ b_5& 0& b_4\end{array}\right),\hfill \\ \hfill & R_3=\left(\begin{array}{ccc}0& b_2+b_5& -b_3-b_6\\ b_3+b_6& -b_1-b_4& 0\\ -b_2-b_5& 0& b_1+b_4\end{array}\right),\hfill \end{array}$$

$R\left(b\right)$ requires satisfy

$$R\left(b\right)M=-{\left(R\left(b\right)M\right)}^T,M^T=M,M_{xt}=0.$$

(30)

Solving (30) obtains

$$M=\left(\begin{array}{cccccc}{\eta }_1& 0& 0& {\eta }_2& 0& 0\\ 0& 0& {\eta }_1& 0& 0& {\eta }_2\\ 0& {\eta }_1& 0& 0& {\eta }_2& 0\\ {\eta }_2& 0& 0& {\eta }_2& 0& 0\\ 0& 0& {\eta }_2& 0& 0& {\eta }_2\\ 0& {\eta }_2& 0& 0& {\eta }_2& 0\end{array}\right),$$

here ${\eta }_2$ and ${\eta }_2$ are constants.

According to Refs. [16,17], a linear functional is defined by

$$\{a,b\}=a^{T}Mb.$$

(31)

Taking

$$U=(-\lambda ,q,r,0,u_1,u_2)\in R^6,$$

$$V=(A,B,C,D,E,G)\in R^6,$$

$$A=\sum _{i\ge 0}{a}_i{\lambda }^{-i},B=\sum _{i\ge 0}{b}_i{\lambda }^{-i},\cdots$$

In terms of (31), it is easy to calculate that

$$\begin{array}{cc}\hfill & \{V,\frac{\partial U}{\partial q}\}=C{\eta }_1+G{\eta }_2,\{V,\frac{\partial U}{\partial r}\}=B{\eta }_1+E{\eta }_2,\hfill \\ \hfill & \{V,\frac{\partial U}{\partial u_1}\}=C{\eta }_2+G{\eta }_2,\{V,\frac{\partial U}{\partial u_2}\}=(B+E){\eta }_2,\hfill \\ \hfill & \{V,\frac{\partial U}{\lambda }\}=A{\eta }_1+D{\eta }_2.\hfill \end{array}$$

Substituting the above results into the quadratic-form identity reads that

$$\frac{\delta }{\delta u}(A{\eta }_1+D{\eta }_2)={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}C{\eta }_1+G{\eta }_2\\ B{\eta }_1+E{\eta }_2\\ C{\eta }_2+G{\eta }_2\\ B{\eta }_2+E{\eta }_2\end{array}\right).$$

(32)

Comparing the coefficients of ${\lambda }^{-n-1}$ of both sides in (32) gives

$$\frac{\delta }{\delta u}(a_{n+1}{\eta }_1+d_{n+1}{\eta }_2)=(-n)\left(\begin{array}{c}{c}_n{\eta }_1+g_n{\eta }_2\\ b_n{\eta }_1+e_n{\eta }_2\\ c_n{\eta }_2+g_n{\eta }_2\\ b_n{\eta }_2+e_n{\eta }_2\end{array}\right).$$

Thus,

$$\left(\begin{array}{c}{c}_n{\eta }_1+g_n{\eta }_2\\ b_n{\eta }_1+e_n{\eta }_2\\ c_n{\eta }_2+g_n{\eta }_2\\ b_n{\eta }_2+e_n{\eta }_2\end{array}\right).=:\frac{\delta H_n}{\delta u},$$

where $H_n=-\frac{a_{n+1}{\eta }_1+d_{n+1}{\eta }_2}{n}$ are the Hamiltonian function. We can take ${\eta }_1=1$, then the integrable coupling (28) can be written as

$$\begin{array}{cc}\hfill u_t& ={\left(\begin{array}{c}q\\ r\\ u_1\\ u_2\end{array}\right)}_t=\left(\begin{array}{cccc}0& \frac{1}{{\eta }_2-1}& 0& \frac{1}{1-{\eta }_2}\\ \frac{1}{1-{\eta }_2}& 0& \frac{1}{{\eta }_2-1}& 0\\ 0& \frac{1}{1-{\eta }_2}& 0& -\frac{1}{{\eta }_2-{\eta }_2^2}\\ \frac{1}{{\eta }_2-1}& 0& \frac{1}{{\eta }_2-{\eta }_2^2}& 0\end{array}\right)\left(\begin{array}{c}{c}_{n+1}{\eta }_1+g_{n+1}{\eta }_2\\ b_{n+1}{\eta }_1+e_{n+1}{\eta }_2\\ c_{n+1}{\eta }_2+g_{n+1}{\eta }_2\\ b_{n+1}{\eta }_2+e_{n+1}{\eta }_2\end{array}\right)\hfill \\ \hfill & =:\overline{J}\left(\begin{array}{c}{c}_{n+1}+g_{n+1}{\eta }_2\\ b_{n+1}+e_{n+1}{\eta }_2\\ c_{n+1}{\eta }_2+g_{n+1}{\eta }_2\\ b_{n+1}{\eta }_2+e_{n+1}{\eta }_2\end{array}\right)=\overline{J}\frac{\delta H_{n+1}}{\delta u}+\left(\begin{array}{c}(r+u_2){\beta }_n{\eta }_2+u_2{\alpha }_n{\eta }_2+{\alpha }_{n}r\\ -(q+u_1){\beta }_n{\eta }_2+u_1{\alpha }_n{\eta }_2+{\alpha }_{n}q\\ (r+u_2){\beta }_n{\eta }_2+u_2{\alpha }_n{\eta }_2+{\alpha }_{n}r{\eta }_2\\ -(q+u_1){\beta }_n{\eta }_2+u_1{\alpha }_n{\eta }_2+q{\alpha }_n{\eta }_2\end{array}\right),\hfill \end{array}$$

(33)

which is the Hamiltonian form of (28). It can be found that

$$\left(\begin{array}{c}{c}_{n+1}+g_{n+1}{\eta }_2\\ b_{n+1}+e_{n+1}{\eta }_2\\ (c_{n+1}+g_{n+1}){\eta }_2\\ (b_{n+1}+e_{n+1}){\eta }_2\end{array}\right)=L\left(\begin{array}{c}{c}_n+g_n{\eta }_2\\ b_n+e_n{\eta }_2\\ (c_n+g_n){\eta }_2\\ (b_n+e_n){\eta }_2\end{array}\right),$$

where

$$\begin{array}{cc}\hfill & L=\hfill \\ \hfill & \begin{array}{c}\hfill \left(\begin{array}{cccc}\partial -r{\partial }^{-1}q& r{\partial }^{-1}r& -r(r+u_2){\partial }^{-1}{u}_1-u_2{\partial }^{-1}q& (r+u_2){\partial }^{-1}{u}_2+u_2{\partial }^{-1}r\\ -q{\partial }^{-1}q& -\partial +q{\partial }^{-1}r& -(q+u_1){\partial }^{-1}{u}_1-u_1{\partial }^{-1}q& (q+u_1){\partial }^{-1}{u}_2+u_1{\partial }^{-1}r\\ (r+u_2){\partial }^{-1}(u_1-u_2)& 0& \partial -(r+u_2){\partial }^{-1}(q+u_1)& (r+u_2){\partial }^{-1}(r+u_2)\\ 0& 0& -(q+u_1){\partial }^{-1}(q+u_1)& -\partial +(q+u_1){\partial }^{-1}(r+u_2)\end{array}\right)\end{array}\hfill \end{array}$$

is a recurrence operator. Hence, (33) can be written again as

$$\begin{array}{cc}\hfill u_t={\left(\begin{array}{c}q\\ r\\ u_1\\ u_2\end{array}\right)}_t& =\overline{J}L\left(\begin{array}{c}{c}_n+g_n{\eta }_2\\ b_n+e_n{\eta }_2\\ (c_n+g_n){\eta }_2\\ (b_n+e_n){\eta }_2\end{array}\right)+\left(\begin{array}{c}(r+u_2){\beta }_n{\eta }_2+u_2{\alpha }_n{\eta }_2+{\alpha }_{n}r\\ -(q+u_1){\beta }_n{\eta }_2+u_1{\alpha }_n{\eta }_2+{\alpha }_{n}q\\ (r+u_2){\beta }_n{\eta }_2+u_2{\alpha }_n{\eta }_2+{\alpha }_{n}r{\eta }_2\\ -(q+u_1){\beta }_n{\eta }_2+u_1{\alpha }_n{\eta }_2+q{\alpha }_n{\eta }_2\end{array}\right)\hfill \\ \hfill & =:P_1+P_2={\mathsf{\Phi }}^n\overline{J}\left(\begin{array}{c}{c}_1+g_1{\eta }_2\\ b_1+e_1{\eta }_2\\ (c_1+g_1){\eta }_2\\ (b_1+e_1){\eta }_2\end{array}\right)+P_2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathsf{\Phi }=& \overline{J}L{\overline{J}}^{-1}\hfill \\ \hfill =& \left(\begin{array}{cccc}0& 1& 0& -1\\ -1& 0& 1& 0\\ 0& -1& 0& \frac{1}{{\eta }_2}\\ 1& 0& -\frac{1}{{\eta }_2}& 0\end{array}\right)\hfill \\ \hfill & \begin{array}{c}\hfill \left(\begin{array}{cccc}\partial -r{\partial }^{-1}q& r{\partial }^{-1}r& -(r+u_2){\partial }^{-1}{u}_1-u_2{\partial }^{-1}q& (r+u_2){\partial }^{-1}{u}_2+u_2{\partial }^{-1}r\\ -q{\partial }^{-1}q& -\partial +q{\partial }^{-1}r& -(q+u_1){\partial }^{-1}{u}_1-u_1{\partial }^{-1}q& (q+u_1){\partial }^{-1}{u}_2+u_1{\partial }^{-1}r\\ (r+u_2){\partial }^{-1}(u_1-u_2)& 0& \partial -(r+u_2){\partial }^{-1}(q+u_1)& (r+u_2){\partial }^{-1}(r+u_2)\\ 0& 0& -(q+u_1){\partial }^{-1}(q+u_1)& \partial +(q+u_1){\partial }^{-1}(r+u_2)\end{array}\right)\end{array}\hfill \\ \hfill & \left(\begin{array}{cccc}0& -\frac{1}{{\eta }_2}\left(\frac{1}{{\eta }_2}-1\right)& 0& 1-\frac{1}{{\eta }_2}\\ \frac{1}{{\eta }_2}\left(\frac{1}{{\eta }_2}-1\right)& 0& 1-\frac{1}{{\eta }_2}& 0\\ 0& 1-\frac{1}{{\eta }_2}& 0& \frac{1}{{\eta }_2}-1\\ \frac{1}{{\eta }_2}-1& 0& \frac{1}{{\eta }_2}-1& 0\end{array}\right).\hfill \end{array}$$

Obviously,

$$\begin{array}{cc}\hfill K_0=& \overline{J}\left(\begin{array}{c}{c}_1+g_1{\eta }_2\\ b_1+e_1{\eta }_2\\ (c_1+g_1){\eta }_2\\ (b_1+e_1){\eta }_2\end{array}\right)\hfill \\ \hfill =& \frac{1}{{\eta }_2-1}\left(\begin{array}{cccc}0& 1& 0& -1\\ -1& 0& 1& 0\\ 0& -1& 0& \frac{1}{{\eta }_2}\\ 1& 0& -\frac{1}{{\eta }_2}& 0\end{array}\right)\left(\begin{array}{c}{\alpha }_{0}r+({\beta }_{0}r+{\beta }_0{u}_2+{\alpha }_0{u}_1){\eta }_2\\ {\alpha }_{0}q+({\beta }_{0}q+{\beta }_0{u}_1+{\alpha }_0{u}_1){\eta }_2\\ ({\alpha }_{0}r+{\beta }_{0}r+{\beta }_0{u}_2+{\alpha }_0{u}_1){\eta }_2\\ ({\alpha }_{0}q+{\beta }_{0}q+{\beta }_0{u}_1+{\alpha }_0{u}_1){\eta }_2\end{array}\right)\hfill \\ \hfill =& \left(\begin{array}{c}-{\alpha }_{0}q\\ {\alpha }_{0}r\\ -({\beta }_{0}q+{\beta }_0{u}_1+{\alpha }_0{u}_1)\\ ({\beta }_{0}r+{\beta }_0{u}_2+{\alpha }_0{u}_1)\end{array}\right)\hfill \end{array}$$

is an initial symmetry of the integrable coupling (33). The isospectral–nonisospectral integrable hierarchy (19) also contains the nonisospectral integrable hierarchy

$$u_t={\left(\begin{array}{c}q\\ r\\ u_1\\ u_2\end{array}\right)}_t=\left(\begin{array}{c}-{\overline{b}}_{m+1}\\ {\overline{c}}_{m+1}\\ -{\overline{e}}_{m+1}\\ {\overline{g}}_{m+1}\end{array}\right).$$

(34)

In what follows, we consider some reductions of (34). Set ${\overline{b}}_0={\overline{c}}_0={\overline{g}}_0={\overline{e}}_0=0,{\overline{a}}_0=-k_0\left(t\right)x,{\overline{d}}_0=-{\gamma }_0\left(t\right)x$, then we have from (17) and (18) that

$$\begin{array}{cc}\hfill & {\overline{b}}_1=k_{0}xq,{\overline{c}}_1=k_{0}xr,{\overline{g}}_1={\gamma }_{0}x(r+u_2)+k_{0}x{u}_2,\hfill \\ \hfill & {\overline{e}}_1={\gamma }_{0}x(q+u_1)+k_{0}x{u}_1,{\overline{a}}_1-k_1\left(t\right)x,{\overline{d}}_1=-{\gamma }_1\left(t\right)x,\hfill \\ \hfill & {\overline{b}}_2=-k_0{\left(xq\right)}_x+k_{1}xq,{\overline{c}}_2=k_0{\left(xr\right)}_x+k_{1}xr,\hfill \\ \hfill & {\overline{g}}_2={\gamma }_0{\left(x(r+u_2)\right)}_x+k_0{\left(x{u}_2\right)}_x+{\gamma }_{1}x(r+u_2)+k_{1}x{u}_2,\hfill \\ \hfill & {\overline{e}}_2=-{\gamma }_0{\left(x(q+u_1)\right)}_x-k_0{\left(x{u}_1\right)}_x+{\gamma }_{1}x(q+u_1)+k_{1}x{u}_1,\hfill \\ \hfill & {\overline{a}}_2=k_{0}xqr+k_0{\partial }^{-1}\left(qr\right)-k_{2}x,\hfill \\ \hfill & {\overline{b}}_3=k_0{\left(xq\right)}_{xx}+k_1{\left(xq\right)}_x-k_{0}x{q}^{2}r-k_{0}q{\partial }^{-1}\left(qr\right)+k_{2}xq,\hfill \\ \hfill & {\overline{c}}_3=k_0{\left(xr\right)}_{xx}+k_1{\left(xr\right)}_x-k_{0}xq{r}^2-k_{0}r{\partial }^{-1}\left(qr\right)+k_{2}xr,\hfill \\ \hfill & {\overline{d}}_2={\gamma }_{0}x(q+u_1)(r+u_2)+{\gamma }_0{\partial }^{-1}(q+u_1)(r+u_2)\hfill \\ \hfill & +k_0{\partial }^{-1}[(q+u_1){\left(x{u}_2\right)}_x+(r+u_2){\left(x{u}_1\right)}_x]\hfill \\ \hfill & +k_0{\partial }^{-1}(u_{1}r+u_{1}x{r}_x+u_{2}q+u_{2}x{q}_x)-{\gamma }_{2}x,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{g}}_3={\gamma }_0{\left(x(r+u_2)\right)}_{xx}+k_0{\left(x{u}_2\right)}_{xx}+{\gamma }_1{\left(x(r+u_2)\right)}_x\hfill \\ \hfill & +k_1{\left(x{u}_2\right)}_x-u_2[k_{0}xqr+k_0{\partial }^{-1}\left(qr\right)-k_{2}x]-(r+u_2){\overline{d}}_2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\overline{e}}_3={\gamma }_0{\left(x(q+u_1)\right)}_{xx}+k_0{\left(x{u}_1\right)}_{xx}-{\gamma }_1{\left(x(q+u_1)\right)}_x\hfill \\ \hfill & -k_1{\left(x{u}_1\right)}_x-u_1[k_{0}xqr+k_0{\partial }^{-1}\left(qr\right)-k_{2}x]-(q+u_1){\overline{d}}_2,\hfill \end{array}$$

(36)

$$\cdots \cdots$$

When $m=1$, (34) reduces to

$$\left\{\begin{array}{c}{q}_t=k_0{\left(xq\right)}_x-k_{1}xq,\hfill \\ r_t=k_0{\left(xr\right)}_x+k_{1}xr,\hfill \\ u_{1t}={\gamma }_0{\left(x(q+u_1)\right)}_x+k_0{\left(x{u}_1\right)}_x-{\gamma }_{1}x(q+u_1)-k_{1}x{u}_1,\hfill \\ u_{2}t={\gamma }_0{\left(x(r+u_2)\right)}_x+k_0{\left(x{u}_2\right)}_x+{\gamma }_{1}x(r+u_2)+k_{1}x{u}_2.\hfill \end{array}\right.$$

(37)

Obviously, When ${\gamma }_0={\gamma }_1=u_1=u_2=0,$ (37) just reduces to

$$\left\{\begin{array}{c}{q}_t=k_0{\left(xq\right)}_x-k_{1}xq,\hfill \\ r_t=k_0{\left(xr\right)}_x+k_{1}xr.\hfill \end{array}\right.$$

(38)

Hence, (37) is a nonisospectral integrable coupling of (38).

When $m=2$, (34) becomes

$$\left\{\begin{array}{c}{q}_t=-k_0{\left(xq\right)}_{xx}-k_1{\left(xq\right)}_x+k_{0}x{q}^{2}r+k_{0}q{\partial }^{-1}\left(qr\right)-k_{2}xq,\hfill \\ r_t=k_0{\left(xr\right)}_{xx}+k_1{\left(xr\right)}_x-k_{0}xq{r}^2-k_{0}r{\partial }^{-1}\left(qr\right)+k_{2}xr,\hfill \\ u_{1t}=-{\overline{e}}_3,\hfill \\ u_{2}t={\overline{g}}_3,\hfill \end{array}\right.$$

(39)

where ${\overline{e}}_3,{\overline{g}}_3$ are presented by (36) and (35), respectively.

When ${\gamma }_0={\gamma }_1={\gamma }_2=0,u_1=u_2=0,$ (39) reduces to

$$\left\{\begin{array}{c}{q}_t=-k_0{\left(xq\right)}_{xx}-k_1{\left(xq\right)}_x+k_{0}x{q}^{2}r+k_{0}q{\partial }^{-1}\left(qr\right)-k_{2}xq,\hfill \\ r_t=k_0{\left(xr\right)}_{xx}+k_1{\left(xr\right)}_x-k_{0}xq{r}^2-k_{0}r{\partial }^{-1}\left(qr\right)+k_{2}xr.\hfill \end{array}\right.$$

(40)

Therefore, (39) is a nonlinear nonisospectral integrable coupling of (40).

Taking $r=0$, (40) again reduces to

$$q_t=-k_0{\left(xq\right)}_{xx}-k_1{\left(xq\right)}_x-k_{2}xq.$$

A simple symmetry of the nonisospectral integrable coupling (34) is given by

$$\begin{array}{cc}\hfill {\tau }_0& =\overline{J}\left(\begin{array}{c}{\overline{c}}_1+{\overline{g}}_1{\eta }_2\\ {\overline{b}}_1+{\overline{e}}_1{\eta }_2\\ ({\overline{c}}_1+{\overline{g}}_1){\eta }_2\\ ({\overline{b}}_1+{\overline{g}}_1){\eta }_2\end{array}\right)\hfill \\ \hfill & =\frac{1}{{\eta }_2-1}\left(\begin{array}{cccc}0& 1& 0& -1\\ -1& 0& 1& 0\\ 0& -1& 0& \frac{1}{{\eta }_2}\\ 1& 0& -\frac{1}{{\eta }_2}& 0\end{array}\right)\left(\begin{array}{c}{k}_{0}xr+[{\gamma }_{0}x(r+u_2)+k_{0}x{u}_2]{\eta }_2\\ k_{0}xq+[{\gamma }_{0}x(q+u_1)+k_{0}x{u}_1]{\eta }_2\\ [k_{0}xr+{\gamma }_{0}x(r+u_2)+k_{0}x{u}_2]{\eta }_2\\ [k_{0}xq+{\gamma }_{0}x(q+u_1)+k_{0}x{u}_1]{\eta }_2\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{c}-k_{0}xq\\ k_{0}xr\\ -{\gamma }_{0}x(q+u_1)-k_{0}x{u}_1\\ {\gamma }_{0}x(r+u_2)+k_{0}x{u}_2\end{array}\right).\hfill \end{array}$$

Case 2: The second kind of integrable model

In the section, we take the basis of the Lie algebra $A_1$:

$$h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),$$

which is enlarged to the following

$$G=:\mathbf{span}\{f_1,f_2,f_3,f_4,f_5,f_6\},$$

where

$$f_1=\frac{1}{2}\left(\begin{array}{cc}h& 0\\ 0& h\end{array}\right),f_2=\left(\begin{array}{cc}e& 0\\ 0& e\end{array}\right),f_3=\frac{1}{2}\left(\begin{array}{cc}f& 0\\ 0& f\end{array}\right),$$

$$f_4=\frac{1}{2}\left(\begin{array}{cc}0& h\\ h& 0\end{array}\right),f_5=\left(\begin{array}{cc}0& e\\ e& 0\end{array}\right),f_6=\frac{1}{2}\left(\begin{array}{cc}0& f\\ f& 0\end{array}\right),$$

along with the following commutative relations

$$\begin{array}{cc}\hfill & [f_1,f_2]=f_2,[f_1,f_3]=-f_3,[f_1,f_4]=0,[f_1,f_5]=f_5,\hfill \\ \hfill & [f_1,f_6]=-f_6,[f_2,f_3]=f_1,[f_2,f_4]=-f_5,[f_2,f_5]=0,\hfill \\ \hfill & [f_2,f_6]=f_4,[f_3,f_4]=f_6,[f_3,f_5]=-f_4,[f_3,f_6]=0,\hfill \\ \hfill & [f_4,f_5]=f_2,[f_4,f_6]=-f_3,[f_5,f_6]=f_1.\hfill \end{array}$$

Denoting

$$G=G_1\oplus G_2,G_1=\mathbf{span}\{f_1,f_2,f_3\},G_2=\mathbf{span}=\{f_4,f_5,f_6\},$$

we find that

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

(41)

Remark 4.

Obviously, $G_1$ and $H_1$ has the same commutators. Hence, if defining the same loop algebras $\widetilde{G_1}$ and $\widetilde{H_1}$ for $G_1$ and $H_1$, and introducing the same spectral equations, then it follows that the same integrable hierarchy can be generated by the $\widetilde{G_1}$ and $\widetilde{H_1}$. However, (41) is different from (7), we conclude that we could obtain various integrable coupling by using the Lie algebra G. In what follows, we shall employ the Lie algebra G for generating another kind of expanding integrable model of (14).

Set

$$\begin{array}{cc}& {\phi }_x=U\phi ,U=-f_1\left(1\right)+q{f}_2\left(0\right)+r{f}_3\left(0\right)+s_1{h}_5\left(0\right)+s_2{h}_6\left(0\right),\hfill \end{array}$$

(42)

$$\begin{array}{cc}& {\phi }_t=V\phi ,V=V_1+V_2,\hfill \\ & V_1=\sum _{i\ge 0}(a_{1i}{f}_1(-i)+a_{2i}{f}_2(-i)+a_{3i}{f}_3(-i)+a_{4i}{f}_4(-i)+a_{5i}{f}_5(-i)+a_{6i}{f}_6(-i)),\hfill \\ & V_2=\sum _{j\ge 0}\sum _{l=1}^6{b}_{l_j}{f}_l(-j),{\lambda }_t=\sum _{j\ge 0}{k}_j\left(t\right){\lambda }^{-j}.\hfill \end{array}$$

(43)

Denoting

$$V_{+}^{(n,m)}=V_{1,+}^{\left(n\right)}+V_{2,+}^{\left(m\right)}=\sum _{l=1}^6\left(\sum _{i=0}^n{a}_{li}{f}_l(-i)+\sum _{j=0}^m{b}_{lj}{f}_l(-j)\right),$$

$${\lambda }_{t,+}^{\left(m\right)}=\sum _{j=0}^m{k}_j\left(t\right){\lambda }^{m-j}={\lambda }^m{\lambda }_t-{\lambda }_{t,-}^{\left(m\right)},$$

then equation

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

admits that

$$\begin{array}{cc}& a_{1i}={\partial }^{-1}(q{a}_{3i}-r{a}_{2i}+s_1{a}_{6i}-s_2{a}_{5i})-{\alpha }_i\left(t\right),\hfill \\ & a_{2,i+1}=-{\left(a_{2i}\right)}_x-q{a}_{1i}-s_1{a}_{4i},\hfill \\ & a_{3,i+1}={\left(a_{3i}\right)}_x-r{a}_{1i}-s_2{a}_{4i},\hfill \\ & a_{4i}={\partial }^{-1}(q{a}_{6i}-r{a}_{5i}+s_1{a}_{3i}-s_2{a}_{2i})-{\beta }_i\left(t\right),\hfill \\ & a_{5,i+1}=-{\left(a_{5i}\right)}_x-q{a}_{4i}-s_1{a}_{1i},\hfill \\ & a_{6,i+1}={\left(a_{6i}\right)}_x-r{a}_{4i}-s_2{a}_{1i},\hfill \end{array}$$

$$\begin{array}{cc}& {\overline{a}}_{1j}={\partial }^{-1}(q{\overline{a}}_{3j}-r{\overline{a}}_{2j}+s_1{\overline{a}}_{6j}-s_2{\overline{a}}_{5j})-k_j\left(t\right)x,\hfill \\ & {\overline{a}}_{2,j+1}=-{\left({\overline{a}}_{2j}\right)}_x-q{\overline{a}}_{1j}-s_1{\overline{a}}_{4j},\hfill \\ & a_{3,j+1}={\left({\overline{a}}_{3j}\right)}_x-r{\overline{a}}_{1j}-s_2{\overline{a}}_{4j},\hfill \\ & {\overline{a}}_{4j}={\partial }^{-1}(q{\overline{a}}_{6j}-r{\overline{a}}_{5j}+s_1{\overline{a}}_{3j}-s_2{\overline{a}}_{1j})-{\gamma }_j\left(t\right),\hfill \\ & {\overline{a}}_{5,j+1}=-{\left({\overline{a}}_{5j}\right)}_x-q{\overline{a}}_x-q{\overline{a}}_{4j}-s_1{\overline{a}}_{1j},\hfill \\ & {\overline{a}}_{6,j+1}={\left({\overline{a}}_{6j}\right)}_x-r{\overline{a}}_{4j}-s_2{\overline{a}}_{1j},i,j\ge 0.\hfill \end{array}$$

Taking

$$\begin{array}{cc}& a_{20}=a_{30}=a_{50}=a_{60}={\overline{a}}_{20}={\overline{a}}_{30}={\overline{a}}_{50}={\overline{a}}_{60}=0,\hfill \\ & a_{10}=-{\alpha }_0\left(t\right),a_{40}=-{\beta }_0\left(t\right),{\overline{a}}_{10}=-k_0\left(t\right)x,{\overline{a}}_{40}=-{\gamma }_0\left(t\right),\hfill \end{array}$$

then we get from the above equations that

$$\begin{array}{cc}& a_{21}={\alpha }_{0}q+{\beta }_0{s}_1,a_{31}={\alpha }_0\gamma +{\beta }_0{s}_2,a_{51}={\alpha }_0{s}_1+{\beta }_{0}q,\hfill \\ & a_{61}={\alpha }_0{s}_2+{\beta }_0\gamma ,a_{11}=-{\alpha }_1\left(t\right),a_{41}=-{\beta }_1\left(t\right),\hfill \end{array}$$

$$\begin{array}{cc}& a_{22}=-{\alpha }_0{q}_x-{\beta }_0{s}_{1x}+{\alpha }_{1}q+{\beta }_1{s}_1,\hfill \\ & a_{32}=-{\alpha }_0{\gamma }_x+{\beta }_0{s}_{2x}+{\alpha }_{1}r+{\beta }_1{s}_2,\hfill \\ & a_{52}=-{\alpha }_0{s}_{1x}-{\beta }_0{q}_x+{\beta }_{1}q+{\alpha }_1{s}_1,\hfill \\ & a_{62}={\alpha }_0{s}_{2x}+{\beta }_0{\gamma }_x+{\beta }_{1}r+{\alpha }_1{s}_1,\hfill \\ & a_{12}={\alpha }_0(qr+s_1{s}_2)+{\beta }_0(q{s}_2+r{s}_1)-{\alpha }_2\left(t\right),\hfill \\ & a_{42}={\alpha }_0(q{s}_2+r{s}_1)+{\beta }_0(qr+s_1{s}_2)-{\beta }_2\left(t\right),\hfill \end{array}$$

$$\begin{array}{cc}& a_{23}={\alpha }_0(q_{xx}-q^{2}r-2q{s}_1{s}_2-r{s}_1{s}_2)+{\beta }_0(s_{1xx}-q^2{s}_2-2qr{s}_1-s_1^2{s}_2)\hfill \\ & -{\alpha }_1{q}_x-{\beta }_1{s}_{1x}+{\alpha }_{2}q+{\beta }_2\gamma ,\hfill \\ & a_{33}={\alpha }_0(r_{xx}-q{r}^2-2r{s}_1{s}_2-q{s}_2^2)+{\beta }_0(s_{2xx}-2qr{s}_2-r^2{s}_1+s_1{s}_2^2)\hfill \\ & +{\alpha }_1{r}_x+{\beta }_1{s}_{2x}+{\alpha }_{2}r+{\beta }_2{s}_2,\hfill \\ & a_{53}={\alpha }_0(s_{1xx}-2qr{s}_1-s_1^2{s}_2-q^2{s}_2)+{\beta }_0(q_{xx}-q^{2}r-2q{s}_1{s}_2-r{s}_1^2)\hfill \\ & -{\alpha }_1{s}_{1x}-{\beta }_1{q}_x+{\alpha }_2{s}_1+{\beta }_{2}q,\hfill \\ & a_{63}={\alpha }_0(s_{2xx}-2qr{s}_2-r^2{s}_1-s_1{s}_2^2)+{\beta }_0(r_{xx}-q{r}^2-2r{s}_1{s}_2-q{s}_2^2)\hfill \\ & +{\alpha }_1{s}_{2x}+{\beta }_1{r}_x+{\beta }_{2}r+{\alpha }_2{s}_2,\hfill \end{array}$$

$$\begin{array}{cc}& {\overline{a}}_{21}=k_{0}xq+{\gamma }_0{s}_1,{\overline{a}}_{31}=k_{0}xr+{\gamma }_0{s}_2,\hfill \\ & {\overline{a}}_{51}={\gamma }_{0}q+k_{0}x{s}_1,{\overline{a}}_{61}={\gamma }_{0}r+k_{0}x{s}_2,{\overline{a}}_{11}=-k_1\left(t\right)x,\hfill \\ & {\overline{a}}_{41}=-{\gamma }_1\left(t\right),{\overline{a}}_{52}=-{\gamma }_0{q}_x-k_0{\left(x{s}_1\right)}_x+q{\gamma }_1+k_{1}x{s}_1,\hfill \\ & {\overline{a}}_{62}={\gamma }_0{\gamma }_x+k_0{\left(x{s}_2\right)}_x+{\gamma }_{1}r+k_{1}x{s}_2,\hfill \\ & {\overline{a}}_{32}=k_0{\left(xr\right)}_x+{\gamma }_0{s}_{2x}+k_{1}xr+{\gamma }_1{s}_2,\hfill \\ & {\overline{a}}_{22}=-k_0{\left(xq\right)}_x-{\gamma }_0{s}_{1x}+k_{1}xq+{\gamma }_1{s}_1,\hfill \\ & {\overline{a}}_{12}=k_0{\partial }^{-1}(qr+s_1{s}_2)+k_{0}x(qr+s_1{s}_2)+{\gamma }_0(q{s}_2+r{s}_1)-k_2\left(t\right)x,\hfill \\ & {\overline{a}}_{42}=k_0{\partial }^{-1}(q{s}_2+r{s}_1)+{\gamma }_0(qr+s_1{s}_2)+k_{0}x(q{s}_2+r{s}_1)-{\gamma }_2\left(t\right),\hfill \end{array}$$

$$\begin{array}{cc}& {\overline{a}}_{23}=k_0{\left(xq\right)}_{xx}+{\gamma }_0{s}_{1xx}-k_1{\left(xq\right)}_x-{\gamma }_1{s}_{1x}-k_{0}q{\partial }^{-1}(qr+s_1{s}_2)\hfill \\ & -k_{0}xq(qr+s_1{s}_2)-{\gamma }_0(q{s}_2+r{s}_1)+k_{2}xq-k_0{s}_1{\partial }^{-1}(q{s}_2+r{s}_1)\hfill \\ & -{\gamma }_0{s}_1(qr+s_1{s}_2)-k_{0}x{s}_1(q{s}_2+r{s}_1)+{\gamma }_2{s}_1,\hfill \\ & {\overline{a}}_{33}=k_0{\left(xr\right)}_{xx}+{\gamma }_0{s}_{2xx}+k_1{\left(xr\right)}_x+{\gamma }_1{s}_{2x}-k_{0}r{\partial }^{-1}(qr+s_1{s}_2)\hfill \\ & -k_{0}xr(qr+s_1{s}_2)-{\gamma }_{0}r(q{s}_2+r{s}_1)+k_{2}xr-k_0{s}_2{\partial }^{-1}(q{s}_2+r{s}_1)\hfill \\ & -{\gamma }_0{s}_2(qr+s_1{s}_2)-k_{0}x{s}_2(q{s}_2+r{s}_1)+{\gamma }_2{s}_2,\hfill \\ & {\overline{a}}_{53}=k_0{\left(x{s}_1\right)}_{xx}+{\gamma }_0{q}_{xx}-{\gamma }_1{q}_x-k_1{\left(x{s}_1\right)}_x-k_{0}q{\partial }^{-1}(q{s}_2+r{s}_1)\hfill \\ & -{\gamma }_{0}q(qr+s_1{s}_2)-k_{0}xq(q{s}_2+r{s}_1)+{\gamma }_{2}q-k_0{s}_1{\partial }^{-1}(qr+s_1{s}_2)\hfill \\ & -k_{0}x{s}_1(qr+s_1{s}_2)-{\gamma }_0{s}_1(q{s}_2+r{s}_1)+k_{2}x{s}_1,\hfill \\ & {\overline{a}}_{63}=k_0{\left(x{s}_2\right)}_{xx}+{\gamma }_0{r}_{xx}+{\gamma }_1{r}_x+k_1{\left(x{s}_2\right)}_x-k_{0}r{\partial }^{-1}(q{s}_2+r{s}_1)\hfill \\ & -{\gamma }_{0}r(qr+s_1{s}_2)-k_{0}xr(q{s}_2+r{s}_2)+{\gamma }_{2}r-k_0{s}_2{\partial }^{-1}(qr+s_1{s}_2)\hfill \\ & -k_{0}x{s}_2(qr+s_1{s}_2)-{\gamma }_0{s}_2(q{s}_2+r{s}_1)+k_{2}x{s}_2,\hfill \\ & \cdots \hfill \end{array}$$

A direct calculation reads

$$\begin{array}{cc}& -{\left(V_{+}^{(n,m)}\right)}_x+[U,V_{+}^{(n,m)}]+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}\hfill \\ & ={\left(V_{-}^{(n,m)}\right)}_x-[U,V_{-}^{(n,m)}]-\frac{\partial U}{\partial \lambda }{\lambda }_{t,-}^{\left(m\right)}\hfill \\ & =(a_{2,n+1}+{\overline{a}}_{2,m+1})f_2\left(0\right)-(a_{3,n+1}+{\overline{a}}_{3,m+1})f_3\left(0\right)\hfill \\ & +(a_{5,n+1}+{\overline{a}}_{5,m+1})f_5\left(0\right)-(a_{6,n+1}+{\overline{a}}_{6,m+1})f_6\left(0\right).\hfill \end{array}$$

Therefore, the nonisospectral zero curvature equation

$$\frac{\partial U}{\partial u}{u}_t+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(m\right)}-V_{+,x}^{(n,m)}+[U,V_{+}^{(n,m)}]=0$$

admits an isospectral-nonisospectral Lax integrable hierarchy

$$u_t={\left(\begin{array}{c}q\\ r\\ s_1\\ s_2\end{array}\right)}_t=\left(\begin{array}{c}-a_{2,n+1}-{\overline{a}}_{2,m+1}\\ a_{3,n+1}+{\overline{a}}_{3,m+1}\\ -a_{5,n+1}-{\overline{a}}_{5,m+1}\\ a_{6,n+1}+{\overline{a}}_{6,m+1}\end{array}\right).$$

(44)

When ${\overline{a}}_{2,p},{\overline{a}}_{3,p},{\overline{a}}_{5,p},{\overline{a}}_{6,p}(p=m+1)=0,$ (44) reduces to the resulting isospectral integrable hierarchy

$$u_t={\left(\begin{array}{c}q\\ r\\ s_1\\ s_2\end{array}\right)}_t=\left(\begin{array}{c}-a_{2,n+1}\\ a_{3,n+1}\\ -a_{5,n+1}\\ a_{6,n+1}\end{array}\right).$$

(45)

Specially, when $s_1=s_2$, (45) reduces to the AKNS hierarchy.

In what follows, we deduce the Hamiltonian structure of (45). The U and $V_1$ in (42) and (43) can be written as

$$\begin{array}{cc}& U=\left(\begin{array}{cc}-h\left(1\right)+qe\left(0\right)+rf\left(0\right)& s_{1}e\left(0\right)+s_{2}f\left(0\right)\\ s_{1}e\left(0\right)+s_{2}f\left(0\right)& -h\left(1\right)+qe\left(0\right)+rf\left(0\right)\end{array}\right),\hfill \\ & V_1=\left(\begin{array}{cc}{a}_{1}h\left(0\right)+a_{2}e\left(0\right)+a_{3}f\left(0\right)& a_{4}h\left(0\right)+a_{5}e\left(0\right)+a_{6}f\left(0\right)\\ a_{4}h\left(0\right)+a_{5}e\left(0\right)+a_{6}f\left(0\right)& a_{1}h\left(0\right)+a_{2}e\left(0\right)+a_{3}f\left(0\right)\end{array}\right),\hfill \end{array}$$

where

$$a_i=\sum _{j\ge 0}{a}_{ij}{f}_i\left(j\right),i=1,2,\cdots ,6.$$

It is easy to calculate that

$$\begin{array}{cc}& \langle V_1,\frac{\partial U}{\partial \lambda }\rangle =-4a_1,\langle V_1,\frac{\partial U}{\partial q}\rangle =4a_2,\langle V_1,\frac{\partial U}{\partial r}\rangle =-4a_3,\hfill \\ & \langle V_1,\frac{\partial U}{\partial s_1}\rangle =-4a_5,\langle V_1,\frac{\partial U}{\partial s_2}\rangle =-4a_6.\hfill \end{array}$$

Substituting the above consequences to the trace identity gives

$$\frac{\delta }{\delta u}(-4a_1)={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}4a_2\\ -4a_3\\ 4a_5\\ -4a_6\end{array}\right).$$

(46)

Comparing the coefficients of ${\lambda }^{-n-1}$ both sides of (46) leads to

$$\frac{\delta }{\delta u}(-a_{1,n+1})=(-n+\gamma )\left(\begin{array}{c}{a}_{2,n}\\ -a_{3,n}\\ a_{5,n}\\ -a_{6,n}\end{array}\right).$$

It is easy to check that $\gamma =0$. Thus, we have

$$\left(\begin{array}{c}{a}_{2,n}\\ -a_{3,n}\\ a_{5,n}\\ -a_{6,n}\end{array}\right)=\frac{\delta H_n}{\delta u},H_n=\frac{a_{1,n+1}}{n}.$$

The integrable hierarchy (46) can be written as Hamiltonian form

$$\begin{array}{cc}\hfill u_t=& {\left(\begin{array}{c}q\\ r\\ s_1\\ s_2\end{array}\right)}_t\left(\begin{array}{c}-a_{2,n+1}\\ a_{3,n+1}\\ -a_{5,n+1}\\ a_{6,n+1}\end{array}\right)=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)\left(\begin{array}{c}{a}_{2,n+1}\\ -a_{3,n+1}\\ a_{5,n+1}\\ -a_{6,n+1}\end{array}\right)\hfill \\ \hfill =& \tilde{J}\left(\begin{array}{c}-a_{2,n+1}\\ a_{3,n+1}\\ -a_{5,n+1}\\ a_{6,n+1}\end{array}\right)=\tilde{J}\frac{\delta H_{n+1}}{\delta u}.\hfill \end{array}$$

Next, we consider reductions of (44).

When $n=2,m=2$, we have that

$$\begin{array}{cc}\hfill q_t=& {\alpha }_0(-q_{xx}+q^{2}r+2q{s}_1{s}_2+r{s}_1{s}_2)+{\beta }_0(-s_{1xx}+q^2{s}_2+2qr{s}_1+s_1^2{s}_2)\hfill \\ \hfill & +{\alpha }_1{q}_x+{\beta }_1{s}_{1x}-{\alpha }_{2}q-{\beta }_{2}r-k_{0}xq{\left(qr\right)}_{xx}-{\gamma }_0{s}_{1xx}+k_1{\left(xq\right)}_x+{\gamma }_1{s}_{1x}\hfill \\ \hfill & +k_{0}q{\partial }^{-1}(qr+s_1{s}_2)+k_{0}xq(qr+s_1{s}_2)+{\gamma }_0(q{s}_2+r{s}_1)-k_{2}xq\hfill \\ \hfill & +k_0{s}_1{\partial }^{-1}(q{s}_2+r{s}_1)+{\gamma }_0{s}_1(qr+s_1{s}_2)+k_{0}x{s}_1(q{s}_2+r{s}_1)-{\gamma }_2{s}_1,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill r_t=& {\alpha }_0(r_{xx}-q{r}^2-2r{s}_1{s}_2-q{s}_2^2)+{\beta }_0(s_{2xx}-2qr{s}_2-r^2{s}_1+s_1{s}_2^2)\hfill \\ \hfill & +{\alpha }_1{r}_x+{\beta }_1{s}_{2x}+{\alpha }_{2}r+{\beta }_2{s}_2+k_0{\left(xr\right)}_{xx}+{\gamma }_0{s}_{2xx}+k_1{\left(xr\right)}_x+{\gamma }_1{s}_{2x}\hfill \\ \hfill & -k_{0}r{\partial }^{-1}(qr+s_1{s}_2)-k_{0}xr(qr+s_1{s}_2)-{\gamma }_0(q{s}_2+r{s}_1)+k_{2}xr\hfill \\ \hfill & -k_0{s}_2{\partial }^{-1}(q{s}_2+r{s}_1)-{\gamma }_0{s}_2(qr+s_1{s}_2)-k_{0}x{s}_2(q{s}_2+r{s}_1)+{\gamma }_2{s}_2,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill s_{1t}=& {\alpha }_0(-s_{1xx}+2qr{s}_1+s_1^2{s}_2+q^2{s}_2)+{\beta }_0(-q_{xx}+q^{2}r+2q{s}_1{s}_2+r{s}_1^2)\hfill \\ \hfill & +{\alpha }_1{s}_{1x}+{\beta }_1{q}_x-{\alpha }_2{s}_1-{\beta }_{2}q-k_0{\left(x{s}_2\right)}_{xx}-{\gamma }_0{r}_{xx}-{\gamma }_1{r}_x-k_1{\left(x{s}_2\right)}_x\hfill \\ \hfill & +k_{0}r{\partial }^{-1}(q{s}_2+r{s}_1)+{\gamma }_{0}r(qr+s_1{s}_2)+k_{0}xr{\partial }^{-1}(q{s}_2+r{s}_1)-{\gamma }_{2}r\hfill \\ \hfill & +k_0{s}_2{\partial }^{-1}(qr+s_1{s}_2)+k_{0}x{s}_2(qr+s_1{s}_2)+{\gamma }_0{s}_2(q{s}_2+r{s}_1)-k_{2}x{s}_2,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill s_{2t}=& {\alpha }_0(s_{2xx}-2qr{s}_2-r^2{s}_1-s_1{s}_2^2)+{\beta }_0(r_{xx}-q{r}^2-2r{s}_1{s}_2-q{s}_2^2)\hfill \\ \hfill & +{\alpha }_1{s}_{2x}+{\beta }_1{r}_x+{\beta }_{2}r+{\alpha }_2{s}_2+k_0{\left(x{s}_2\right)}_{xx}+{\gamma }_0{r}_{xx}+{\gamma }_1{r}_x+k_1{\left(x{s}_2\right)}_x\hfill \\ \hfill & -k_{0}r{\partial }^{-1}(q{s}_2+r{s}_1)-{\gamma }_{0}r(qr+s_1{s}_2)-k_{0}xr(q{s}_2+r{s}_1)\hfill \\ \hfill & +{\gamma }_{2}r-k_{0}x{s}_2{\partial }^{-1}(qr+s_1{s}_2)-k_{0}x{s}_2(qr+s_1{s}_2)+k_{2}x{s}_2.\hfill \end{array}$$

(50)

Taking ${\beta }_0=s_1=s_2=0$, (47) and (48) reduce to

$$\left\{\begin{array}{cc}\hfill q_t=& {\alpha }_0(-q_{xx}+q^{2}r)+{\alpha }_1{q}_x-{\alpha }_{2}q-{\beta }_{2}r-k_0{\left(xq\right)}_{xx}\hfill \\ \hfill & +k_1{\left(xq\right)}_x+k_{0}q{\partial }^{-1}\left(qr\right)+k_{0}x{q}^{2}r-k_{2}xq,\hfill \\ \hfill r_t=& {\alpha }_0(r_{xx}-q{r}^2)+{\alpha }_1{r}_x+{\alpha }_{2}r+k_0{\left(sr\right)}_{xx}+k_1{\left(xr\right)}_x\hfill \\ \hfill & -k_{0}r{\partial }^{-1}\left(qr\right)-k_{0}xq{r}^2+k_{2}xr.\hfill \end{array}\right.$$

(51)

If taking ${\alpha }_1={\alpha }_2={\beta }_2=k_i(i=0,1,2)=0,{\alpha }_0=1$, (51) becomes

$$\{\begin{array}{cc}\hfill & q_t=-q_{xx}+q^{2}r,\hfill \\ \hfill & r_t=r_{xx}-q{r}^2,\hfill \end{array}$$

which can be transformed to

$$i{q}_t=q_{x}x-q{\left|q\right|}^2$$

(52)

by using the transformation $r\to q*,t\to -it$. Equation (52) is obviously nonlinear Schrödinger equation.

When ${\alpha }_0={\alpha }_1={\alpha }_2={\beta }_2=0$, (51) turns to

$$\left\{\begin{array}{c}{q}_t=-k_0{\left(xq\right)}_{xx}+k_1{\left(xq\right)}_x+k_{0}q{\partial }^{-1}\left(qr\right)+k_{0}x{q}^{2}r-k_{2}xq,\hfill \\ r_t=k_0{\left(xr\right)}_{xx}+k_1{\left(xr\right)}_x-k_{0}r{\partial }^{-1}\left(qr\right)-k_{0}xq{r}^2+k_{2}xr,\hfill \end{array}\right.$$

(53)

which is a non-local integrable system with variable coefficients.

In particular, when $k_1=k_2=0,k_0=1,$ (53) presents

$$\left\{\begin{array}{c}{q}_t=-{\left(xq\right)}_{xx}+q{\partial }^{-1}\left(qr\right)+x{q}^{2}r,\hfill \\ r_t={\left(xr\right)}_{xx}-r{\partial }^{-1}\left(qr\right)-xq{r}^2.\hfill \end{array}\right.$$

(54)

If taking $r=q\ast ,t\to -it,$ (54) reads that

$$i{q}_t={\left(xq\right)}_{xx}+q{\partial }^{-1}{\left|q\right|}^2+xq{\left|q\right|}^2,$$

which is a non-local nonlinear Schrödinger equation with variable coefficients.

Remark 5.

The first two equations in (39) are the same with (53). When $m=2$, the nonisospectral integrable hierarchy of (44) presents

$$\left\{\begin{array}{c}{q}_t=-{\overline{a}}_{2,3},\hfill \\ r_t={\overline{a}}_{3,3},\hfill \\ s_{1t}=-{\overline{a}}_{5,3},\hfill \\ s_{2t}={\overline{a}}_{6,3}.\hfill \end{array}\right.$$

(55)

Specially, when ${\beta }_0={\beta }_1={\beta }_2={\gamma }_0={\gamma }_1={\gamma }_2=0$, (55) reduces to

$$\begin{array}{cc}\hfill q_t=& -k_0{\left(xq\right)}_{xx}+k_1{\left(xq\right)}_x+k_{0}q{\partial }^{-1}(qr+s_1{s}_2)+k_{0}xq(qr+s_1{s}_2)\hfill \\ & -k_{2}xq+k_0{s}_1{\partial }^{-1}(q{s}_2+r{s}_1)+k_{0}x{s}_1(q{s}_2+r{s}_1),\hfill \\ \hfill r_t=& k_0{\left(xr\right)}_{xx}+k_1{\left(xr\right)}_x-k_{0}r{\partial }^{-1}(qr+s_1{s}_2)-k_{0}xr(qr+s_1{s}_2)\hfill \\ & +k_{2}xr-k_0{s}_2{\partial }^{-1}(q{s}_2+r{s}_1)-k_{0}x{s}_2(q{s}_2+r{s}_1),\hfill \\ \hfill s_{1t}=& -k_0{\left(x{s}_2\right)}_{xx}-k_1{\left(x{s}_2\right)}_x+k_{0}r{\partial }^{-1}(q{s}_2+r{s}_1)+k_{0}xr(q{s}_2+r{s}_1)\hfill \\ & +k_0{s}_2{\partial }^{-1}(qr+s_1{s}_2)+k_{0}x{s}_2(qr+s_1{s}_2)-k_{2}x{s}_2,\hfill \\ \hfill s_{2t}=& k_0{\left(x{s}_2\right)}_{xx}+k_1{\left(x{s}_2\right)}_x-k_{0}r{\partial }^{-1}(q{s}_2+r{s}_1)-k_{0}xr(q{s}_2+r{s}_1)\hfill \\ & -k_0{s}_2{\partial }^{-1}(qr+s_1{s}_2)-k_{0}x{s}_2(qr+s_1{s}_2)+k_{2}x{s}_2,\hfill \end{array}$$

which are different from (39). Hence, the isospectral–nonisospectral integrable hierarchy (44) is different from the hierarchy (19), they are all integrable couplings of the AKNS hierarchy (14).

4. Conclusions

In the paper, we obtained an isospectral–nonisospectral AKNS hierarchy by using a third-order matrix Lie algebras. In order to discover its expanding integrable models, i.e., integrable couplings, we turned the matrix Lie algebra into a $2\times 2$ matrix Lie algebra so that it was enlarged into two higher order semi-simple Lie algebras for which two kinds of nonisospectral expanding integrable models were generated. Specially, we obtained nonlinear integrable couplings which has been required to overcome. An interesting result reads that we united the well-known KdV equation and the NLS as an integrable model. The aim for publishing the paper lies in a few aspects. One purpose reads how to transfer higher-order matrix Lie algebras into lower-order matrix Lie algebras. The second one presents how to generate nonlinear integrable couplings. The third one manifests how to generate multiply nonisospectral integrable couplings. The approach presented in the paper can be used to discuss other integrable systems.