Generating of Nonisospectral Integrable Hierarchies via the Lie-Algebraic Recursion Scheme

Abstract

In the paper, we introduce an efficient method for generating non-isospectral integrable hierarchies, which can be used to derive a great many non-isospectral integrable hierarchies. Based on the scheme, we derive a non-isospectral integrable hierarchy by using Lie algebra and the corresponding loop algebra. It follows that some symmetries of the non-isospectral integrable hierarchy are also studied. Additionally, we also obtain a few conserved quantities of the isospectral integrable hierarchies.

1. Introduction

We have known that there exist two main approaches for constructing nonlinear systems integrable by the inverse scattering transform: the one of the Lax representation ($L_t=[A,L]$) and the one of the zero curvature representation ($U_t-V_x+[U,V]=0$) [1,2]. In [3], the authors introduced the spectral transform technique to solve certain classes of nonlinear evolution equations, and gave a thorough account also of the non-isospectral deformations of KdV-like equations [4,5]. Magri once proposed one approach for generating integrable systems [6], which was called the Lax-pair method [7,8]. Based on it, Tu [9] proposed a method for generating integrable Hamiltonian hierarchies by making use of a trace identity, which was called the Tu scheme [10,11]. Through making use of the Tu scheme, some integrable systems and the corresponding Hamiltonian structures as well as other properties were obtained, such as the works in [12,13,14,15,16]. It is well known that many different methods for generating isospectral integrable equations have been proposed [17,18,19]. However, as far non-isospectral integrable equations are concerned, fewer works were presented, as far as we know. In [20,21], the author proposed a method of constructing its corresponding non-isospearal ${\lambda }_t={\lambda }^n(n\ge 0)$ hierarchy of evolution equations closely related to $\tau$-symmetries. Generally speaking, integrable systems correspond to the isospectral (${\lambda }_t=0$) case, and mastersymmetries of integrable systems correspond to the non-isospectral ${\lambda }_t={\lambda }^n(n\ge 0)$ case. In [22], the author adopted the Lenard series method to obtain some non-isospectral integrable hierarchies under the case ${\lambda }_t={\lambda }^{m+1}M$, and found that the same spectral problem can produce two different hierarchies of soliton evolution equations.

In this article, we apply an efficient scheme to generating non-isospectral integrable hierarchies of evolution equations under the case where ${\lambda }_t={\displaystyle \sum _{j=0}^n}{k}_j\left(t\right){\lambda }^{n-j}$. Obviously, this case is a generalized expression for the case ${\lambda }_t={\lambda }^n$ [23,24]. By taking different values of the parameters in the non-isospectral integrable hierarchies, we can obtain many integrable equations, such as the coupled equations. Under obtaining non-isospectral integrable systems, their properties including Darboux transformations, exact solutions, and so on, could be investigated; a lot of such work has been done, such as the papers [25,26,27,28,29,30,31,32,33,34].

2. A Non-Isospectral Integrable Hierarchy

In this section, we derive a non-isospectral integrable hierarchy by using the Lie algebra, and obtain a Hamiltonian construction of the hierarchy via the trace identity proposed by Tu [9]. In the following, the steps for generating non-isospectral integrable hierarchies of evolution equations present

Step 1: Introducing the spectral problems

$${\psi }_x=U\psi ,U=R+u_1{e}_1\left(n\right)+\cdots +u_q{e}_q\left(n\right),$$

(1)

$${\psi }_t=V\psi ,V=A_1{e}_1\left(n\right)+\cdots +A_p{e}_p\left(n\right),$$

(2)

$${\lambda }_t=\sum _{i\ge 0}{k}_i\left(t\right){\lambda }^{-N_{i}i},$$

(3)

where the potential functions $u_1,\cdots ,u_q\in S$(the Schwartz space), and $R\left(n\right)$, $e_1\left(n\right),\cdots ,e_p\left(n\right)$∈$\tilde{G}$ satisfy that

(a)

R, $e_1,\cdots ,e_p$ are linear independent,

(b)

R is pseudoregular,

(c)

$deg\left(R\left(n\right)\right)\ge deg\left(e_i\left(n\right)\right)$, $i=1,2,\dots ,p$.

Step 2: Solving the following stationary zero curvature equation for $A_i,i=1,2,\dots ,p$:

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

(4)

It follows that one can get the compatibility condition of Equations (1) and (2)

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

(5)

Equation (4) can be broken down into

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

(6)

where

$${\lambda }_{t,+}^{\left(m\right)}={\lambda }^{N_{i}m}{\lambda }_t-{\lambda }_{t,-}^{\left(m\right)}=\sum _{i=\mu }^m{k}_i\left(t\right){\lambda }^{N_{i}m-N_{i}i+x},x=0,1,\cdots ,N_i-1;m<n.$$

Step 3: We search for a modified term ${\Delta }_n$ so that, for

$$V^{\left(n\right)}={\left({\lambda }^{N_{i}n}V\right)}_{+}+{\Delta }_n=:V_{+}^{\left(n\right)}+{\Delta }_n,$$

$$-V_x^{\left(n\right)}+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}+[U,V^{\left(n\right)}]=B_1{e}_1+\cdots +B_q{e}_q,$$

where $B_i(i=1,2,\dots ,q)\in C$.

Step 4: The non-isospectral integrable hierarchies of evolution equations could be deduced via the non-isospectral zero curvature equation

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

(7)

Step 5: The Hamiltonian structures of the hierarchies Equation (7) are sought out according to the trace identity given by Tu [9]. We will show the specific calculation process in the following:

A basis of the Lie algebra A is given by

$$A=span\{h,e,f\}$$

with $h=\frac{1}{2}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right),$ $e=\frac{1}{2}\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right),$ $f=\frac{1}{2}\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right),$ and the corresponding loop algebra is taken by

$$\tilde{A}=span\{h\left(n\right),e\left(n\right),f\left(n\right)\},$$

where $h\left(n\right)=h{\lambda }^{2n}$, $e\left(n\right)=e{\lambda }^{2n-1}$, $f\left(n\right)=f{\lambda }^{2n-1}$. It is easy to find that the commutator of $\tilde{A}$ is as follows:

$$[h\left(n\right),e\left(m\right)]=f{\lambda }^{2n+2m-1}=f(m+n),[h\left(n\right),f\left(m\right)]=e(m+n),[e\left(n\right),f\left(m\right)]=h(m+n-1),m,n\in Z,$$

where the gradations of $h\left(n\right)$, $e\left(n\right)$, and $f\left(n\right)$ are given by

$$degh\left(n\right)=2n,dege\left(n\right)=2n-1,degf\left(n\right)=2n-1,n\in Z.$$

Consider the following non-isospectral problems based on $\tilde{A}$

$${\psi }_x=U\psi ,U=h\left(1\right)+qe\left(1\right)+rf\left(1\right)=\frac{1}{2}\left(\begin{array}{cccc}{\lambda }^2& (r+q)\lambda & 0& 0\\ (r-q)\lambda & -{\lambda }^2& 0& 0\\ 0& 0& {\lambda }^2& (r+q)\lambda \\ 0& 0& (r-q)\lambda & -{\lambda }^2\end{array}\right),$$

(8)

$${\psi }_t=V\psi ,V=ah\left(0\right)+be\left(1\right)+cf\left(1\right)=\frac{1}{2}\left(\begin{array}{cccc}a& (b+c)\lambda & 0& 0\\ (-b+c)\lambda & -a& 0& 0\\ 0& 0& a& (b+c)\lambda \\ 0& 0& (-b+c)\lambda & -a\end{array}\right),$$

(9)

where ${\overline{i}}^2=-1$, $a={\displaystyle \sum _{i\ge 0}}{a}_i{\lambda }^{-2i}$, $b={\displaystyle \sum _{i\ge 0}}{b}_i{\lambda }^{-2i}$, $c={\displaystyle \sum _{i\ge 0}}{c}_i{\lambda }^{-2i}$.

It follows that we have

$$\begin{array}{cc}\hfill \frac{\partial U}{\partial \lambda }{\lambda }_t& =\frac{1}{2}\left(\begin{array}{cccc}2\lambda & r+q& 0& 0\\ r-q& -2\lambda & 0& 0\\ 0& 0& 2\lambda & r+q\\ 0& 0& r-q& -2\lambda \end{array}\right)\sum _{i\ge 0}{k}_i\left(t\right){\lambda }^{-2i+1}\hfill \\ \hfill & =\sum _{i\ge 0}{k}_i\left(t\right)[2h(1-i)+qe(1-i)+rf(1-i)].\hfill \end{array}$$

Furthermore, the following equation can be derived by taking ${\lambda }_t={\displaystyle \sum _{i\ge 0}}{k}_i\left(t\right){\lambda }^{1-2i}$ with Equation (6),

$$\left\{\begin{array}{c}{a}_{ix}=q{c}_{i+1}-r{b}_{i+1}+2k_{i+1}\left(t\right),\hfill \\ b_{ix}=c_{i+1}-r{a}_i+k_i\left(t\right)q,\hfill \\ c_{ix}=b_{i+1}-q{a}_i+k_i\left(t\right)r,\hfill \end{array}\right.$$

(10)

that is,

$$\left\{\begin{array}{c}{a}_{ix}=q{b}_{ix}-r{c}_{ix}-q^2{k}_i\left(t\right)+r^2{k}_i\left(t\right)+2k_{i+1}\left(t\right),\hfill \\ c_{i+1}=b_{ix}+r{a}_i-q{k}_i\left(t\right),\hfill \\ b_{i+1}=c_{ix}+q{a}_i-r{k}_i\left(t\right).\hfill \end{array}\right.$$

(11)

We take the initial values

$$b_0=k_0{\partial }^{-1}q,c_0=k_0{\partial }^{-1}r.$$

Then, Equation (11) admits that

$$a_0=2k_1\left(t\right)x+{\beta }_0\left(t\right),$$

$$b_1=2k_1\left(t\right)qx,c_1=2k_1\left(t\right)rx,$$

$$a_1=k_1\left(t\right)x(q^2-r^2)+2k_2\left(t\right)x+{\beta }_1\left(t\right),$$

$$b_2=k_1\left(t\right)(r+2x{r}_x)+qx(k_1\left(t\right)q^2-k_1\left(t\right)r^2+2k_2\left(t\right)),$$

$$\begin{gathered} c_2=k_1\left(t\right)(q+2x{q}_x)+rx(k_1\left(t\right)q^2-k_1\left(t\right)r^2+2k_2\left(t\right)), \\ \cdots \end{gathered}$$

where ${\beta }_0\left(t\right){\beta }_1\left(t\right)=0$ is an integral constant. Denoting that

$$V_{+}^{\left(n\right)}=\sum _{i=0}^n(a_{i}h(n-i)+b_{i}e(n+1-i)+c_{i}f(n+1-i)),$$

$$V_{-}^{\left(n\right)}=\sum _{i=n+1}^{\infty }(a_{i}h(n-i)+b_{i}e(n+1-i)+c_{i}f(n+1-i)),$$

$${\lambda }_{t,+}^{\left(n\right)}=\sum _{i=0}^n{K}_i\left(t\right){\lambda }^{2n-2i+1},{\lambda }_{t,-}^{\left(n\right)}=\sum _{i=n+1}^{\infty }{K}_i\left(t\right){\lambda }^{2n-2i+1},$$

By using Equations (8) and (9), the gradations of the left-hand side of Equation (6) are derived as:

$$deg{V}_{+}^{\left(n\right)}=:(0,1,1)\ge 0,deg\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}=:(2,1,1)\ge 1,deg\left([U,V_{+}^{\left(n\right)}]\right)=:(2,1,1;0,1,1)\ge 1,$$

which signifies that the minimum gradation of the left-hand side of Equation (6) is zero. Similarly, the gradations of the right-hand side of Equation (6) are also obtained as follows:

$$deg{V}_{-}^{\left(n\right)}=:(-2,-1,-1)\le -1,deg\frac{\partial U}{\partial \lambda }{\lambda }_{t,-}^{\left(n\right)}=:(0,-1,-1)\le 0,deg\left([U,V_{-}^{\left(n\right)}]\right)=:(2,1,1;-2,-1,-1)\le 1,$$

which indicates that the maximum gradation of the right-hand side of Equation (6) is 1. By taking these terms which have the gradations 0 and 1, one has

$$\begin{array}{c}\hfill V_{-,x}^{\left(n\right)}-\frac{\partial U}{\partial \lambda }{\lambda }_{t,-}^{\left(n\right)}-[U,V_{-}^{\left(n\right)}]=-b_{n+1}f\left(1\right)-c_{n+1}e\left(1\right)-q{c}_{n+1}h\left(0\right)+r{b}_{n+1}h\left(0\right)-2K_{n+1}\left(t\right)h\left(0\right),\end{array}$$

that is,

$$\begin{array}{c}\hfill -V_{+,x}^{\left(n\right)}+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}+[U,V_{+}^{\left(n\right)}]=-b_{n+1}f\left(1\right)-c_{n+1}e\left(1\right)-q{c}_{n+1}h\left(0\right)+r{b}_{n+1}h\left(0\right)-2K_{n+1}\left(t\right)h\left(0\right).\end{array}$$

(12)

In what follows, we takes modified term ${\Delta }_n=-a_{n}h\left(0\right)$ so that, for $V^{\left(n\right)}=V_{+}^{\left(n\right)}-a_{n}h\left(0\right)$ to obtain the non-isospectral integrable hierarchies, we have from Equation (13) that

$$\begin{array}{c}\hfill -V_x^{\left(n\right)}+\frac{\partial U}{\partial \lambda }{\lambda }_{t,+}^{\left(n\right)}+[U,V^{\left(n\right)}]=(-c_{n+1}+r{a}_n)e\left(1\right)+(-b_{n+1}+q{a}_n)f\left(1\right).\end{array}$$

Thus, Equation (7) admits the non-isospectral integrable hierarchy

$$\begin{array}{cc}\hfill u_{t_n}& ={\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}=\left(\begin{array}{c}{c}_{n+1}-r{a}_n\\ b_{n+1}-q{a}_n\end{array}\right)=\left(\begin{array}{c}{b}_{nx}-K_n\left(t\right)q\\ c_{nx}-K_n\left(t\right)r\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}0& \partial \\ \partial & 0\end{array}\right)\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)-K_n\left(t\right)\left(\begin{array}{c}q\\ r\end{array}\right)\hfill \\ \hfill & =:J_1\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)-K_n\left(t\right)\left(\begin{array}{c}q\\ r\end{array}\right),\hfill \end{array}$$

(13)

or

$$\begin{array}{cc}\hfill u_{t_n}& ={\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}=\left(\begin{array}{c}r{\partial }^{-1}r{b}_{n+1}+(1-r{\partial }^{-1}q)c_{n+1}-2r{K}_{n+1}\left(t\right)x\\ -q{\partial }^{-1}q{c}_{n+1}+(1+q{\partial }^{-1}r)b_{n+1}-2q{K}_{n+1}\left(t\right)x\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}1-r{\partial }^{-1}q& r{\partial }^{-1}r\\ -q{\partial }^{-1}q& 1+q{\partial }^{-1}r\end{array}\right)\left(\begin{array}{c}{c}_{n+1}\\ b_{n+1}\end{array}\right)+2K_{n+1}\left(t\right)x\left(\begin{array}{c}-r\\ -q\end{array}\right)\hfill \\ \hfill & =:J_2\left(\begin{array}{c}{c}_{n+1}\\ b_{n+1}\end{array}\right)+2K_{n+1}\left(t\right)x\left(\begin{array}{c}-r\\ -q\end{array}\right),\hfill \end{array}$$

(14)

where

$$J_1=\left(\begin{array}{cc}0& \partial \\ \partial & 0\end{array}\right),J_2=\left(\begin{array}{cc}1-r{\partial }^{-1}q& r{\partial }^{-1}r\\ -q{\partial }^{-1}q& 1+q{\partial }^{-1}r\end{array}\right).$$

From Equation (11), we infer that

$$\begin{array}{cc}\hfill \left(\begin{array}{c}{c}_{n+1}\\ b_{n+1}\end{array}\right)& =\left(\begin{array}{cc}-r{\partial }^{-1}r\partial & \partial +r{\partial }^{-1}q\partial \\ \partial -q{\partial }^{-1}r\partial & q{\partial }^{-1}q\partial \end{array}\right)\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)+K_n\left(t\right)\left(\begin{array}{c}r{\partial }^{-1}(-q^2+r^2)-q\\ q{\partial }^{-1}(-q^2+r^2)-r\end{array}\right)+2K_{n+1}\left(t\right)x\left(\begin{array}{c}r\\ q\end{array}\right)\hfill \\ \hfill & =:L\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)+K_n\left(t\right)Q+2K_{n+1}\left(t\right)xR,\hfill \end{array}$$

(15)

where

$$L=\left(\begin{array}{cc}-r{\partial }^{-1}r\partial & \partial +r{\partial }^{-1}q\partial \\ \partial -q{\partial }^{-1}r\partial & q{\partial }^{-1}q\partial \end{array}\right),Q=\left(\begin{array}{c}r{\partial }^{-1}(-q^2+r^2)-q\\ q{\partial }^{-1}(-q^2+r^2)-r\end{array}\right),R=\left(\begin{array}{c}r\\ q\end{array}\right).$$

Therefore, Equation (13) can be written as

$$\begin{array}{cc}\hfill u_{t_n}& ={\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}=J_1{L}^n\left(\begin{array}{c}{K}_0{\partial }^{-1}r\\ K_0{\partial }^{-1}q\end{array}\right)+J_1\sum _{i=0}^{n-1}\left(L^i{K}_{n-1-i}\left(t\right)Q\right)+2J_1\sum _{i=0}^{n-1}{L}^i{K}_{n-i}\left(t\right)xR-K_n\left(t\right)\left(\begin{array}{c}q\\ r\end{array}\right)\hfill \\ & ={\mathsf{\Phi }}^n{K}_0\left(\begin{array}{c}q\\ r\end{array}\right)+\sum _{i=0}^{n-1}{\mathsf{\Phi }}^i{J}_1{K}_{n-1-i}\left(t\right)Q+2\sum _{i=0}^{n-1}{K}_{n-i}\left(t\right){\mathsf{\Phi }}^i\partial \left(\begin{array}{c}xq\\ xr\end{array}\right)-K_n\left(t\right)\left(\begin{array}{c}q\\ r\end{array}\right),\hfill \end{array}$$

(16)

where

$$\mathsf{\Phi }=J_{1}L{J}_1^{-1}=\left(\begin{array}{cc}{q}_x{\partial }^{-1}q+q^2& \partial -q_x{\partial }^{-1}r-qr\\ \partial +r_x{\partial }^{-1}q+qr& -r_x{\partial }^{-1}r-r^2\end{array}\right).$$

(17)

When $n=1$, the non-isospectral integrable hierarchy Equation (16) becomes

$$\left\{\begin{array}{c}{q}_t=K_1(2x{q}_x+q),\hfill \\ r_t=K_1(2x{r}_x+r).\hfill \end{array}\right.$$

(18)

When $n=2$, the non-isospectral integrable hierarchy Equation (16) reduces to

$$\left\{\begin{array}{c}{q}_t=K_1{(q^{3}x-q{r}^{2}x+r+2r_{x}x)}_x+2K_2{\left(qx\right)}_x+K_2(2x{q}_x+q),\hfill \\ r_t=K_1{(-r^{3}x+r{q}^{2}x+q+2q_{x}x)}_x+K_2(2x{r}_x+r)\hfill \end{array}\right.$$

(19)

Furthermore, we focus on a format of Hamiltonian constructure of the hierarchy Equation (16) via the trace identity proposed by Tu [9]. Denoting the trace of the square matrices A and B by $<A,B>=tr\left(AB\right)$.

Equations (8) and (9) admit that

$$<V,\frac{\partial U}{\partial q}>=-b{\lambda }^2,<V,\frac{\partial U}{\partial r}>=c{\lambda }^2,<V,\frac{\partial U}{\partial \lambda }>=cr\lambda +2a\lambda -bq\lambda ,$$

which can be substituted into the trace identity

$$\frac{\delta }{\delta u}(<V,\frac{\partial U}{\partial \lambda }>)={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}<V,\frac{\partial U}{\partial q}>\\ <V,\frac{\partial U}{\partial r}>\end{array}\right)$$

gives rise to

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

(20)

It follows that one can get the following equation by comparing the two sides of the above formula

$$\frac{\delta }{\delta u}(2a_n-q{b}_n+r{c}_n)=(2-2n+\gamma )\left(\begin{array}{c}-b_n\\ c_n\end{array}\right).$$

(21)

Inserting the initial values of Equations (11) into (21), we obtain $\gamma =0$. Hence, we have

$$\left(\begin{array}{c}-b_n\\ c_n\end{array}\right)=\frac{\delta H_n}{\delta u}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right)=:M_1\left(\begin{array}{c}{c}_n\\ b_n\end{array}\right),$$

where

$$H_n=\frac{2\overline{i}{a}_n-q{b}_n-r{c}_n}{2n-2},M_1^{-1}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),M_1=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).$$

Hence, the hierarchy Equations (13) and (14) can be written as

$$u_{t_n}={\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}=J_1{M}_1^{-1}\frac{\delta H_n}{\delta u}-K_n\left(t\right)\left(\begin{array}{c}q\\ r\end{array}\right)=J_2{M}_1^{-1}\frac{\delta H_{n+1}}{\delta u}+2K_{n+1}\left(t\right)x\left(\begin{array}{c}-r\\ -q\end{array}\right).$$

(22)

It is remarkable that, when $K_n\left(t\right)=K_{n+1}\left(t\right)=0$, Equation (22) is the Hamiltonian structure of the corresponding isospectral integrable hierarchy of Equation (16).

3. Discussion on Symmetries and Conserved Quatities

In this section, we consider the K symmetries and $\tau$ symmetries of the hierarchy Equation (16), and obtain some conserved quantities of the hierarchy Equation (16) from the obtained symmetries. The way to find K symmetries and $\tau$ symmetries comes from Li and Zhu [14], who applied the isospectral and non-isospectral integrable AKNS hierarchy to construct K symmetries and $\tau$ symmetries which constitute an infinite-dimensional Lie algebra. In the following, we show the specific process.

One can find that the $\mathsf{\Phi }$ presented in Equation (17) satisfies

$${\mathsf{\Phi }}^{\prime }\left[\mathsf{\Phi }f\right]g-{\mathsf{\Phi }}^{\prime }\left[\mathsf{\Phi }g\right]f=\mathsf{\Phi }\{{\mathsf{\Phi }}^{\prime }\left[f\right]g-{\mathsf{\Phi }}^{\prime }\left[g\right]f\},$$

for $\forall f,g\in S$. Therefore, $\mathsf{\Phi }$ is the hereditary symmetry of Equation (16). In addition, we can also prove the following relation holding:

Proposition 1.

$${\mathsf{\Phi }}^{\prime }\left[K_0\right]=[K_0^{\prime },\mathsf{\Phi }],$$

(23)

where $K_0=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)=u_{t_0}$.

In fact,

$${\mathsf{\Phi }}^{\prime }\left[K_0\right]=\partial \left(\begin{array}{cc}{q}_x{\partial }^{-1}q+q{\partial }^{-1}{q}_x& -q_x{\partial }^{-1}r-q{\partial }^{-1}{r}_x\\ r_x{\partial }^{-1}q+r{\partial }^{-1}{q}_x& -r_x{\partial }^{-1}r-r{\partial }^{-1}{r}_x\end{array}\right),$$

and thus

$${\mathsf{\Phi }}^{\prime }\left[K_0\right]=\left(\begin{array}{cc}{q}_{xx}{\partial }^{-1}q+{\left(q^2\right)}_x+q_x{\partial }^{-1}{q}_x& -q_{xx}{\partial }^{-1}r-{\left(qr\right)}_x-q_x{\partial }^{-1}{r}_x\\ r_{xx}{\partial }^{-1}q+{\left(qr\right)}_x+r_x{\partial }^{-1}{q}_x& -r_{xx}{\partial }^{-1}r-{\left(r^2\right)}_x-r_x{\partial }^{-1}{r}_x\end{array}\right),$$

$$\begin{array}{cc}\hfill K_0^{\prime }\mathsf{\Phi }& =\left(\begin{array}{cc}\partial & 0\\ 0& \partial \end{array}\right)\left(\begin{array}{cc}{q}_x{\partial }^{-1}q+q^2& \partial -q_x{\partial }^{-1}r-qr\\ \partial +r_x{\partial }^{-1}q+qr& -r_x{\partial }^{-1}r-r^2\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}{q}_{xx}{\partial }^{-1}q+3q{q}_x+q^2\partial & {\partial }^2-q_{xx}{\partial }^{-1}r-2q_{x}r-q{r}_x-qr\partial \\ {\partial }^2+r_{xx}{\partial }^{-1}q+2r_{x}q+r{q}_x+qr\partial & -r_{xx}{\partial }^{-1}r-3r{r}_x-r^2\partial \end{array}\right).\hfill \end{array}$$

$$\mathsf{\Phi }{K}_0^{\prime }=\left(\begin{array}{cc}{q}_x{\partial }^{-1}q\partial +q^2\partial & {\partial }^2-q_x{\partial }^{-1}r\partial -qr\partial \\ {\partial }^2+r_x{\partial }^{-1}q\partial +qr\partial & -r_x{\partial }^{-1}r\partial -r^2\partial \end{array}\right)$$

We therefore verified that Equation (23) is correct. Owing to the Φ is a hereditary symmetry, one finds

$${\mathsf{\Phi }}^{\prime }\left[K_m\right]=[K_m^{\prime },\mathsf{\Phi }],$$

which means Φ is a strong symmetry, where $K_m={\mathsf{\Phi }}^m\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right).$

Proposition 2.

$${\mathsf{\Phi }}^{\prime }\left[xu\right]+\mathsf{\Phi }{\left(xu\right)}^{\prime }-{\left(xu\right)}^{\prime }\mathsf{\Phi }=HI,$$

(24)

where $u=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)$, $H=\left(\begin{array}{cc}0& \partial \\ \partial & 0\end{array}\right)$ and I is an identity matrix.

In fact,

$${\mathsf{\Phi }}^{\prime }\left[xu\right]=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),$$

where

$$\left\{\begin{array}{c}A=q_x{\partial }^{-1}q+x{q}_{xx}{\partial }^{-1}q+2x{q}_{x}q+q_x{\partial }^{-1}x{q}_x,\hfill \\ B=-(q_x{\partial }^{-1}r+x{q}_{xx}{\partial }^{-1}r+x{q}_{x}r+xq{r}_x+q_x{\partial }^{-1}x{r}_x),\hfill \\ C=r_x{\partial }^{-1}q+x{r}_{xx}{\partial }^{-1}q+x{r}_{x}q+xr{q}_x+r_x{\partial }^{-1}x{q}_x,\hfill \\ D=-(r_x{\partial }^{-1}r+x{r}_{xx}{\partial }^{-1}r+2x{r}_{x}r+r_x{\partial }^{-1}x{r}_x).\hfill \end{array}\right.$$

$$\mathsf{\Phi }{\left(xu\right)}^{\prime }=\left(\begin{array}{cc}x{q}^2\partial +xq{q}_x-q_x{\partial }^{-1}(q+x{q}_x)& -xqr\partial +\partial +x{\partial }^2-xr{q}_x+q_x{\partial }^{-1}(r+x{r}_x)\\ xqr\partial +\partial +x{\partial }^2+xq{r}_x-r_x{\partial }^{-1}(q+x{q}_x)& -x{r}^2\partial -xr{r}_x+r_x{\partial }^{-1}(r+x{r}_x)\end{array}\right),$$

$${\left(xu\right)}^{\prime }\mathsf{\Phi }=\left(\begin{array}{cc}x{q}_{xx}{\partial }^{-1}q+3xq{q}_x+x{q}^2\partial & x{\partial }^2-x{q}_{xx}{\partial }^{-1}r-2xr{q}_x-xq{r}_x-xqr\partial \\ x{\partial }^2+x{r}_{xx}{\partial }^{-1}q+2xq{r}_x+xr{q}_x+xqr\partial & -x{r}_{xx}{\partial }^{-1}r-3xr{r}_x-x{r}^2\partial \end{array}\right),$$

where

$${\left(xu\right)}^{\prime }\left[\sigma \right]=\frac{d}{dϵ}{\mid }_{ϵ=0}\left(\begin{array}{c}x{(q+ϵ{\sigma }_1)}_x\\ x{(q+ϵ{\sigma }_2)}_x\end{array}\right)=x\partial \left(\begin{array}{c}{\sigma }_1\\ {\sigma }_2\end{array}\right)⟹{\left(xu\right)}^{\prime }=\left(\begin{array}{cc}x\partial & 0\\ 0& x\partial \end{array}\right).$$

Thus, Equation (24) holds.

Proposition 3.

$$[K_1,xu]=[\mathsf{\Phi }u,xu]=Hu+K_1,$$

(25)

where $u=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)$, $H=\left(\begin{array}{cc}0& \partial \\ \partial & 0\end{array}\right)$ and $K_1=\mathsf{\Phi }u$.

In fact,

$$\mathsf{\Phi }u=\left(\begin{array}{c}{r}_{xx}+\frac{1}{2}{q}_x(q^2-r^2)-qr{r}_x+q^2{q}_x\\ q_{xx}+\frac{1}{2}{r}_x(q^2-r^2)+qr{q}_x-r^2{r}_x\end{array}\right),$$

$${\left(\mathsf{\Phi }u\right)}^{\prime }=\left(\begin{array}{cc}\frac{1}{2}(q^2-r^2)\partial +3q{q}_x+q^2\partial -r{r}_x& {\partial }^2-qr\partial -{\left(qr\right)}_x\\ {\partial }^2+qr\partial +{\left(qr\right)}_x& \frac{1}{2}(q^2-r^2)\partial -3r{r}_x-r^2\partial +q{q}_x\end{array}\right),$$

$${\left(\mathsf{\Phi }u\right)}^{\prime }\left(\begin{array}{c}x{q}_x\\ x{r}_x\end{array}\right)=\left(\begin{array}{c}\frac{1}{2}(q^2-r^2)\partial \left(x{q}_x\right)+3xq{q}_x^2+q^2\partial \left(x{q}_x\right)-xr{r}_x{q}_x+{\partial }^2\left(x{r}_x\right)-qr\partial \left(x{r}_x\right)-x{r}_x{\left(qr\right)}_x\\ {\partial }^2\left(x{q}_x\right)+qr\partial \left(x{q}_x\right)+x{q}_x{\left(qr\right)}_x+\frac{1}{2}(q^2-r^2)\partial \left(x{r}_x\right)-3xr{r}_x^2-r^2\partial \left(x{r}_x\right)+x{r}_{x}q{q}_x\end{array}\right),$$

Then, we have

$${\left(xu\right)}^{\prime }\left[\mathsf{\Phi }u\right]=\left(\begin{array}{c}x\partial (r_{xx}+\frac{1}{2}{q}_x(q^2-r^2)-qr{r}_x+q^2{q}_x)\\ x\partial (q_{xx}+\frac{1}{2}{r}_x(q^2-r^2)+qr{q}_x-r^2{r}_x)\end{array}\right),$$

$$[\mathsf{\Phi }u,xu]={\left(\mathsf{\Phi }u\right)}^{\prime }\left[xu\right]-{\left(xu\right)}^{\prime }\left[\mathsf{\Phi }u\right]=\left(\begin{array}{cc}0& \partial \\ \partial & 0\end{array}\right)\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)+\left(\begin{array}{c}{r}_{xx}+\frac{1}{2}{q}_x(q^2-r^2)-qr{r}_x+q^2{q}_x\\ q_{xx}+\frac{1}{2}{r}_x(q^2-r^2)+qr{q}_x-r^2{r}_x\end{array}\right)=Hu+K_1.$$

We therefore verified that Equation (25) is correct.

Proposition 4.

$$[K_m,K_n]=0,m,n=0,1,2,\cdots$$

(26)

where $K_m={\mathsf{\Phi }}^{m}u$, $K_n={\mathsf{\Phi }}^{n}u$.

Proposition 5.

$$[{\mathsf{\Phi }}^{m}xu,xu]=m{\mathsf{\Phi }}^{m-1}\left(xu\right).$$

The proofs of Propositions 4 and 5 were presented in [23].

From the above results, one have

$$[{\mathsf{\Phi }}^{m}xu,{\mathsf{\Phi }}^{n}xu]=(m-n){\mathsf{\Phi }}^{m+n-1}\left(xu\right),m=0,1,2,\cdots ;n=0,1,2,\cdots .$$

We can find that $\{{\mathsf{\Phi }}^{n}u,{\mathsf{\Phi }}^{m}xu\}$ can not constitute a Lie algebra from Equation (25). However, $\{{\mathsf{\Phi }}^{n}u,n=0,1,2,\cdots \}$ and $\{{\mathsf{\Phi }}^{n}xu,n=0,1,2,\cdots \}$ constitute the infinite-dimensional Lie algebra, respectively based on the above analysis.

Now, we will derive some conserved quantities of Tu isospectral hierarchy

$$u_{t_n}={\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}={\mathsf{\Phi }}^n\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right).$$

(27)

Definition 1

([14,16]). If we have known the integrable hierarchy $u_t=K_n\left(u\right)$, then the v satisfied the following equation:

$$\frac{dv}{dt}+K^{\prime *}v=0,$$

(28)

which is called the conserved covariance, where $K^{\prime }$ is the linearized operator of K, and $K^{\prime *}$ denotes a conjugate operator of $K^{\prime }$.

Proposition 6

([13,16]). If σ is a symmetry of Equation $u_t=K_n\left(u\right)$, v is its conserved covariance, then we have

$${\int }_{-\infty }^{\infty }v\sigma dx=<v,\sigma >,$$

which is independent of time t, that is, $\frac{d}{dt}<v,\sigma >=0$.

Definition 2

([13,14,16]). If $F^{\prime }f=<v,f>$, for $\forall f\in S$, then v is called the gradient of the functional F, which is denoted by $v=\frac{\delta F}{\delta u}$.

Proposition 7

([16]). If $v^{\prime }=v^{\prime *}$, then v is the gradient of the following functional

$$F={\int }_0^1<v\left(\lambda u\right),u>d\lambda .$$

(29)

According to the symbols above, we can deduce:

Proposition 8

([13,14]). If I is a conserved quality of the hierarchy $u_t=K_n\left(u\right)$, and the conserved covariance v satisfies

$$I^{\prime }{K}_n=<v,K_n>,$$

then one has

$$\frac{\partial I}{\partial t}+<v,K_n>=0,$$

that is,

$$\frac{\partial v}{\partial t}+K_n^{\prime *}v+v^{\prime }{K}_n=0.$$

Therefore, we deduce the following conserved quantities related to the integrable hierarchy $u_t=K_n\left(u\right)$

$$I_m={\int }_0^1<{\partial }_x^{-1}{K}_m\left(\lambda u\right),u>d\lambda .$$

(30)

Moreover, a few conserved quantities are also derived for the integrable hierarchy Equation (27) as follows:

$$K_0={\mathsf{\Phi }}^{0}u=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}-r_x\\ q_x\end{array}\right),$$

$$I_0={\int }_0^1<{\partial }_x^{-1}{K}_0\left(\lambda u\right),u>d\lambda ={\int }_0^1<{\left[\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{q}_x\lambda \\ r_x\lambda \end{array}\right)\right]}^T,\left(\begin{array}{c}q\\ r\end{array}\right)>d\lambda ={\int }_{-\infty }^{\infty }(q_{x}r-r_{x}q)dx,$$

$$K_1=\mathsf{\Phi }u=\left(\begin{array}{c}{r}_{xx}+\frac{1}{2}{q}_x(q^2-r^2)-qr{r}_x+q^2{q}_x\\ q_{xx}+\frac{1}{2}{r}_x(q^2-r^2)+qr{q}_x-r^2{r}_x\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}-q_{xx}-\frac{1}{2}{r}_x(q^2-r^2)-qr{q}_x+r^2{r}_x\\ r_{xx}+\frac{1}{2}{q}_x(q^2-r^2)-qr{r}_x+q^2{q}_x\end{array}\right),$$

$$\begin{array}{cc}\hfill I_1=& {\int }_0^1<{\left[\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{r}_{xx}\lambda +\frac{1}{2}{q}_x(q^2-r^2){\lambda }^3-qr{r}_x{\lambda }^3+q^2{q}_x{\lambda }^3\\ q_{xx}\lambda +\frac{1}{2}{r}_x(q^2-r^2){\lambda }^3+qr{q}_x{\lambda }^3-r^2{r}_x{\lambda }^3\end{array}\right)\right]}^T,\left(\begin{array}{c}q\\ r\end{array}\right)>d\lambda \hfill \\ & ={\int }_{-\infty }^{\infty }[\frac{1}{2}(-q{q}_{xx}+r{r}_{xx})+\frac{1}{8}(q^2-r^2)(q_{x}r-r_{x}q)]dx,\hfill \\ & ⋮\end{array}$$

$$I_k={\int }_{-\infty }^{\infty }<\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)K_k\left(\lambda u\right),\left(\begin{array}{c}q\\ r\end{array}\right)>d\lambda .$$