A Few Kinds of Loop Algebras and Some Applications

Abstract

In this paper, we search for some approaches for generating (1+1)-dimensional, (2+1)-dimensional and (3+1)-dimensional integrable equations by making use of various Lie algebras and the corresponding loop algebras under the frame of the Tu scheme. The well-known modified KdV equation, the heat conduction equation, the nonlinear Schrödinger equation, the (2+1)-dimensional cylindrical dissipative Zaboloskaya–Khokhlov equation and the (3+1)-dimensional heavenly equation are obtained, respectively. In addition, some new isospectral integrable hierarchies and their nonisospectral integrable hierarchies are singled out. All the Lie algebras and their loop algebras presented in the paper can be extensively applied to investigate other integrable hierarchies of evolution equations.

1. Introduction

Searching for new integrable systems has been an important topic in soliton theory and integrable theory. Since Magri proposed the method for generating integrable equations by the use of Lax pairs [1], many integrable hierarchies of evolution equations have been obtained, for example, the works [2,3,4,5,6,7,8,9,10]. As we all know that many different methods for producing integrable equations have been presented, such as the ways of Lie algebras [11], the symmetry approaches [12,13,14], the Dbar method [15,16,17], and so on, in this paper, we only focus on the approach proposed by Tu Guizhang [18] based on the Lax pairs and Lie algebras to investigate integrable systems and their corresponding some properties, which is called the Tu scheme [19]. In the Lie algebra $A_1$, the following commutation relations are not found:

$$[h,e]=f,[e,f]=h,[f,h]=e.$$

(1)

where $h,e,f\in A_1,[a,b]=ab-ba,a,b\in A_1.$

However, in the Lie algebra $A_2$, we can find the commuting relation (1). For example, in subalgebra $A_{21}$ of the Lie algebra $A_2$, assume that

$$h=\left(\begin{array}{cc}\overline{O}& -{\alpha }^T\\ \alpha & \overline{O}\end{array}\right),e=\left(\begin{array}{cc}\overline{O}& {\beta }^T\\ -\beta & \overline{O}\end{array}\right),f=\left(\begin{array}{cc}-\alpha & \overline{O}\\ {\beta }^T& \overline{O}\end{array}\right),$$

where $\overline{O}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),\alpha =(0,1),\beta =(1,0),$ we just have the relations (1).

Now, we make use of the Lie subalgebra $A_{21}$ to construct a high-order matrix Lie algebra which belongs to the Lie algebra $A_3$:

$$H=:span\{H_1,H_2,\dots ,H_6\},$$

where

$$H_1=\left(\begin{array}{cc}h& 0\\ 0& h\end{array}\right),H_2=\left(\begin{array}{cc}0& e\\ e& 0\end{array}\right),H_3=\left(\begin{array}{cc}0& f\\ f& 0\end{array}\right),$$

$$H_4=\left(\begin{array}{cc}e& 0\\ 0& e\end{array}\right),H_5=\left(\begin{array}{cc}f& 0\\ 0& f\end{array}\right),H_6=\left(\begin{array}{cc}0& h\\ h& 0\end{array}\right).$$

It is easy to verify that

$$[H_1,H_2]=H_3,[H_2,H_3]=H_1,[H_3,H_1]=H_2,[H_1,H_4]=H_5,$$

$$[H_1,H_5]=-H_4,[H_1,H_6]=[H_2,H_4]=0,[H_2,H_5]=H_6,$$

$$[H_2,H_6]=-H_5,[H_3,H_4]=-H_6,[H_3,H_5]=0,$$

$$[H_3,H_6]=H_4,[H_4,H_5]=H_1,[H_4,H_6]=-H_3,[H_5,H_6]=H_2.$$

In addition, we find that the Lie algebra $A_{21}=span\{h,e,f\}$ is isomorphic to the Lie algebra $\{H_1,H_2,H_3\}$, which means that the two Lie algebras have the same commutator.

A simplest loop algebra $\tilde{H}$ of the Lie algebra H can be given by

$$\widetilde{H_I}=:span\{H_1\left(n\right),H_2\left(n\right),\cdots ,H_6\left(n\right)\},$$

where $H_i\left(n\right)=H_i\otimes {\lambda }^n=:H_i{\lambda }^n,i=1,2,\cdots ,6;n\in N.$

Thus, the corresponding loop algebra ${\tilde{A}}_{21}$ of the Lie algebra $A_{21}$ presents that

$$\widetilde{A_{21}}=:span\{H_1\left(n\right),H_2\left(n\right),H_3\left(n\right)\}\cong span\{h\left(n\right),e\left(n\right),f\left(n\right)\}.$$

According to the isomorphic idea, we introduce a map from the linear space $A_{21}$ to a vector space $\overrightarrow{e}=span\{\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3}\}$ as follows:

$$h\to \overrightarrow{e_1},e\to \overrightarrow{e_2},f\to \overrightarrow{e_3}.$$

In terms of the commutator in $A_{21}$, we naturally define the following vector product:

$$\overrightarrow{e_1}\times \overrightarrow{e_2}=\overrightarrow{e_3},\overrightarrow{e_2}\times \overrightarrow{e_3}=\overrightarrow{e_1},\overrightarrow{e_3}\times \overrightarrow{e_1}=\overrightarrow{e_2}.$$

(2)

It can be verified that

$$\overrightarrow{e_i}\times \overrightarrow{e_j}=-\overrightarrow{e_j}\times \overrightarrow{e_i},$$

$$\overrightarrow{e_i}\times (\overrightarrow{e_j}\times \overrightarrow{e_k})+\overrightarrow{e_j}\times (\overrightarrow{e_k}\times \overrightarrow{e_i})+\overrightarrow{e_k}\times (\overrightarrow{e_i}\times \overrightarrow{e_j})=\overrightarrow{0},i,j,k=1,2,\cdots ,6.$$

That is, the Jacobi identity holds. Therefore the vector space $\overrightarrow{e}$ along with (2) constitutes a Lie algebra. It follows from the Lie algebra H that an enlarged Lie algebra $\overrightarrow{E}$ is given by

$$\overrightarrow{E}=:span\{\overrightarrow{e_1},\overrightarrow{e_2},\cdots ,\overrightarrow{e_6}\}$$

which has the following operations:

$$\overrightarrow{e_1}\times \overrightarrow{e_2}=\overrightarrow{e_3},\overrightarrow{e_2}\times \overrightarrow{e_3}=\overrightarrow{e_1},\overrightarrow{e_3}\times \overrightarrow{e_1}=\overrightarrow{e_2},\overrightarrow{e_1}\times \overrightarrow{e_4}=\overrightarrow{e_5},\overrightarrow{e_1}\times \overrightarrow{e_5}=-\overrightarrow{e_4},$$

$$\overrightarrow{e_1}\times \overrightarrow{e_6}=\overrightarrow{0},\overrightarrow{e_2}\times \overrightarrow{e_4}=\overrightarrow{0},\overrightarrow{e_2}\times \overrightarrow{e_5}=\overrightarrow{e_6},\overrightarrow{e_2}\times \overrightarrow{e_6}=-\overrightarrow{e_5},\overrightarrow{e_3}\times \overrightarrow{e_4}=-\overrightarrow{e_6},$$

$$\overrightarrow{e_3}\times \overrightarrow{e_5}=\overrightarrow{0},\overrightarrow{e_3}\times \overrightarrow{e_6}=\overrightarrow{e_4},\overrightarrow{e_4}\times \overrightarrow{e_5}=\overrightarrow{e_1},\overrightarrow{e_4}\times \overrightarrow{e_5}=-\overrightarrow{e_3},\overrightarrow{e_5}\times \overrightarrow{e_6}=\overrightarrow{e_2}.$$

Denote $\overrightarrow{E_1}$ and $\overrightarrow{E_1}$ by

$$\overrightarrow{E_1}=:span\{\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3}\},\overrightarrow{E_2}=:span\{\overrightarrow{e_4},\overrightarrow{e_5},\overrightarrow{e_6}\},$$

then it is easy to find that

$$\overrightarrow{E}=\overrightarrow{E_1}\oplus \overrightarrow{E_2},\overrightarrow{E_1}\cong A_{21},\overrightarrow{E_1}\times \overrightarrow{E_2}\subset \overrightarrow{E_2},\overrightarrow{E_2}\times \overrightarrow{E_2}\subset \overrightarrow{E_1},$$

(3)

which can be used to investigate the integrable couplings of evolution equations [20,21,22,23,24,25].

According to the isomorphic idea again, due to (2), we have

$$\overrightarrow{e_1}\times \overrightarrow{e_2}=-\overrightarrow{e_2}\times \overrightarrow{e_1},\overrightarrow{e_2}\times \overrightarrow{e_3}=-\overrightarrow{e_3}\times \overrightarrow{e_2},\overrightarrow{e_3}\times \overrightarrow{e_1}=-\overrightarrow{e_1}\times \overrightarrow{e_3}.$$

A map is constructed:

$$\overrightarrow{e}\to \overrightarrow{f}=:\{{\partial }_x,{\partial }_y,{\partial }_z\},\overrightarrow{e_1}\to {\partial }_x,\overrightarrow{e_2}\to {\partial }_y,\overrightarrow{e_3}\to {\partial }_z$$

and an operator in the set $\overrightarrow{f}$ is defined by

$${\partial }_x{\partial }_y={\partial }_y{\partial }_x,{\partial }_x{\partial }_z={\partial }_z{\partial }_x,{\partial }_y{\partial }_z={\partial }_z{\partial }_y.$$

Then, $\overrightarrow{f}$ becomes an associative algebra. Now we can enlarge the algebra $\overrightarrow{f}$ into the following associative algebra:

$$\overrightarrow{F}=:\{{\partial }_x,{\partial }_y,{\partial }_z,{\partial }_t,{\partial }_{\lambda }\},$$

where $\lambda$ is a parameter, and ${\partial }_t$ and ${\partial }_{\lambda }$ satisfy the associative laws ${\partial }_t\overrightarrow{f}=\overrightarrow{f}{\partial }_t,{\partial }_{\lambda }\overrightarrow{f}=\overrightarrow{f}{\partial }_{\lambda }.$

In what follows, we shall make use of the Lie algebras as shown above to investigate new integrable hierarchies with the help of the Tu scheme. In Section 2, applications of the two loop algebras of the Lie algebra $A_{21}$ obtain two types of integrable hierarchies. One is an isospectral hierarchy, that is, the spectral parameter ${\lambda }_t=0$. The other one is a nonisospectral case, i.e., ${\lambda }_t\ne 0.$ By applying the two loop algebras ${\tilde{H}}_I$ and ${\tilde{H}}_{II}$, we produce two expanding integrable models, which reduce to a modified Korteweg–De Vries (KdV) equation and the Ablowitz–Kaup–Newell–Segur (AKNS) system, respectively. In Section 3, we present a kind of vector Lie algebra $\overrightarrow{E}$ and show some of its new applications. In Section 4, we utilize an associative algebra $\overrightarrow{F}$ to give a method for producing (2+1)-dimensional integrable systems. Finally, a conclusion is presented.

2. Applications of the Lie Algebra H

In the previous section, we showed a loop algebra $\widetilde{H_I}$ of the Lie algebra H. Now, another loop algebra of the Lie algebra H is given by

$$\widetilde{H_{\mathsf{\Pi }}}=span\{h(0,n),h(1,n),e(0,n),e(1,n),f(0,n),f(1,n)\},$$

where

$$h(0,n)=h{\lambda }^{2n},h(1,n)=h{\lambda }^{2n+1},e(0,n)=e{\lambda }^{2n},$$

$$e(1,n)=e{\lambda }^{2n+1},f(0,n)=f{\lambda }^{2n},f(1,n)=f{\lambda }^{2n+1},$$

where the commutative relations are as follows:

$$\left[h\right(0,m),e(0,n\left)\right]=f(0,m+n),\left[h\right(0,m),f(0,n\left)\right]=-e(0,m+n),$$

$$\left[h\right(0,m),e(1,n\left)\right]=f(1,m+n),\left[h\right(0,m),f(1,n\left)\right]=-e(1,m+n),$$

$$\left[h\right(1,m),e(0,n\left)\right]=f(1,m+n),\left[h\right(1,m),e(1,n\left)\right]=f(0,m+n+1),$$

$$\left[h\right(1,m),f(0,n\left)\right]=-e(1,m+n),\left[h\right(1,m),f(1,n\left)\right]=-e(0,m+n+1),$$

$$\left[e\right(0,m),f(0,n\left)\right]=h(0,m+n),\left[e\right(0,m),f(1,n\left)\right]=h(1,m+n),$$

$$\left[e\right(1,m),f(0,n\left)\right]=h(1,m+n),\left[e\right(1,m),f(1,n\left)\right]=h(0,m+n+1).$$

It is easy to find that the loop algebras $\widetilde{H_I}$ and $\widetilde{H_{\mathsf{\Pi }}}$ are different. Note

$$K=:span\{H_1,H_2,H_3\},L=:span\{H_4,H_5,H_6\},$$

then we have

$$H=K\oplus L,[K,K]\subset K,[k,L]\subset L,[L,L]\subset K.$$

(4)

However, for the loop algebra $\widetilde{H_{\mathsf{\Pi }}}$, relation (4) does not hold.

2.1. Applications of the Loop Algebra $\widetilde{A_{21}}$

We shall adopt the loop algebra $\widetilde{A_{21}}$ to generate isospectral integrable hierarchies and Hamiltonian structure as well as nonisospectral integrable hierarchies by the use of the Tu scheme. We first utilize the subloop algebra $\widetilde{A_{21}}\subset \widetilde{H_I}$ to generate integrable hierarchies.

2.1.1. Isospectral Integrable Hierarchy

For set

$${\phi }_x=U\phi ,{\phi }_t=V\phi ,$$

(5)

its compatibility condition presents

$$U_t-V_x+[U,V]=0,$$

(6)

which is a well-known zero curvature equation.

Let

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}U=h\left(1\right)+qe\left(0\right)+rf(-1)+sh(-1),\hfill \\ V={\displaystyle \sum _{m\ge 0}}(a_{m}h(-m)+b_{m}e(-m)+c_{m}f(-m)).\hfill \end{array}\right.\end{array}\end{array}$$

(7)

According to the steps of the Tu scheme, the stationary zero curvature equation

$$V_x=[U,V]$$

(8)

admits the following recursive relations:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{c}_{m+1}=-b_{m,x}+r{a}_{m-1}-s{c}_{m-1},\hfill \\ b_{m+1}=c_{m,x}+q{a}_m-s{b}_{m-1},m\ge 1,\hfill \\ a_{m,x}=q{c}_m-r{b}_{m-1}.\hfill \end{array}\right.\end{array}\end{array}$$

(9)

Taking $b_0=c_0=a_1=c_1=0,a_0=\alpha \left(t\right),$ then from (9), one infers that

$$b_2=0,c_2=-{\alpha }_1{q}_x+{\alpha }_{1}r,b_1={\alpha }_{1}q,a_2=-{\displaystyle \frac{\alpha }{2}}{q}^2+{\alpha }_2\left(t\right),$$

$$c_3=0,b_3=-{\alpha }_1(q_{xx}+{\displaystyle \frac{1}{2}}{q}^3)+{\alpha }_{2}q-{\alpha }_1(qs-r_x),a_3={\alpha }_3\left(t\right),$$

$$c_4={\alpha }_1(q_{xxx}+{\displaystyle \frac{3}{2}}{q}^2{q}_x)-{\alpha }_2{q}_x+{\alpha }_1{\left(qs\right)}_x-{\displaystyle \frac{{\alpha }_1}{2}}r{q}^2+{\alpha }_{2}r+{\alpha }_1{q}_{x}s-{\alpha }_{1}rs-{\alpha }_1{r}_{xx},$$

$$b_4={\alpha }_{3}q,a_4={\alpha }_1(q{q}_{xx}-{\displaystyle \frac{1}{2}}{q}_x^2)+{\displaystyle \frac{3{\alpha }_1}{8}}{q}^4-{\displaystyle \frac{{\alpha }_2}{2}}{q}^2+{\alpha }_1{q}^{2}s+{\alpha }_1(r{q}_x-r_{x}q)+{\alpha }_4\left(t\right),\cdots$$

where ${\alpha }_1\left(t\right),{\alpha }_2\left(t\right),{\alpha }_3\left(t\right),{\alpha }_4\left(t\right)$ are all arbitrary functions in t.

Denoting that

$$\begin{array}{cc}{V}_{+}^{\left(n\right)}\hfill & ={\displaystyle \sum _{m=0}^n}\left(a_{m}h(-m)+b_{m}e(-m)+c_{m}f(-m)\right){\lambda }^n\hfill \\ \hfill & ={\lambda }^{n}V-V_{-}^{\left(n\right)}={\lambda }^{n}V-{\displaystyle \sum _{m=n+1}^{\infty }}\left(a_{m}h(n-m)+b_{m}e(n-m)+c_{m}f(n-m)\right).\hfill \end{array}$$

Equation (8) can decouple to the following equation:

$$-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]={\left(V_{-}^{\left(n\right)}\right)}_x-[U,V_{-}^{\left(n\right)}].$$

(10)

A direct calculation reads that

$$\begin{array}{cc}-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]\hfill & =a_{n+1,x}h(-1)+b_{n+1,x}e(-1)+c_{n+1,x}f(-1)-b_{n+1}f\left(0\right)+c_{n+1}e\left(0\right)\hfill \\ \hfill & -q{a}_{n+1}f(-1)-q{c}_{n+1}h(-1)-b_{n+2}f(-1)+c_{n+2}e(-1).\hfill \end{array}$$

(11)

When $n=2k+1,$ (11) reduces to

$$\begin{array}{cc}-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]\hfill & =(a_{2k+2,x}-q{c}_{2k+2}h(-1)+(-b_{2k+3}\hfill \\ \hfill & +c_{2k+2,x}+q{a}_{2k+2})f(-1)+c_{2k+2}e\left(0\right).\hfill \end{array}$$

Thus, the zero-curvature equation

$$U_t-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]=0$$

gives rise to

$${\left(\begin{array}{c}q\\ r\\ s\end{array}\right)}_{t_{2k+2}}=\left(\begin{array}{c}-c_{2k+2}\\ -c_{2k+2,x}-q{a}_{2k+2}+b_{2k+3}\\ q{c}_{2k+2}-a_{2k+2,x}\end{array}\right)=\left(\begin{array}{c}-c_{2k+2}\\ -s{b}_{2k+1}\\ r{b}_{2k+1}\end{array}\right).$$

(12)

When $k=0,$ the integrable hierarchy (12) reduces to

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{q}_{t_2}={\alpha }_1(q_x-r),\hfill \\ r_{t_2}=-{\alpha }_{1}qs,\hfill \\ s_{t_2}=-{\alpha }_{1}qr,\hfill \end{array}\right.\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_{tt}={\alpha }_1{q}_{xt}+{\alpha }_1^{2}qs,\hfill \\ s_t=-{\alpha }_{1}q{q}_x+q{q}_t.\hfill \end{array}\right.\end{array}\end{array}$$

(13)

When $k=1,r=s=0,{\alpha }_1\left(t\right)=-2,$ from (12), we have the following modified KdV equation:

$$q_{t_4}=2q_{xxx}+3q^2{q}_x.$$

Let $r=0,s=1,$ and similarly, we obtain a new integrable equation

$$q_t=q^2{q}_x+2q_x.$$

(14)

Remark 1.

The integrable hierarchy (12), as we know, is new, which can single out some new integrable equations or integrable systems.

In what follows, we search for the Hamiltonian structure of the hierarchy (12). It is easy to compute that

$$U=\left(\begin{array}{ccc}0& -{\displaystyle \frac{r}{\lambda }}& q\\ {\displaystyle \frac{r}{\lambda }}& 0& -\lambda -{\displaystyle \frac{s}{\lambda }}\\ -q& \lambda +{\displaystyle \frac{s}{\lambda }}& 0\end{array}\right),V=\left(\begin{array}{ccc}0& -c& b\\ c& 0& -a\\ -b& a& 0\end{array}\right),$$

$$<V,{\displaystyle \frac{\partial U}{\partial q}}>=:trace\left(\begin{array}{ccc}0& -c& b\\ c& 0& -a\\ -b& a& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right)=-2b,$$

$$<V,{\displaystyle \frac{\partial U}{\partial r}}>=-{\displaystyle \frac{2c}{\lambda }},<V,{\displaystyle \frac{\partial U}{\partial s}}>=-{\displaystyle \frac{2a}{\lambda }},<V,{\displaystyle \frac{\partial U}{\partial \lambda }}>={\displaystyle \frac{2rc}{{\lambda }^2}}+{\displaystyle \frac{2as}{{\lambda }^2}}-2a.$$

Substituting the above results into the well-known trace identity [18] yields

$${\displaystyle \frac{\delta }{\delta u}}\left({\displaystyle \frac{2rc+2as}{{\lambda }^2}}-2a\right)={\lambda }^{-\gamma }{\displaystyle \frac{\partial }{\partial \lambda }}{\lambda }^{\gamma }\left(\begin{array}{c}-2b\\ -{\displaystyle \frac{2c}{\lambda }}\\ -{\displaystyle \frac{2a}{\lambda }}\end{array}\right),$$

(15)

where

$${\displaystyle \frac{\delta }{\delta u}}={({\displaystyle \frac{\delta }{\delta q}},{\displaystyle \frac{\delta }{\delta r}},{\displaystyle \frac{\delta }{\delta s}})}^T.$$

Comparing the coefficients of ${\lambda }^{-m-1},m=2k+3,\gamma =0,$ we have

$$-(2k+3)\left(\begin{array}{c}{b}_{2k+3}\\ c_{2k+2}\\ a_{2k+2}\end{array}\right)={\displaystyle \frac{\delta }{\delta u}}(a_{2k+4}-r{c}_{2k}-s{a}_{2k}),$$

that is,

$$\left(\begin{array}{c}{b}_{2k+3}\\ c_{2k+2}\\ a_{2k+2}\end{array}\right)={\displaystyle \frac{\delta }{\delta u}}\left({\displaystyle \frac{r{c}_{2k}+s{a}_{2k}-a_{2k+4}}{2k+3}}\right)=:{\displaystyle \frac{\delta H_{2k}}{\delta u}}.$$

Therefore, the integrable hierarchy (12) can be expressed by the Hamiltonian form:

$$u_t={\left(\begin{array}{c}q\\ r\\ s\end{array}\right)}_t=J\left(\begin{array}{c}{b}_{2k+3}\\ c_{2k+2}\\ a_{2k+2}\end{array}\right)=J{\displaystyle \frac{\delta H_{2k}}{\delta u}},$$

(16)

where the Hamiltonian operator is given by

$$J=\left(\begin{array}{ccc}0& -1& 0\\ 1& -\partial & -q\\ 0& q& -\partial \end{array}\right).$$

2.1.2. Nonisospectral Integrable Hierarchy

We shall use the subloop algebra $\widetilde{A_{21}}$ to investigate the nonisospectral integrable hierarchy of evolution equations.

If the Lax pair (5) is nonisospectral, then the spectral parameter $\lambda$ depends on time t, that is, ${\lambda }_t={\displaystyle \frac{\partial \lambda }{\partial t}}\ne 0.$ It follows that the compatibility condition of (5) reads that

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

(17)

Suppose that ${\lambda }_t={\displaystyle \sum _{m\ge 0}}{k}_m\left(t\right){\lambda }^{-m}$; it is easy to have

$${\displaystyle \frac{\partial U}{\partial \lambda }}{\lambda }_t={\displaystyle \sum _{m\ge 0}}[k_m\left(t\right)h(-m)-k_m\left(t\right)sh(-m-2)-k_m\left(t\right)rf(-m-2)].$$

Thus, the equation

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

leads to the following:

$$\left\{\begin{array}{c}{b}_{m+2}=c_{m+1,x}-r{k}_{m-1}+q{a}_{m+1}-s{b}_m,\\ c_{m+2}=-b_{m+1,x}+r{a}_m-s{c}_m,\\ a_{m+1,x}=q{c}_{m+1}-r{b}_m+k_{m+1}s-k_{m+1},m\ge 0.\end{array}\right.$$

(18)

Taking $b_0=c_0=a_1=c_1=0,a_0=\alpha \left(t\right),$ then (18) can give the following results:

$$\begin{array}{cc}{b}_2\hfill & =0,c_2=-\alpha q_x+\alpha r,b_1=\alpha q,a_2=-{\displaystyle \frac{\alpha }{2}}{q}^2+k_1{\partial }_x^{-1}s-k_2\left(t\right)x,\hfill \\ b_3\hfill & =-\alpha q_{xx}+\alpha r_x-k_0\left(t\right)r-{\displaystyle \frac{\alpha }{2}}{q}^3+k_1\left(t\right)q{\partial }_x^{-1}s-k_2\left(t\right)xq-\alpha sq,\hfill \\ c_3\hfill & =0,a_3=k_1\left(t\right){\partial }_x^{-1}s-k_3\left(t\right)x,b_4=-(r-q{\partial }_x^{-1}s)k_1\left(t\right)-k_3\left(t\right)xq,\hfill \\ c_4\hfill & =\alpha q_{xxx}-\alpha r_{xx}+k_0{r}_x+{\displaystyle \frac{3\alpha }{2}}{q}^2{q}_x+k_1{\left(q{\partial }_x^{-1}s\right)}_x+k_2\left(t\right){\left(xq\right)}_x+\alpha {\left(sq\right)}_x\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{2}}r{q}^2+k_{1}r{\partial }_x^{-1}s-k_{2}xr+\alpha s{q}_x-\alpha sq,\hfill \\ a_4\hfill & =\alpha q{q}_{xx}-{\displaystyle \frac{\alpha }{2}}{q}_x^2+\alpha r{q}_x-\alpha r_{x}q+{\displaystyle \frac{3\alpha }{8}}{q}^4+\alpha s{q}^2-{\displaystyle \frac{\alpha }{2}}{r}^2+2k_1{\partial }_x^{-1}\left(qr{\partial }_x^{-1}s\right)\hfill \\ \hfill & +k_2{\partial }_x^{-1}\left(q{\left(xq\right)}_x\right)+k_1{\partial }_x^{-1}q{\left(q{\partial }_x^{-1}s\right)}_x+k_0{\partial }_x^{-1}(q{r}_x+r^2),\cdots \hfill \end{array}$$

Noting

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

then one has that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]-{\displaystyle \frac{\partial U}{\partial \lambda }}{\lambda }_{t,+}^{\left(n\right)}\hfill \\ \hfill =& {\left(V_{-}^{\left(n\right)}\right)}_x-[U,V_{-}^{\left(n\right)}]+{\displaystyle \frac{\partial U}{\partial \lambda }}{\lambda }_{t,-}^{\left(n\right)}\hfill \\ \hfill =& (a_{n+1,x}-q{c}_{n+1}+k_{n+1})h(-1)+(b_{n+1,x}+c_{n+2})e(-1)\hfill \\ \hfill +& (c_{n+1,x}-b_{n+2}+q{a}_{n+1})f(-1)-b_{n+1}f\left(0\right)+c_{n+1}e\left(0\right).\hfill \end{array}\end{array}$$

(19)

Taking $n=2k+1,V^{\left(n\right)}=V_{+}^{\left(n\right)},$ then the zero-curvature equation

$$U_t-V_{+}^{\left(n\right)}+[U,V^{\left(n\right)}]-{\displaystyle \frac{\partial U}{\partial \lambda }}{\lambda }_{t,+}^{\left(n\right)}=0$$

admits the following nonisospectral integrable hierarchy:

$${\left(\begin{array}{c}q\\ r\\ s\end{array}\right)}_{t_{2k+1}}=\left(\begin{array}{c}-c_{2k+2}\\ b_{2k+3}-c_{2k+2,x}-q{a}_{2k+2}\\ -a_{2k+2,x}+q{c}_{2k+2}-k_{2k+2}\end{array}\right)=\left(\begin{array}{c}{b}_{2k+1,x}-r{a}_{2k}+s{c}_{2k}\\ -s{b}_{2k+1}-k_{2k}r\\ r{b}_{2k+1}-k_{2k}s\end{array}\right).$$

(20)

When $k=1,$ from (20), we obtain that

$$\begin{array}{cc}{q}_t\hfill & =-\alpha q_{xxx}+\alpha r_{xx}-k_0{r}_x-{\displaystyle \frac{3\alpha }{2}}{q}^2{q}_x-k_1{\left(q{\partial }_x^{-1}s\right)}_x-k_2{\left(xq\right)}_x\hfill \\ \hfill & -\alpha {\left(sq\right)}_x+{\displaystyle \frac{\alpha }{2}}r{q}^2-k_{1}r{\partial }_x^{-1}s+k_{2}xr-\alpha sq+\alpha sr,\hfill \\ r_t\hfill & =\alpha s{q}_{xx}-\alpha s{r}_x+k_{0}rs+{\displaystyle \frac{\alpha }{2}}{q}^{3}r+k_{1}qs{\partial }_x^{-1}s\hfill \\ \hfill & +k_{2}xqs+\alpha s^{2}q-k_{2}r,\hfill \\ s_t\hfill & =-\alpha r{q}_{xx}+\alpha r{r}_x-k_0{r}^2-{\displaystyle \frac{\alpha }{2}}{q}^{3}r-k_{1}qr{\partial }_x^{-1}s\hfill \\ \hfill & -k_{2}xqr+\alpha qrs-k_{2}s.\hfill \end{array}$$

In particular, let $r=s=0,$ and the above reduced equations becomes such that

$$q_t=-\alpha q_{xxx}-{\displaystyle \frac{3\alpha }{2}}{q}^2{q}_x-k_2{\left(xq\right)}_x,$$

(21)

which is a nonisospectral modified KdV equation.

2.2. Applications of the Loop Algebra $\widetilde{H_I}$

Since the loop algebra $\widetilde{H_I}$ and its subalgebras K and L satisfy the relation (4), we can generate multi-component integrable hierarchies by utilizing the Tu scheme. Such integrable hierarchies have the same feature which presents that

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

(22)

If the first equation is integrable, then so is the second one. Such coupled integrable systems are called integrable coupling. In what follows, we want to investigate a kind of integrable coupling of the hierarchy (12) based on the frame of the Tu scheme.

Set

$${\psi }_x=\mathcal{U}\psi ,\mathcal{U}=H_1\left(1\right)+q{H}_2\left(0\right)+r{H}_3(-1)+s{H}_1(-1)+u{H}_4(-1)+v{H}_5\left(0\right),$$

(23)

$${\psi }_t=\mathcal{V}\psi ,$$

$$\mathcal{V}={\displaystyle \sum _{m\ge 0}}[a_m{H}_1(-m)+b_m{H}_2(-m)+c_m{H}_3(-m)+d_m{H}_4(-m)+e_m{H}_5(-m)+f_m{H}_6(-m)].$$

(24)

The partial differential equation

$${\mathcal{V}}_x=[\mathcal{U},\mathcal{V}].$$

(25)

shows us the relations between the potential functions $q,r,s,u,v$ and the functions $a_m,b_m,c_m,d_m,e_m$ and $f_m$ as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{m+1,x}=q{c}_{m+1}-r{b}_m+u{e}_m-v{d}_{m+1},\hfill \\ b_{m+1,x}=-c_{m+2}+r{a}_m-s{c}_m+v{f}_{m+1}-w{e}_m,\hfill \\ c_{m+1,x}=b_{m+2}-q{a}_{m+1}+s{b}_m-u{f}_m+w{d}_m,\hfill \\ d_{m+1,x}=-e_{m+2}+r{f}_m-s{e}_m+v{a}_{m+1}-w{c}_m,\hfill \\ e_{m+1,x}=d_{m+2}-q{f}_{m+1}+s{d}_m-u{a}_m+w{b}_m,\hfill \\ f_{m+1,x}=q{e}_{m+2}-r{d}_m+u{c}_m-v{b}_{m+1}.\hfill \end{array}\right.\end{array}\end{array}$$

(26)

Let $b_0=c_0=a_1=c_1=0,a_0=\alpha \left(t\right),$ then from (26), one has that

$$\begin{array}{cc}{d}_0\hfill & =d_1=f_0=e_0=0,b_2=0,c_2=\alpha r-\alpha q_x+\beta v,f_1=\beta \left(t\right),e_2=0,\hfill \\ d_2\hfill & =\alpha v_x+\alpha u+\beta q,a_2=-{\displaystyle \frac{\alpha }{2}}(q^2+v^2)+{\alpha }_2\left(t\right),f_2=0,c_3=-\alpha vw,a_3=0,\hfill \\ d_3\hfill & =-\alpha qw,b_3=\alpha r_x-\alpha q_{xx}+\beta v_x-{\displaystyle \frac{\alpha }{2}}q(q^2+v^2)+{\alpha }_{2}q-\alpha qs+\beta u,\hfill \\ e_3\hfill & =-\alpha v_{xx}-\alpha u_x-\beta q_x+\beta r-\alpha sv-{\displaystyle \frac{\alpha }{2}}v(q^2+v^2)+{\alpha }_{2}v,\hfill \\ f_3\hfill & =\alpha (v{q}_x-v_{x}q)-{\displaystyle \frac{\beta }{2}}{q}^2-{\displaystyle \frac{\beta }{2}}{v}^2-\alpha rv-\alpha qu,\cdots \hfill \end{array}$$

Equation (25) can be decomposed to the following:

$$-{\left({\mathcal{V}}_{+}^{\left(n\right)}\right)}_x+[\mathcal{U},{\mathcal{V}}_{+}^{\left(n\right)}]={\left({\mathcal{V}}_{-}^{\left(n\right)}\right)}_x-[\mathcal{U},{\mathcal{V}}_{-}^{\left(n\right)}].$$

A direct calculation implies that

$$\begin{array}{cc}-{\left({\mathcal{V}}_{+}^{\left(n\right)}\right)}_x+[\mathcal{U},{\mathcal{V}}_{+}^{\left(n\right)}]\hfill & =-(r{b}_n+u{e}_n)H_1(-1)+(r{a}_n-s{c}_n-w{e}_n)H_2(-1)\hfill \\ \hfill & +c_{n+1}{H}_2\left(0\right)+(s{b}_n-u{f}_n+w{d}_n)H_3(-1)-b_{n+1}{H}_3\left(0\right)\hfill \\ \hfill & -d_{n+1}{H}_5\left(0\right)+(r{f}_n-s{e}_n-w{c}_n)H_4(-1)+e_{n+1}{H}_4\left(0\right)\hfill \\ \hfill & +(s{d}_n-u{a}_n+w{b}_n)H_5(-1)+(u{c}_n-r{d}_n)H_6(-1).\hfill \end{array}$$

When $n=2k+1,w=0,$ the above expression can reduce to

$$\begin{array}{cc}\hfill & -{\left({\mathcal{V}}_{+}^{\left(n\right)}\right)}_x+[\mathcal{U},{\mathcal{V}}_{+}^{\left(n\right)}]\hfill \\ =\hfill & c_{2k+2}{H}_2\left(0\right)+(s{b}_{2k+1}-u{f}_{2k+1})H_3(-1)\hfill \\ +\hfill & (u{e}_{2k+1}-r{b}_{2k+1})H_1(-1)\hfill \\ +\hfill & (r{f}_{2k+1}-s{e}_{2k+1})H_4(-1)-d_{2k+2}{H}_5\left(0\right).\hfill \end{array}$$

Set ${\mathcal{V}}^{\left(n\right)}={\mathcal{V}}_{+}^{\left(n\right)},$ then the zero-curvature equation

$${\mathcal{U}}_t-{\mathcal{V}}_{+}^{\left(n\right)}+[\mathcal{U},{\mathcal{V}}^{\left(n\right)}]=0$$

(27)

admits the following integrable hierarchy:

$${\left(\begin{array}{c}q\\ r\\ s\\ u\\ v\end{array}\right)}_t=\left(\begin{array}{c}-c_{2k+2}\\ -s{b}_{2k+1}+u{f}_{2k+1}\\ r{b}_{2k+1}-u{e}_{2k+1}\\ -r{f}_{2k+1}+s{e}_{2k+1}\\ d_{2k+2}\end{array}\right).$$

(28)

Obviously, when $f_{2k+1}=e_{2k+1}=d_{2k+2}=0,$ the hierarchy (28) reduces to the integrable hierarchy (12). Therefore, (28) is just an integrable coupling of (12) according to definition (22). Through reductions in the hierarchy (28), we can obtain some explicit integrable systems; here, we omit them. Similar to the discussion above, we can obtain the nonisospectral integrable coupling of the hierarchy (12), which is omitted as well.

2.3. Applications of the Loop Algebra $\widetilde{H_{II}}$

By using the loop algebra $\widetilde{H_{II}}$ along with the Tu scheme, multi-component potential integrable systems can be generated via zero-curvature equations.

Set

$${\phi }_x=U\phi ,$$

$$U=h(0,1)+qh(0,0)+re(1,0)+ue(0,0)+vf(0,0),$$

(29)

$${\phi }_t=V\phi ,$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V& ={\displaystyle \sum _{m\ge 0}}[a_0\left(m\right)h(0,-m)+b_0\left(m\right)e(0,-m)+c_0\left(m\right)f(0,-m)\hfill \\ \hfill & +a_1\left(m\right)h(1,-m)+b_1\left(m\right)e(1,-m)+c_1\left(m\right)f(1,-m).\hfill \end{array}\end{array}$$

(30)

The stationary compatibility condition

$$V_x=[U,V]$$

(31)

leads to the following:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{0x}\left(m\right)=u{c}_0\left(m\right)-v{b}_0\left(m\right),\hfill \\ b_{0x}\left(m\right)=-c_0(m+1)+v{a}_0\left(m\right)-q{c}_0\left(m\right),\hfill \\ c_{0x}\left(m\right)=b_0(m+1)-u{a}_0\left(m\right)+q{b}_0\left(m\right),\hfill \\ a_{1x}\left(m\right)=r{c}_0\left(m\right)+u{c}_1\left(m\right)-v{b}_1\left(m\right),\hfill \\ b_{1x}\left(m\right)=-c_1(m+1)+v{a}_1\left(m\right)-q{c}_0\left(m\right),\hfill \\ c_{1x}\left(m\right)=b_1(m+1)-r{a}_0\left(m\right)-u{a}_1\left(m\right)+q{b}_1\left(m\right).\hfill \end{array}\right.\end{array}\end{array}$$

(32)

Taking $a_0\left(0\right)=a_1\left(0\right)=\alpha \left(constant\right),b_0\left(0\right)=c_0\left(0\right)=c_1\left(0\right)=b_1\left(0\right)=0,$ then from (32), one obtains that

$$\begin{array}{cc}\hfill & b_1\left(1\right)=\alpha (r+u),c_1\left(1\right)=\alpha v,b_0\left(1\right)=\alpha u,c_0\left(1\right)=\alpha v,a_0\left(1\right)=a_1\left(1\right)=0,\hfill \\ \hfill & b_1\left(2\right)=\alpha (v_x-qr-qu),c_1\left(2\right)=-\alpha (r_x+u_x+qv),b_0\left(2\right)=\alpha (v_x-qu),\hfill \\ \hfill & c_0\left(2\right)=-\alpha (u_x+qv),a_0\left(2\right)=-{\displaystyle \frac{\alpha }{2}}(u^2+v^2),a_1\left(2\right)=-{\displaystyle \frac{\alpha }{2}}(u^2+v^2)-\alpha ur,\hfill \\ \hfill & b_1\left(3\right)=\alpha [-r_{xx}-u_{xx}-{\left(qv\right)}_x-{\displaystyle \frac{3}{2}}{u}^{2}r-{\displaystyle \frac{1}{2}}{v}^{2}r-{\displaystyle \frac{1}{2}}{u}^3-{\displaystyle \frac{1}{2}}u{v}^2-q{v}_x+q^{2}r+q^{2}u],\hfill \\ \hfill & b_0\left(3\right)=-\alpha [u_{xx}+{\left(qv\right)}_x++{\displaystyle \frac{1}{2}}u(u^2+v^2)+q{v}_x-q^{2}u],\hfill \\ \hfill & c_0\left(3\right)=\alpha [-v_{xx}+{\left(qu\right)}_x-{\displaystyle \frac{1}{2}}v(u^2+v^2)+q{u}_x+q{v}^2],\cdots \hfill \end{array}$$

Denoting

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{+}^{\left(n\right)}& =\sum _{m=0}^n[a_0\left(m\right)h(0,-m)+b_0\left(m\right)e(0,-m)+c_0\left(m\right)f(0,-m)\hfill \\ \hfill & +a_1\left(m\right)h(1,-m)+b_1\left(m\right)e(1,-m)+c_1\left(m\right)f(1,-m)]{\lambda }^{2n}\hfill \\ \hfill & =:\sum _{m=0}^{n}Q(m,n),\hfill \\ \hfill V_{-}^{\left(n\right)}& ={\lambda }^{2n}V-V_{+}^{\left(n\right)}=\sum _{m=n+1}^{\infty }Q(m,n),\hfill \end{array}\end{array}$$

then (31) can be written as

$$-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]={\left(V_{-}^{\left(n\right)}\right)}_x-[U,V_{-}^{\left(n\right)}].$$

(33)

By careful computation, we have

$$-{\left(V_{+}^{\left(n\right)}\right)}_x+[U,V_{+}^{\left(n\right)}]=-b_0(n+1)f(0,0)+c_0(n+1)e(0,0)-b_1(n+1)f(1,0)+c_1(n+1)e(1,0).$$

Let

$${\mathsf{\Delta }}_n=:-{\displaystyle \frac{1}{r}}{b}_1(n+1),V^{\left(n\right)}=:V_{+}^{\left(n\right)}+{\mathsf{\Delta }}_n,$$

then one infers that

$$U_t-V_{+}^{\left(n\right)}+[U,V^{\left(n\right)}]=0$$

gives rise to

$${\left(\begin{array}{c}q\\ u\\ v\\ r\end{array}\right)}_t=\left(\begin{array}{c}-{\left({\displaystyle \frac{1}{r}}{b}_1(n+1)\right)}_x\\ -c_0(n+1)-{\displaystyle \frac{u}{r}}{b}_1(n+1)\\ b_0(n+1)-b_1(n+1)\\ -c_1(n+1)\end{array}\right).$$

(34)

We consider some reductions in the integrable hierarchy (34). When $n=1,\alpha =1,$ we obtain that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_t=q_x-{\left({\displaystyle \frac{v_x}{r}}\right)}_x+{\left({\displaystyle \frac{qu}{r}}\right)}_x,\hfill \\ u_t=u_x+qv-{\displaystyle \frac{u}{r}}(v_x-qr-u),\hfill \\ v_t=qr,\hfill \\ r_t=r_x+u_x+qv.\hfill \end{array}\right.\end{array}\end{array}$$

(35)

Let $r=1,$ then the system (35) becomes

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{u}_t=-u{v}_x+u{v}_t+u^2,\hfill \\ v_{tt}=v_{xt}-v_{xx}+{\left(u{v}_t\right)}_x,\hfill \\ u_x+v{v}_t=0,\hfill \end{array}\right.\hfill \end{array}$$

which is a new integrable system.

When $n=2,$ (34) reduces to

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{q}_t\hfill & =3\alpha u{u}_x+\alpha v{v}_x-2\alpha q{q}_x-{\left[{\displaystyle \frac{1}{r}}(\alpha r_{xx}+\alpha u_{xx}+\alpha {\left(qv\right)}_x+{\displaystyle \frac{\alpha }{2}}{u}^3+{\displaystyle \frac{\alpha }{2}}u{v}^2+\alpha q{v}_x-\alpha q^{2}u)\right]}_x,\hfill \\ u_t\hfill & =\alpha v_{xx}-\alpha {\left(qu\right)}_x+{\displaystyle \frac{\alpha }{2}}v(u^2+v^2)-\alpha q{u}_x-\alpha q^{2}v+{\displaystyle \frac{3\alpha }{2}}{u}^3+{\displaystyle \frac{\alpha }{2}}u{v}^2-\alpha u{q}^2\hfill \\ \hfill & +{\displaystyle \frac{\alpha u}{r}}[r_{xx}+u_{xx}+{\left(qv\right)}_x+{\displaystyle \frac{1}{2}}{u}^3+{\displaystyle \frac{1}{2}}u{v}^2+q{v}_x-q^{2}u],\hfill \\ v_t\hfill & =\alpha r_{xx}+{\displaystyle \frac{3\alpha }{2}}{u}^{2}r+{\displaystyle \frac{\alpha }{2}}{v}^{2}r+\alpha q^{2}r,\hfill \\ r_t\hfill & =\alpha v_{xx}-\alpha {\left(qr\right)}_x-\alpha {\left(qu\right)}_x+{\displaystyle \frac{\alpha v}{2}}(u^2+v^2)+\alpha uvr-\alpha q{u}_x-\alpha q^{2}v,\hfill \end{array}\right.\end{array}\end{array}$$

(36)

which is a rational integrable system.

If $b_1(n+1)=c_1(n+1)=0,n=0,1,2,\cdots ,$ then the integrable hierarchy (34) reduces to the well-known AKNS hierarchy

$${\left(\begin{array}{c}u\\ v\end{array}\right)}_t=\left(\begin{array}{c}-c_0(n+1)\\ b_0(n+1)\end{array}\right).$$

(37)

Therefore, the integrable hierarchy (35) is a kind of expanding integrable model of the AKNS hierarchy. However, it is not an integrable coupling of (37) because the loop algebra $\widetilde{H_{\mathsf{\Pi }}}$ does not posses the property (4). In the paper, we only take the Lax pair (29), (30) as an example to illustrate the explicit application of the loop algebra $\widetilde{H_{\mathsf{\Pi }}}$. Actually, the loop algebra $\widetilde{H_{\mathsf{\Pi }}}$ has extensive applications in the aspect of generating integrable hierarchies with multi-component potential functions.

3. Applications of the Lie Algebra $\overrightarrow{E}$

For the Lie algebra $\overrightarrow{E}$, we construct two loop algebras $\overrightarrow{E_I}$ and $\overrightarrow{E_{\mathsf{\Pi }}}$:

$$\overrightarrow{E_I}=span\{\overrightarrow{e_1}\left(n\right),\overrightarrow{e_2}\left(n\right),\cdots ,\overrightarrow{e_6}\left(n\right)\},e_i\left(n\right)=e_i{\lambda }^n,i=1,2,\cdots ,6,$$

$$\overrightarrow{e_i}\left(m\right)\times \overrightarrow{e_j}\left(n\right)=\mu \overrightarrow{e_k}(m+n),i,j,k=1,2,\cdots ,6;m,n\in N,\mu =\pm 1.$$

$$\overrightarrow{E_{\mathsf{\Pi }}}=span\{\overrightarrow{e_1}\left(n\right),\overrightarrow{e_2}\left(n\right),\cdots ,\overrightarrow{e_6}\left(n\right)\},\overrightarrow{e_1}\left(n\right)=\overrightarrow{e_1}\otimes {\lambda }^{2n},\overrightarrow{e_2}\left(n\right)=\overrightarrow{e_2}\otimes {\lambda }^{2n+1},$$

$$\overrightarrow{e_3}\left(n\right)=\overrightarrow{e_3}\otimes {\lambda }^{2n+1},\overrightarrow{e_4}\left(n\right)=\overrightarrow{e_4}\otimes {\lambda }^{2n},\overrightarrow{e_5}\left(n\right)=\overrightarrow{e_5}\otimes {\lambda }^{2n},\overrightarrow{e_6}\left(n\right)=\overrightarrow{e_6}\otimes {\lambda }^{2n+1},n\in N$$

According to the commutator of the Lie algebra $\overrightarrow{E}$, we have

$$\overrightarrow{e_1}\left(n\right)\times \overrightarrow{e_2}\left(m\right)=\overrightarrow{e_3}(m+n),\overrightarrow{e_2}\left(n\right)\times \overrightarrow{e_3}\left(m\right)=\overrightarrow{e_1}(m+n+1),$$

$$\overrightarrow{e_3}\left(n\right)\times \overrightarrow{e_1}\left(m\right)=\overrightarrow{e_2}(m+n),\overrightarrow{e_1}\left(n\right)\times \overrightarrow{e_4}\left(m\right)=\overrightarrow{e_5}(m+n),$$

$$\overrightarrow{e_1}\left(m\right)\times \overrightarrow{e_5}\left(n\right)=-\overrightarrow{e_4}(m+n),\overrightarrow{e_1}\left(m\right)\times \overrightarrow{e_6}\left(n\right)=\overrightarrow{0},$$

$$\overrightarrow{e_2}\left(m\right)\times \overrightarrow{e_4}\left(n\right)=\overrightarrow{0},\overrightarrow{e_2}\left(m\right)\times \overrightarrow{e_5}\left(n\right)=\overrightarrow{e_6}(m+n),$$

$$\overrightarrow{e_2}\left(m\right)\times \overrightarrow{e_6}\left(n\right)=-\overrightarrow{e_5}(m+n+1),\overrightarrow{e_3}\left(m\right)\times \overrightarrow{e_4}\left(n\right)=-6\overrightarrow{e_3}(m+n),$$

$$\overrightarrow{e_3}\left(m\right)\times \overrightarrow{e_5}\left(n\right)=\overrightarrow{0},\overrightarrow{e_3}\left(m\right)\times \overrightarrow{e_6}\left(n\right)=\overrightarrow{e_4}(m+n+1),$$

$$\overrightarrow{e_4}\left(m\right)\times \overrightarrow{e_5}\left(n\right)=\overrightarrow{e_1}(m+n),\overrightarrow{e_4}\left(m\right)\times \overrightarrow{e_6}\left(n\right)=-\overrightarrow{e_3}(m+n),$$

$$\overrightarrow{e_5}\left(m\right)\times \overrightarrow{e_6}\left(n\right)=\overrightarrow{e_2}(m+n).$$

For arbitrary vectors $\overrightarrow{U},\overrightarrow{V},\overrightarrow{\varphi }\in \overrightarrow{E}$. We assume that

$${\overrightarrow{\varphi }}_x=\overrightarrow{U}\times \overrightarrow{\varphi },\overrightarrow{{\varphi }_t}=\overrightarrow{V}\times \overrightarrow{\varphi },$$

(38)

the compatibility condition of (38) is the following zero-curvature equation:

$$\overrightarrow{U_t}-\overrightarrow{V_x}+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0}.$$

(39)

In fact,

$$\overrightarrow{{\varphi }_{xt}}=\overrightarrow{U_t}\times \overrightarrow{\varphi }+\overrightarrow{U}\times \overrightarrow{{\varphi }_t}=\overrightarrow{U_t}\times \overrightarrow{\varphi }+\overrightarrow{U}\times (\overrightarrow{V}\times \overrightarrow{\varphi })=\overrightarrow{U_t}\times \overrightarrow{\varphi }+(\overrightarrow{U}·\overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{U}·\overrightarrow{V})\overrightarrow{\varphi }),$$

$$\overrightarrow{{\varphi }_{tx}}=\overrightarrow{V_x}\times \overrightarrow{\varphi }+\overrightarrow{V}\times \overrightarrow{{\varphi }_x}=\overrightarrow{V_x}\times \overrightarrow{\varphi }+\overrightarrow{V}\times (\overrightarrow{U}\times \overrightarrow{\varphi })=\overrightarrow{V_x}\times \overrightarrow{\varphi }+(\overrightarrow{V}·\overrightarrow{\varphi })\overrightarrow{U}-(\overrightarrow{V}·\overrightarrow{U})\overrightarrow{\varphi }).$$

Thus,

$$\overrightarrow{{\varphi }_{xt}}=\overrightarrow{{\varphi }_{tx}}\Rightarrow \overrightarrow{U_t}\times \overrightarrow{\varphi }-\overrightarrow{V_x}\times \overrightarrow{\varphi }-(\overrightarrow{U}·\overrightarrow{V}-\overrightarrow{V}·\overrightarrow{U})\overrightarrow{\varphi }+(\overrightarrow{U}\times \overrightarrow{V})\times \overrightarrow{\varphi }=\overrightarrow{0}.$$

Since $\overrightarrow{\varphi }$ is an arbitrary function, we obtain

$$\overrightarrow{U_t}-\overrightarrow{V_x}+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0}.$$

3.1. Applications of the Loop Algebra $\overrightarrow{E_I}$

Consider an isospectral problem

$$\overrightarrow{{\phi }_x}=\overrightarrow{U}\times \overrightarrow{\phi },$$

(40)

$$\overrightarrow{U}=-\overrightarrow{e_1}\left(1\right)+\overrightarrow{e_2}\left(0\right)-w\overrightarrow{e_3}\left(0\right)+{\displaystyle \frac{v}{2}}\overrightarrow{e_1}\left(0\right)+u\overrightarrow{e_4}\left(0\right)+s\overrightarrow{e_5}\left(0\right).$$

The time part for evolution of $\overrightarrow{\phi }$ is given by

$$\overrightarrow{{\phi }_t}=\overrightarrow{V}\times \overrightarrow{\phi },$$

(41)

$$\overrightarrow{V}={\displaystyle \sum _{j=0}^{\infty }}\left(a_m\overrightarrow{e_1}\left(0\right)+b_m\overrightarrow{e_2}\left(0\right)+c_m\overrightarrow{e_3}\left(0\right)+d_m\overrightarrow{e_4}\left(0\right)+e_m\overrightarrow{e_5}\left(0\right)+f_m\overrightarrow{e_6}\left(0\right)\right){\lambda }^{-m}.$$

In terms of the steps of the Tu scheme, the stationary zero-curvature equation

$${\overrightarrow{V}}_x=\overrightarrow{U}\times \overrightarrow{V}$$

(42)

leads to the following relations:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{m,x}=c_m+b_{m}w+u{e}_m-s{d}_m,\hfill \\ b_{m,x}=c_{m+1}-w{a}_m-{\displaystyle \frac{v}{2}}{c}_m+s{f}_m,\hfill \\ c_{m,x}=-b_{m+1}-a_m+{\displaystyle \frac{v}{2}}{b}_m-u{f}_m,\hfill \\ d_{m,x}=e_{m+1}-w{f}_m-{\displaystyle \frac{v}{2}}{e}_m+s{a}_m,\hfill \\ e_{m,x}=-d_{m+1}-f_m+{\displaystyle \frac{v}{2}}{d}_m-u{a}_m,\hfill \\ f_{m,x}=e_m+w{d}_m+u{c}_m-s{b}_m.\hfill \end{array}\right.\end{array}\end{array}$$

(43)

Taking $a_0={\alpha }_0\left(t\right),b_0=c_0=e_0=d_0,$ then we obtain from (43) that

$$c_1=\alpha w,b_1=-{\alpha }_0,e_1=-{\alpha }_{0}s,d_1=-{\alpha }_{0}u,a_1=:{\alpha }_1\left(t\right),f_1=:{\beta }_1\left(t\right),$$

$$c_2={\alpha }_{1}w+{\displaystyle \frac{{\alpha }_0}{2}}wv-{\beta }_{1}s,b_2=-{\alpha }_0{w}_x-{\alpha }_1-{\displaystyle \frac{{\alpha }_0}{2}}v-{\beta }_{1}u,$$

$$e_2=-{\alpha }_0{u}_x+{\beta }_{1}w-{\displaystyle \frac{{\alpha }_0}{2}}vs-{\alpha }_{1}s,d_2={\alpha }_0{s}_x-{\beta }_1-{\displaystyle \frac{{\alpha }_0}{2}}uv-{\alpha }_{1}u,$$

$$a_2=-{\displaystyle \frac{{\alpha }_0}{2}}{w}^2-{\displaystyle \frac{{\alpha }_0}{2}}{s}^2-{\displaystyle \frac{{\alpha }_0}{2}}{u}^2+{\alpha }_2\left(t\right),f_2=-{\alpha }_{0}u+{\alpha }_{0}sw+{\beta }_2\left(t\right),\cdots$$

where ${\alpha }_0\left(t\right),{\alpha }_1\left(t\right),{\alpha }_2\left(t\right),{\beta }_1\left(t\right),{\beta }_2\left(t\right),\cdots$ are all integral constants.

Equation (42) can be decomposed to

$$-{\left({\overrightarrow{V}}_{+}^{\left(n\right)}\right)}_x+\overrightarrow{U}\times {\overrightarrow{V}}_{+}^{\left(n\right)}={\left({\overrightarrow{V}}_{-}^{\left(n\right)}\right)}_x-\overrightarrow{U}\times {\overrightarrow{V}}_{-}^{\left(n\right)},$$

where

$${\overrightarrow{V}}_{+}^{\left(n\right)}={\displaystyle \sum _{j=0}^n}(a_m\overrightarrow{e_1}(-m)+b_m\overrightarrow{e_2}(-m)+c_m\overrightarrow{e_3}(-m)+d_m\overrightarrow{e_4}(-m)+e_m\overrightarrow{e_5}(-m)+f_m\overrightarrow{e_6}(-m)){\lambda }^n,$$

$${\overrightarrow{V}}_{-}^{\left(n\right)}={\lambda }^n\overrightarrow{V}-{\overrightarrow{V}}_{+}^{\left(n\right)}.$$

It is easy to find that

$$-{\left({\overrightarrow{V}}_{+}^{\left(n\right)}\right)}_x+\overrightarrow{U}\times {\overrightarrow{V}}_{+}^{\left(n\right)}=b_{n+1}{e}_3\left(0\right)-c_{n+1}{e}_2\left(0\right)+d_{n+1}{e}_3\left(0\right)-e_{n+1}{e}_4\left(0\right).$$

Set

$${\overrightarrow{V}}^{\left(n\right)}={\overrightarrow{V}}_{+}^{\left(n\right)}+\overrightarrow{{\mathsf{\Delta }}_n},\overrightarrow{{\mathsf{\Delta }}_n}=-{\displaystyle \frac{c_{n+1}}{w}},$$

then the form of the zero-curvature equation

$${\overrightarrow{U}}_t-{\overrightarrow{V}}_x^{\left(n\right)}+\overrightarrow{U}\times {\overrightarrow{V}}^{\left(n\right)}=\overrightarrow{0}$$

admits the following integrable hierarchy:

$${\left(\begin{array}{c}w\\ v\\ u\\ s\end{array}\right)}_t=\left(\begin{array}{c}{b}_{n+1}+{\displaystyle \frac{c_{n+1}}{w}}\\ -2{\left({\displaystyle \frac{c_{n+1}}{w}}\right)}_x\\ e_{n+1}+{\displaystyle \frac{s{c}_{n+1}}{w}}\\ -d_{n+1}-{\displaystyle \frac{u{c}_{n+1}}{w}}\end{array}\right).$$

(44)

When $n=2,$ from (44) we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w_{t_2}& =-{\alpha }_1{w}_x-{\displaystyle \frac{{\alpha }_0}{2}}{\left(uw\right)}_x+{\beta }_1{s}_x+{\alpha }_0(w^2+s^2)-{\displaystyle \frac{{\alpha }_0}{2}}v{w}_x\hfill \\ \hfill & -{\displaystyle \frac{{\beta }_1}{2}}uv-{\alpha }_{0}suw-{\beta }_{2}u-{\alpha }_0{\displaystyle \frac{w_{xx}}{w}}-{\displaystyle \frac{{\alpha }_0{v}_x}{2w}}\hfill \\ \hfill & -{\displaystyle \frac{{\beta }_1{u}_x}{w}}+{\displaystyle \frac{{\alpha }_{0}su}{w}}-{\alpha }_{0}s-{\displaystyle \frac{{\beta }_{2}s}{w}}+{\displaystyle \frac{{\alpha }_0{v}^2}{2}}-{\displaystyle \frac{{\beta }_{1}vs}{2w}},\hfill \\ \hfill v_{t_2}& =[{\displaystyle \frac{2{\alpha }_0{w}_{xx}}{w}}+{\alpha }_0{\displaystyle \frac{v_x}{w}}+{\displaystyle \frac{2{\beta }_1{u}_x}{w}}+{\alpha }_0(w^2+s^2+u^2)-2{\alpha }_2\hfill \\ \hfill & -{\displaystyle \frac{{\alpha }_0}{2}}{v}^2+{\displaystyle \frac{{\beta }_{1}vs}{2w}}-2{\alpha }_0{\displaystyle \frac{su}{w}}+2{\alpha }_0{s}^2+{\displaystyle \frac{2{\beta }_{2}s}{w}}{]}_x.\hfill \end{array}\end{array}$$

In particular, let $u=s=0,{\alpha }_1={\alpha }_2=0,$ then the above equations reduce to

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{w}_{t_2}=-{\displaystyle \frac{{\alpha }_0}{2}}v{w}_x-{\alpha }_0{\displaystyle \frac{w_{xx}}{w}}+{\alpha }_0{w}^2-{\displaystyle \frac{{\alpha }_0}{2}}{\left(vw\right)}_x+{\displaystyle \frac{{\alpha }_0}{2}}{v}^2,\hfill \\ v_{t_2}=-2{\alpha }_0{(-{\displaystyle \frac{w_{xx}}{w}}-{\displaystyle \frac{v_x}{2w}}-{\displaystyle \frac{1}{2}}{w}^2+{\displaystyle \frac{1}{4}}{v}^2)}_x.\hfill \end{array}\right.\end{array}\end{array}$$

(45)

Set $w=1,$ then (45) becomes such that

$$v_{t_2}={\alpha }_{0}v{v}_x,v^2-v_x+2=0$$

which gives rise to

$$v_{t_2}={\displaystyle \frac{{\alpha }_0}{2}}{v}_{xx}.$$

(46)

When ${\alpha }_0>0,$ Equation (46) is just the well-known heat conduction equation. Other reducing equations from the integrable hierarchy (44) do not further discuss them again.

3.2. Applications of the Loop Algebra $\overrightarrow{E_{\mathsf{\Pi }}}$

3.2.1. An Isospectral Integrable Hierarchy

Consider the following spectral problems:

$$\overrightarrow{{\varphi }_x}=\overrightarrow{U}\times \overrightarrow{\varphi },$$

(47)

$$\overrightarrow{U}=\overrightarrow{e_1}\left(1\right)+q\overrightarrow{e_2}\left(0\right)+r\overrightarrow{e_3}\left(0\right)+u\overrightarrow{e_4}\left(0\right)+v\overrightarrow{e_5}\left(0\right),$$

$$\overrightarrow{{\varphi }_t}=\overrightarrow{V}\times \overrightarrow{\varphi },$$

(48)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overrightarrow{V}& =\sum _{m\ge 0}[(a_m\overrightarrow{e_1}(-m)+b_m\overrightarrow{e_2}(-m)+c_m\overrightarrow{e_3}(-m)+d_m\overrightarrow{e_4}(-m)+e_m\overrightarrow{e_5}(-m)+f_m\overrightarrow{e_6}(-m)]\hfill \\ \hfill & =:\sum _{m\ge 0}P(m,n).\hfill \end{array}\end{array}$$

The vector equation

$$\overrightarrow{V_x}=\overrightarrow{U}\times \overrightarrow{V}$$

is equivalent to the following equations:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{c}_{m+1}=-b_{m,x}+r{a}_m+v{f}_m,\hfill \\ b_{m+1}=c_{m,x}+q{a}_m+u{f}_m,\hfill \\ e_{m+1}=-d_{m,x}+r{f}_{m+1}+v{a}_m,\hfill \\ d_{m+1}=e_{m,x}+q{f}_{m+1}+u{a}_m,\hfill \\ f_{m+1,x}=-q{d}_{m,x}-r{e}_{m,x}-u{b}_{m,x}-v{c}_{m,x}\hfill \end{array}\right.\end{array}\end{array}$$

(49)

Taking $c_0=b_0=e_0=d_0=0,a_0={\alpha }_0\left(t\right),f_0={\beta }_0\left(t\right),$ then we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_1& =:{\beta }_0\left(t\right),e_1={\beta }_{0}r+{\alpha }_{0}v,a_1=:{\alpha }_1\left(t\right),c_2=-{\alpha }_0{q}_x-{\beta }_0{u}_x+{\alpha }_{1}r+{\beta }_{0}v,\hfill \\ \hfill b_2& ={\alpha }_0{r}_x+{\beta }_0{v}_x+{\alpha }_{1}q+{\beta }_{0}u,f_2={\displaystyle \frac{{\beta }_0}{2}}(q^2+r^2+u^2+v^2)-{\alpha }_0(rv+qu),\hfill \\ \hfill e_2& =-{\beta }_0{q}_x-{\alpha }_0{u}_x-{\displaystyle \frac{{\beta }_0}{2}}r(q^2+r^2+u^2+v^2)-{\alpha }_{0}r(rv+qu)+{\alpha }_{1}v,\hfill \\ \hfill d_2& ={\beta }_0{r}_x+{\alpha }_0{v}_x-{\displaystyle \frac{{\beta }_0}{2}}q(q^2+r^2+u^2+v^2)-{\alpha }_{0}q(rv+qu)+{\alpha }_{1}u,\hfill \\ \hfill \cdots \end{array}\end{array}$$

Set ${\overrightarrow{V}}^{\left(n\right)}={\displaystyle \sum _{j=0}^n}P(m,n){\lambda }^{2n},$ then the zero-curvature equation

$$\overrightarrow{U}-{\overrightarrow{V}}_x^{\left(n\right)}+\overrightarrow{U}\times {\overrightarrow{V}}^{\left(n\right)}=\overrightarrow{0}$$

yields the following isospectral integrable hierarchy

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=\left(\begin{array}{c}-c_{n+1}\\ b_{n+1}\\ r{f}_{n+1}-e_{n+1}\\ d_{n+1}-q{f}_{n+1}\end{array}\right)=\left(\begin{array}{c}-c_{n+1}\\ b_{n+1}\\ d_{n,x}-v{a}_n\\ e_{n,x}+u{a}_n\end{array}\right).$$

(50)

When $n=2,u=v=0,$ the hierarchy (50) reduces to

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_t={\alpha }_0{r}_{xx}+{\alpha }_1{q}_x+{\displaystyle \frac{{\alpha }_0}{2}}r(q^2+r^2)+{\alpha }_{2}r,\hfill \\ r_t=-{\alpha }_0{q}_{xx}+{\alpha }_1{r}_x-{\displaystyle \frac{{\alpha }_0}{2}}q(q^2+r^2)+{\alpha }_{2}q.\hfill \end{array}\right.\end{array}\end{array}$$

(51)

Let $r=\pm iq,$ then the integrable system (51) reduces to

$$q_t=\pm i{\alpha }_0{q}_{xx}+{\alpha }_1{q}_x\mp i{\alpha }_{2}q,$$

(52)

which is a simplified linear Schrödinger equation.

Set ${\alpha }_0=i,$ then (51) becomes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_t=i{r}_{xx}+{\alpha }_1{q}_x+{\displaystyle \frac{i}{2}}r(q^2+r^2)+{\alpha }_{2}r,\hfill \\ r_t=-i{q}_{xx}+{\alpha }_1{r}_x-{\displaystyle \frac{i}{2}}q(q^2+r^2)+{\alpha }_{2}q.\hfill \end{array}\right.\end{array}\end{array}$$

(53)

When $r=\overline{q},{\alpha }_2=0,$ the integrable system (53) reduces to

$$q_t=\pm i{\overline{q}}_{xx}+{\alpha }_1{q}_x+{\displaystyle \frac{i}{2}}({\overline{q}}^3+{\left|q\right|}^{2}q),$$

(54)

which is a nonlinear Schrödinger-type equation. As we know, (54) is perhaps a new equation.

3.2.2. A Nonisospectral Integrable Hierarchy

Assume that

$${\overrightarrow{\chi }}_x=\overrightarrow{U}\times \overrightarrow{\chi },{\overrightarrow{\chi }}_t=\overrightarrow{V}\times \overrightarrow{\chi },$$

(55)

where $\overrightarrow{\chi },\overrightarrow{U},\overrightarrow{V}\in \overrightarrow{E_{\mathsf{\Pi }}}.$ Then, under the condition ${\lambda }_t\ne 0,$ the compatibility condition of (55) yields the following nonisospectral-type zero-curvature equation

$${\displaystyle \frac{\partial \overrightarrow{U}}{\partial \lambda }}{\lambda }_t+{\displaystyle \frac{\partial \overrightarrow{U}}{\partial u}}{u}_t-{\overrightarrow{V}}_x+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0},$$

(56)

where u is a potential function in the vector function $\overrightarrow{U}$. We suppose that

$$\overrightarrow{U}={\overrightarrow{e}}_1\left(1\right)+q{\overrightarrow{e}}_2\left(0\right)+r{\overrightarrow{e}}_3\left(0\right)+u{\overrightarrow{e}}_4\left(0\right)+v{\overrightarrow{e}}_5\left(0\right),$$

$$\overrightarrow{V}=:{\displaystyle \sum _{m\ge 0}}P(m,n),{\lambda }_t={\displaystyle \sum _{j\ge 0}}{k}_j\left(t\right){\lambda }^{-2j+1},$$

and it is easy to see that

$${\displaystyle \frac{\partial \overrightarrow{U}}{\partial \lambda }}{\lambda }_t={\displaystyle \sum _{j\ge 0}}\left(2k_j{\overrightarrow{e}}_1(1-j)+k_{j}q{\overrightarrow{e}}_2(-j)+k_{j}r{\overrightarrow{e}}_3(-j)\right).$$

The stationary zero-curvature equation

$${\displaystyle \frac{\partial \overrightarrow{U}}{\partial \lambda }}{\lambda }_t-{\overrightarrow{V}}_x+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0},$$

(57)

yields that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{c}_{j+1}=-b_{j,x}+r{a}_j+v{f}_j+q{k}_j,\hfill \\ b_{j+1}=c_{j,x}+q{a}_j+u{f}_j-r{k}_j,\hfill \\ e_{j+1}=-d_{j,x}+r{f}_{j+1}+v{a}_j,\hfill \\ d_{j+1}=e_{j,x}+q{f}_{j+1}+u{a}_j,\hfill \\ f_{j+1,x}=-q{d}_{j,x}-r{e}_{j,x}-u{b}_{j,x}-v{c}_{j,x}+(qu+vr)k_j.\hfill \end{array}\right.\end{array}\end{array}$$

(58)

Set $c_0=b_0=e_0=d_0=0,a_0={\sigma }_0\left(t\right),f_0={\delta }_0\left(t\right),$ then in term of (58), one infers that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill c_1& ={\sigma }_{0}r+{\delta }_{0}v+q{k}_0,b_1={\sigma }_{0}q+{\delta }_{0}u-r{k}_0,f_1=k_0{\partial }_x^{-1}(qu+vr)+{\delta }_0,\hfill \\ \hfill d_1& =k_{0}q{\partial }_x^{-1}(qu+vr)+{\delta }_{0}q+{\sigma }_{0}u,e_1=k_{0}r{\partial }^{-1}(qu+vr)+{\delta }_{0}r+{\sigma }_{0}v,\hfill \\ \hfill a_1& =k_0{\partial }^{-1}[q^2+r^2+(ur-qv){\partial }^{-1}(qu+vr)]+2k_{2}x\hfill \\ \hfill & =:k_{0}M+2k_{2}x,\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_x& =q^2+r^2+(ur-qv){\partial }^{-1}(qu+vr),\hfill \\ \hfill f_2& =-k_0{\partial }_x^{-1}[{\displaystyle \frac{1}{2}}{(q^2+r^2)}_x{\partial }_x^{-1}(qu+vr)+(q^2+r^2)(qu+vr)-u{r}_x+v{q}_x]\hfill \\ \hfill & +k_1{\partial }_x^{-1}(qu+vr)-{\displaystyle \frac{{\delta }_0}{2}}(q^2+r^2+u^2+v^2)-{\sigma }_0(qu+vr)\hfill \\ \hfill & =:N-{\displaystyle \frac{{\delta }_0}{2}}(q^2+r^2+u^2+v^2)-{\sigma }_0(qu+vr),\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill N& =-k_0{\partial }_x^{-1}[{\displaystyle \frac{1}{2}}{(q^2+r^2)}_x{\partial }_x^{-1}(qu+vr)+(q^2+r^2)(qu+vr)-u{r}_x+v{q}_x]\hfill \\ \hfill & +k_1{\partial }_x^{-1}(qu+vr),\hfill \\ \hfill c_2& =-{\sigma }_0{q}_x-{\delta }_0{u}_x-k_0{r}_x+k_{0}rM+2k_{2}xr+k_{0}v{\partial }_x^{-1}(qu+vr)+{\delta }_{0}v+k_{1}q,\hfill \\ \hfill b_2& ={\sigma }_0{r}_x+{\delta }_0{v}_x+k_0{q}_x+k_{0}qM+2k_{2}xq+k_{0}u{\partial }^{-1}(qu+vr),\hfill \\ \hfill e_2& =-k_0{q}_x{\partial }_x^{-1}(qu+vr)-k_{0}q(qu+vr)-{\delta }_0{q}_x-{\sigma }_0{u}_x+rN\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_0}{2}}(q^2+r^2+u^2+v^2)-{\sigma }_{0}r(qu+vr)+k_{0}vM+2k_{0}xv,\hfill \\ \hfill d_2& =k_0{r}_x{\partial }_x^{-1}(qu+vr)+k_{0}r(qu+vr)+{\delta }_0{r}_x+{\sigma }_0{v}_x-qN\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_0}{2}}q(q^2+r^2+u^2+v^2)-{\sigma }_{0}q(qu+vr)+k_{0}uM+2k_{0}xu,\hfill \\ \hfill a_2& =-{\displaystyle \frac{{\delta }_0}{2}}(q^2+r^2+u^2+v^2)-{\sigma }_0(qu+vr)-k_{0}qr+2k_{3}x+Q,\hfill \end{array}\end{array}$$

here,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Q_x& =k_{0}qv{\partial }_x^{-1}(qu+vr)+{\delta }_{0}qv+k_1{q}^2-k_{0}ur{\partial }_x^{-1}(qu+vr)-{\sigma }_{0}ur+k_1{r}^2\hfill \\ \hfill & -k_{0}u{q}_x{\partial }_x^{-1}(qu+vr)-k_{0}uq(qu+vr)+(ur-qv)N\hfill \\ \hfill & +{\displaystyle \frac{{\delta }_0}{2}}(qv-ur)(q^2+r^2+u^2+v^2)+{\sigma }_0(qv-ur)(qu+vr)-k_{0}vr(qu+vr),\hfill \\ \hfill c_3& =-{\sigma }_0{r}_{xx}-{\delta }_0{v}_{xx}-k_0{q}_{xx}-k_0{\left(qM\right)}_x-2k_0{\left(xq\right)}_x-k_0{\left[u{\partial }_x^{-1}(qu+vr)\right]}_x+k_1{r}_x\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_0}{2}}r(q^2+r^2+u^2+v^2)-{\sigma }_{0}r(qu+vr)-k_{0}q{r}^2+2k_{3}xr+rQ+vN\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_0}{2}}v(q^2+r^2+u^2+v^2)-{\sigma }_{0}v(qu+vr)+k_{2}q,\hfill \\ \hfill b_3& =-{\sigma }_0{q}_{xx}-{\delta }_0{u}_{xx}-k_0{r}_{xx}+k_0{\left(rM\right)}_x+2k_0{\left(xr\right)}_x+k_0{\left[u{\partial }_x^{-1}(qu+vr)\right]}_x+{\delta }_0{v}_x\hfill \\ \hfill & +k_1{q}_x-{\displaystyle \frac{{\delta }_0}{2}}q(q^2+r^2+u^2+v^2)-{\sigma }_{0}q(qu+vr)-k_0{q}^{2}r+2k_{3}xq+qQ-k_{2}r.\hfill \end{array}\end{array}$$

Similar to the discussion in Section 3.2.1, we obtain a nonisospectral hierarchy

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_{t_n}=-c_{n+1},\hfill \\ r_{t_n}=b_{n+1},\hfill \\ u_{t_n}=d_{n,x}-v{a}_n,\hfill \\ v_{t_n}=e_{n,x}+u{a}_n.\hfill \end{array}\right.\end{array}\end{array}$$

(59)

When $n=2,$ the integrable hierarchy (59) reduces to

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{q}_t=-c_3,\hfill \\ r_t=b_3,\hfill \\ u_t=d_{2,x}-v{a}_2,\hfill \\ v_t=e_{2,x}+u{a}_2,\hfill \end{array}\right.\end{array}\end{array}$$

(60)

where $c_3,b_3,d_2,e_2,a_2$ are presented as above. In particular, set $u=v=0,$ and (60) becomes such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{q}_t\hfill & =-{\sigma }_0{r}_{xx}+k_0{q}_{xx}+k_0{\left[q{\partial }_x^{-1}(q^2+r^2)\right]}_x+2k_0{\left(xq\right)}_x-k_1{r}_x\hfill \\ \hfill & +{\displaystyle \frac{{\delta }_0}{2}}r(q^2+r^2)++k_{0}q{r}^2-2k_{3}xr-k_{2}q-r{k}_1{\partial }_x^{-1}(q^2+r^2),\hfill \\ r_t\hfill & =-{\sigma }_0{q}_{xx}-k_0{r}_{xx}+k_0{\left[r{\partial }_x^{-1}(q^2+r^2)\right]}_x+2k_2{\left(xr\right)}_x+k_1{q}_x\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_0}{2}}q(q^2+r^2)-k_0{q}^{2}r+2k_{3}xq-k_{2}r+q{\partial }_x^{-1}{k}_1(q^2+r^2).\hfill \end{array}\right.\end{array}\end{array}$$

(61)

Let $r=\overline{q},k_0=k_1=0,{\sigma }_0=i,{\overline{k}}_3=i,$ then the integrable system (61) reduces to

$$q_t=i{\overline{q}}_{xx}+2k_2{\left(xq\right)}_x+{\displaystyle \frac{i}{2}}\overline{q}(q^2+{\overline{q}}^2)-2ix\overline{q}-k_{2}q,$$

(62)

which is a nonisospectral nonlinear Schrödinger-type equation.

3.2.3. Hamiltonian Structure of the Integrable Hierarchy (50)

Make a linear map: ${\tilde{E}}_{\mathsf{\Pi }}\to R^6,\forall \overrightarrow{e}={\displaystyle \sum _{i=1}^6}{a}_i\overrightarrow{e_i}\in \widetilde{E_{\mathsf{\Pi }}}\to \overrightarrow{a}=(a_1,a_2,a_3,a_4,a_5,a_6)\in R^6.$ Obviously, f is an isomorphic map between $\widetilde{E_{\mathsf{\Pi }}}$ and $R^6$. For arbitrary vectors $\overrightarrow{a}=(a_1,a_2,a_3,a_4,a_5,a_6),\overrightarrow{b}=(b_1,b_2,b_3,b_4,b_5,b_6)\in R^6,$ we define a linear operation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overrightarrow{a}\times \overrightarrow{b}=& (a_2{b}_3-a_3{b}_2+a_4{b}_5-a_5{b}_4,a_3{b}_1-a_1{b}_3+a_5{b}_6-a_6{b}_5,a_1{b}_2-a_2{b}_1+a_6{b}_4-a_4{b}_6,\hfill \\ \hfill & a_5{b}_1-a_1{b}_5+a_3{b}_6-a_6{b}_3,a_1{b}_4-a_4{b}_1+a_6{b}_2-a_2{b}_6,a_2{b}_5-a_5{b}_2+a_4{b}_3-a_3{b}_4),\hfill \end{array}\end{array}$$

(63)

and it can be verified that $R^6$ is between a Lie algebra along with (63). We again rewrite (63) as the following form:

$$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}R\left(b\right).$$

(64)

If $F={\left(f_{ij}\right)}_{6\times 6}=F^T,$ a solution to the matrix equation

$$R\left(b\right)F=-{\left(R\left(b\right)F\right)}^T$$

(65)

is obtained:

$$F=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 1& 0& 0& 0& 0& 1\end{array}\right).$$

In what follows, we want to search for the Hamiltonian structure of the hierarchy (50) with the help of quadratic-form identity [20,21] and (65). The vectors $\overrightarrow{U},\overrightarrow{V}$ in (47) and (48) can be written as

$$\overrightarrow{U}=({\lambda }^2,\lambda q,\lambda r,u,v,0),\overrightarrow{V}=(a,b\lambda ,c\lambda ,d,e,f\lambda ).$$

A direct computation implies that

$$\{\overrightarrow{V},{\displaystyle \frac{\partial \overrightarrow{U}}{\partial q}}\}=:\overrightarrow{V}F{({\displaystyle \frac{\partial \overrightarrow{U}}{\partial q}})}^T=\lambda (\lambda b+d),\{\overrightarrow{V},{\displaystyle \frac{\partial \overrightarrow{U}}{\partial r}}\}=\lambda (\lambda c+e),$$

$$\{\overrightarrow{V},{\displaystyle \frac{\partial \overrightarrow{U}}{\partial u}}\}=\lambda b+d,\{\overrightarrow{V},{\displaystyle \frac{\partial \overrightarrow{U}}{\partial v}}\}=\lambda c+e,$$

$$\{\overrightarrow{V},{\displaystyle \frac{\partial \overrightarrow{U}}{\partial \lambda }}\}=2\lambda (a+f\lambda )+q(\lambda b+d)+r(\lambda c+e).$$

Inserting the above results into the quadratic-form identity yields that

$${\displaystyle \frac{\delta }{\delta \overrightarrow{u}}}[2\lambda (a+f\lambda )+q(\lambda b+d)+r(\lambda c+e)]={\lambda }^{-\gamma }{\displaystyle \frac{\partial }{\partial \lambda }}{\lambda }^{\gamma }\left(\begin{array}{c}\lambda (\lambda b+d)\\ \lambda (\lambda c+e)\\ \lambda b+d\\ \lambda c+e\end{array}\right).$$

(66)

Comparing the coefficients of ${\lambda }^{-2n}$ in (66) indicates that

$${\displaystyle \frac{\delta }{\delta \overrightarrow{u}}}(2f_{n+1}+q{d}_n+r{e}_n)=(-2n+1+\gamma )\left(\begin{array}{c}{d}_n\\ e_n\\ b_n\\ c_n\end{array}\right).$$

(67)

Substituting the initial values in (50), we obtain $\gamma =0.$ Therefore, (67) becomes

$$\left(\begin{array}{c}{d}_n\\ e_n\\ b_n\\ c_n\end{array}\right)={\displaystyle \frac{\delta H_n}{\delta \overrightarrow{u}}},H_n={\displaystyle \frac{2f_{n+1}+q{d}_n+r{e}_n}{-2n+1}}.$$

(68)

Hence, the integrable hierarchy (50) can be written as

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=J\left(\begin{array}{c}{d}_n\\ e_n\\ b_n\\ c_n\end{array}\right)=J{\displaystyle \frac{\delta H_n}{\delta \overrightarrow{u}}},$$

(69)

where

$$J=\left(\begin{array}{cccc}r{\partial }^{-1}v+v{\partial }^{-1}r& r{\partial }^{-1}u-v{\partial }^{-1}q& \partial +r{\partial }^{-1}r+v{\partial }^{-1}v& -r{\partial }^{-1}q-v{\partial }^{-1}u\\ -q{\partial }^{-1}v-u{\partial }^{-1}r& q{\partial }^{-1}u+u{\partial }^{-1}q& -q{\partial }^{-1}r-u{\partial }^{-1}v& \partial +q{\partial }^{-1}q+u{\partial }^{-1}u\\ \partial +v{\partial }^{-1}v& -v{\partial }^{-1}u& v{\partial }^{-1}r& -v{\partial }^{-1}q\\ -u{\partial }^{-1}v& \partial +u{\partial }^{-1}u& -u{\partial }^{-1}r& u{\partial }^{-1}q\end{array}\right).$$

Since J is not a Hamiltonian operator, (69) is only a quasi-Hamiltonian structure.

4. Applications of the Associative Algebra $\overrightarrow{F}$

In this section, we would like to utilize the algebra $\overrightarrow{F}$ to generate high-dimensional integrable equations. In Ref. [26], the authors consider a nonlinear Lax pair

$${\psi }_y=\mathcal{L}({\psi }_x,u),{\psi }_t=\mathcal{B}({\psi }_x,u),$$

(70)

which is used to generate a type of (2+1)-dimensional integrable systems. The Lax pair (70) has the compatibility condition ${\psi }_{yt}={\psi }_{ty},$ which is equivalent to the following zero-curvature-type equation by using Poisson geometry

$$L_t-B_y+{\{L,B\}}_p=0,$$

(71)

where $L=\mathcal{L}(p,u),B=\mathcal{B}(p,u),p={\psi }_x,$ and ${\{L,B\}}_p={\displaystyle \frac{\partial L}{\partial p}}{\displaystyle \frac{\partial B}{\partial x}}-{\displaystyle \frac{\partial L}{\partial \lambda }}{\displaystyle \frac{\partial B}{\partial p}}$ is called the Poisson bracket. In Refs. [27,28], some (2+1)-dimensional integrable systems are obtained.

Denoting ${\{H,F\}}_c$ by

$${\{H,F\}}_c=H{\displaystyle \frac{\partial F}{\partial z}}+{\displaystyle \frac{\partial H}{\partial p}}{\displaystyle \frac{\partial F}{\partial x}}-p{\displaystyle \frac{\partial H}{\partial p}}{\displaystyle \frac{\partial F}{\partial z}}-(H\leftrightarrow F),$$

(72)

we give a Lax pair

$${\varphi }_y={\{L,\varphi \}}_c,{\varphi }_t={\{B,\varphi \}}_c,$$

(73)

where $\varphi =\varphi (x,y,z,t,p)$. Then, the compatibility condition of (73) leads to the following zero-curvature-type equation:

$$L_t-B_y+{\{L,B\}}_c=0,$$

(74)

which is used to generate (3+1)-dimensional integrable systems. In the section, we directly construct appropriate vector fields by utilizing the associative algebra $\overrightarrow{F}$ to investigate (2+1)-dimensional and (3+1)-dimensional integrable systems.

Set

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{L}_1=\partial y+(\lambda +v_x)\partial x-u_x\partial \lambda ,\hfill \\ L_2=\partial t+({\lambda }^2+v_x\lambda +u-v_y)\partial x+(-u_x\lambda +u_y)\partial \lambda ,\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(75)

then the commutator equation

$$[L_1,L_2]=0$$

leads to the following (2+1)-dimensional integrable systems:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{v}_{xt}=-v_{yy}-v_x{v}_{xy}-u{v}_{xx}+v_y{v}_{xx},\hfill \\ u_{xt}=-u_{yy}-v_x{u}_{xy}-u_x^2-u{u}_{xx}+v_y{u}_{xx}.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(76)

If $v=\alpha \left(constant\right),$ (76) reduces to

$$u_{xt}=-u_{yy}-u_x^2+u{u}_{xx},$$

(77)

which is a special cylindrical dissipative Zaboloskaya–Khokhlov equation (cd ZK equation).

In the following, we consider two general (3+1)-dimensional vector fields:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{L}_1=\partial z+(\lambda +a)\partial x+b\partial y,\hfill \\ L_2=\partial t+(h{\lambda }^2+c\lambda +d)\partial y+(f\lambda +e)\partial x,\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(78)

where $a,b,c,d,h,e,f$ are all functions in $x,y,z,t,$ which will be determined. The commutator equation

$$[L_1,L_2]=0$$

can give rise to the following equations by vanishing the coefficients $\partial x,\partial y,\partial z,\partial t$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{f}_x-h{a}_y=0,\hfill \\ f_y+a{f}_x+e_x+b{f}_y-c{a}_y-a_{x}f=0,\hfill \\ e_z+a{e}_x+b{e}_y-a_t-d{a}_y-a_{x}e=0,\hfill \\ h_x=0,\hfill \\ h_z+c_x+b{h}_y-h{b}_y=0,\hfill \\ c_z+d_x+a{c}_x+b{c}_y-c{b}_y-b_{x}f=0,\hfill \\ d_z+a{d}_x+b{d}_y-b_t-d{b}_y-b_{x}e=0.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(79)

We only consider the case where $h=0,f_x=0.$

Case 1: $f=1,$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{e}_x-c{a}_y-a_x=0,\hfill \\ e_z+a{e}_x+b{e}_x-d{a}_y-a_{x}e=a_t,\hfill \\ c_x=0,\hfill \\ c_z+d_x+b{c}_y-c{b}_y-b_x=0,\hfill \\ d_z+a{d}_x+b{d}_y-d{b}_y-b_{x}e=b_t.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(80)

Let $c=1,$ and we have

$$d_x=b_y+b_x\Rightarrow d=b+{\int }^x{b}_{y}dx,$$

$$e_x=a_y+a_x\Rightarrow e=a+{\int }^x{a}_{y}dx,$$

Set $b=u_x,a=v_x,$ then $d=u_x+u_y,e=v_x+v_y.$ Substituting these consequences into (80) yields a new (3+1)-dimensional integrable systems

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{v}_{xt}=v_{xz}+v_{yz}+(u_x+v_x)(v_{xx}+v_{xy})-v_{xy}(u_x+u_y)-v_{xx}(v_x+v_y),\hfill \\ u_{xt}=u_{xz}+u_{yz}+v_x(u_{xx}+u_{xy})-u_y(u_{xy}+u_{yy})-u_{xx}(v_x+v_y).\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(81)

Let $c=0,e=a,d=b,$ then (80) reduces to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{a}_t=a_z+b(a_x-a_y),\hfill \\ b_t=b_z+b(b_x-b_y),\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(82)

where $a,b$ are arbitrary differential functions in $x,y,z,t.$

Case 2: $f=0.$ Equation (80) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{a}_t=e_z+a{e}_x+b{e}_y-d{a}_y-a_{x}e,\hfill \\ b_t=d_z+a{d}_x+b{d}_y-d{b}_y-b_{x}e.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(83)

Let $c=1,a={\theta }_{xy},e={\theta }_{yy},b=-{\theta }_{xx},$ then from (83), we obtain $d=-{\theta }_{xy},$ and hence, (83) gives rise to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\theta }_{xyt}& ={\theta }_{xyz}+{\theta }_{xy}{\theta }_{xyy}-{\theta }_{xx}{\theta }_{yyy}+{\theta }_{xy}{\theta }_{xyy}-{\theta }_{xxy}{\theta }_{yy}\hfill \\ \hfill & \Rightarrow {\theta }_{xt}={\theta }_{yz}+\int {\theta }_{xy}d{\theta }_{xy}-\int {\theta }_{xx}{\theta }_{yyy}dy+\int {\theta }_{xy}{\theta }_{xyy}dy-\int {\theta }_{yy}d{\theta }_{xx}\hfill \\ \hfill & ={\theta }_{yz}+{\displaystyle \frac{1}{2}}{\theta }_{xy}^2-\int {\theta }_{xx}{\theta }_{yyy}d\theta +\int {\theta }_{xy}d{\theta }_{xy}-{\theta }_{xx}{\theta }_{xy}+\int {\theta }_{xx}{\theta }_{yyy}d\theta \hfill \\ \hfill & ={\theta }_{yz}+{\theta }_{xy}^2-{\theta }_{xx}{\theta }_{yy}.\hfill \\ \hfill {\theta }_{xxt}& ={\theta }_{xyz}+{\theta }_{xy}{\theta }_{xxy}-{\theta }_{xx}{\theta }_{xyy}+{\theta }_{xy}{\theta }_{xxy}-{\theta }_{xxx}{\theta }_{yy}\hfill \\ \hfill & \Rightarrow {\theta }_{xt}={\theta }_{yz}+\int {\theta }_{xy}d{\theta }_{xy}-\int {\theta }_{xx}{\theta }_{xyy}dx+\int {\theta }_{xy}d{\theta }_{xy}-\int {\theta }_{xxx}{\theta }_{yy}dx\hfill \\ \hfill & ={\theta }_{yz}+{\theta }_{xy}^2-\int {\theta }_{xx}d{\theta }_{yy}-\int {\theta }_{xxx}{\theta }_{yy}dx\hfill \\ \hfill & ={\theta }_{yz}+{\theta }_{xy}^2-{\theta }_{xx}{\theta }_{yy}.\hfill \end{array}\end{array}$$

(84)

Thus, we obtain the well-known heavenly equation.

Let $c=0,h=f=0,$ then from (83), we have

$$e_x=0\Rightarrow e=:\alpha (y,z,t),$$

$$d_x=0\Rightarrow d=:\beta (y,z,t),$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left\{\begin{array}{c}{a}_t=e_z+b{\alpha }_y-\beta a_y-\alpha a_x,\hfill \\ b_t=d_z+b{\beta }_y-\beta b_y-\alpha b_x.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(85)

For the linear (3+1)-dimensional integrable system (84), it may be interesting to argue its different reductions. Here, we omit them.

5. Conclusions

In the paper, we applied a few loop algebras to obtain some integrable systems and Hamiltonian structures, including the well-known modified KdV equation, the heat conduction equation, the nonlinear Schrödinger equation, the (2+1)-dimensional cylindrical dissipative Zaboloskaya–Khokhlov equation, and the (3+1)-dimensional heavenly equation, and so on. There is an open problem on how to obtain new solutions of these equations through the Dbar dressing method, which will be further investigated in forthcoming days.