Generating nonisospectral integrable hierarchies via a new scheme

Abstract

In the paper, an efficient and straightforward method for generating nonisospectral integrable hierarchies is introduced. It follows that we consider the application related to Lie algebra $\operatorname{gl}(3)$ based on the method. Then, we derive a nonisospectral integrable hierarchy whose some new symmetries are also investigated. In addition, a few conserved quantities of the nonisospectral integrable hierarchies are also obtained.

1 Introduction

We know that one approach for generating integrable systems was proposed by Magri [1], which was called the Lax-pair method [2, 3]. Based on it, Tu [4] proposed a method for generating integrable Hamiltonian hierarchies, which was called the Tu scheme by Ma [5]. 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 [6–10]. It is well known that many different methods for generating isospectral integrable equations have been proposed [11–15]. However, as nonisospectral integrable equations are concerned, fewer works have been presented, as far as we know. Ma [16, 17] applied Lax equations to work out some nonisospectral integrable hierarchy under the case of $\lambda _{t}=\lambda ^{n}$ ($n>0$). Qiao [18] adopted the Lenard series method to obtain some nonisospectral integrable hierarchies under the case $\lambda _{t}=\lambda ^{m+1}M$. The aim of this paper is to apply an efficient scheme to generate nonisospectral integrable hierarchies of evolution equations under the case where $\lambda _{t}=\sum_{j=0}^{n}k_{j}(t)\lambda ^{n-j}$. Obviously, this case is a generalized expression for the case $\lambda _{t}=\lambda ^{n}$ [19, 20]. Under obtaining nonisospectral integrable systems, some of their properties, including Darboux transformations, exact solutions, and so on, could be studied [21–26]. We first recall some fundamental facts.

Let G be a finite-dimensional Lie algebra over the complex set C, $\widetilde{G}=G\otimes C[\lambda , \lambda ^{-1}]$ be the corresponding loop algebra, where $C[\lambda , \lambda ^{-1}]$ stands for a set of Laurent polynomials in the parameter λ. Suppose that $\{e_{1},\ldots,e_{p}\}$ is a basis of G, then the basis of the loop algebra G̃ can be chosen as $\{e_{1}(n),\ldots,e_{p}(n)\}$, where $e_{i}(n)=e_{i}\lambda ^{N_{i}n}$, $N_{i}=1,2,\ldots$ , $n\in Z$.

Definition 1

One basis element $R\in \widetilde{G}$ is called pseudoregular if the following conditions hold:

• (1) $\widetilde{G}=\operatorname{Ker} \operatorname{ad} R\oplus \operatorname{Im} \operatorname{ad} R$,

• (2) $\ker \operatorname{ad} R$ is commutative, where

$\operatorname{Ker} \operatorname{ad} R=\{x\mid x\in \widetilde{G}, [x,R]=0\}$, $\operatorname{Im} \operatorname{ad} R=\{x\mid \exists y\in \widetilde{G}, x=[y,R]\}$.

Definition 2

For any basis element $e_{i}(n)$ ($i=1,2,\ldots,p$), we define its gradation by

$$ \deg \bigl(e_{i}(n)\bigr)=N_{i}n. $$

(1)

Obviously, for $\forall g\in \widetilde{G}$, g can be expressed by $g=\sum_{n}k_{n}e_{i}(n)=:\sum_{n}g_{n}$, $k_{n}$ are constants. We can decompose g into two parts as follows:

$$ g_{+}=\sum_{n\geq \mu }g_{n}, \qquad g_{-}=\sum_{n< \mu }g_{n}, $$

and call $g_{+}$ the positive part of g, $\mu \in Z$ is some chosen integer.

In the following, the steps for generating nonisospectral integrable hierarchies of evolution equations are presented.

Step 1: By using the loop algebra G̃, we introduce the spectral problems

$$\begin{aligned}& \psi _{x}=U\psi ,\quad U=R+u_{1}e_{1}(n)+ \cdots +u_{q}e_{q}(n), \end{aligned}$$

(2)

$$\begin{aligned}& \psi _{t}=V\psi ,\quad V=A_{1}e_{1}(n)+ \cdots +A_{p}e_{p}(n), \end{aligned}$$

(3)

$$\begin{aligned}& \lambda _{t}=\sum_{i\geq 0}k_{i}(t) \lambda ^{-N_{i}i}, \end{aligned}$$

(4)

where the potential functions $u_{1},\ldots,u_{q}\in S$ (the Schwartz space), and $R(n)$, $e_{1}(n),\ldots, e_{p}(n)\in \widetilde{G}$ satisfy that

• (a) R, $e_{1},\ldots,e_{p}$ are linear independent,

• (b) R is pseudoregular,

• (c) $\deg (R(n))\geq \deg (e_{i}(n))$, $i=1,2,\ldots,p$.

Step 2: Solving the following stationary zero curvature equation for $A_{i}$, $i=1,2,\ldots,p$:

$$ V_{x}=\frac{\partial U}{\partial \lambda }\lambda _{t}+[U,V]. $$

(5)

It follows that one can get the compatibility condition of (2) and (3)

$$ \frac{\partial U}{\partial u}u_{t}+ \frac{\partial U}{\partial \lambda }\lambda _{t}-V_{x}+[U,V]=0. $$

(6)

Equation (6) can be broken down into

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

(7)

where

$$ \lambda _{t,+}^{(m)}=\lambda ^{N_{i}m}\lambda _{t}-\lambda _{t,-}^{(m)}= \sum _{i=\mu }^{m}k_{i}(t)\lambda ^{N_{i}m-N_{i}i+x}, \quad x=0,1,\ldots,N_{i}-1; m< n. $$

Step 3: Choose $\triangle _{n}\in \widetilde{G}$ so that

$$\begin{aligned}& V^{(n)}=\bigl(\lambda ^{N_{i}n}V\bigr)_{+}+\triangle _{n}=:V_{+}^{(n)}+ \triangle _{n}, \\& -V_{x}^{(n)}+\frac{\partial U}{\partial \lambda }\lambda _{t,+}^{(n)}+ \bigl[U,V^{(n)}\bigr]=B_{1}e_{1}+ \cdots +B_{q}e_{q}, \end{aligned}$$

where $B_{i}\ (i=1,2,\ldots,q) \in C$.

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

$$ \frac{\partial U}{\partial u}u_{t}+ \frac{\partial U}{\partial \lambda }\lambda _{t,+}^{(n)}-V_{x}^{(n)}+ \bigl[U,V^{(n)}\bigr]=0. $$

(8)

Step 5: The Hamiltonian structures of hierarchies (8) are sought out according to the trace identity given by Tu [4].

2 A nonisospectral integrable hierarchy of evolution equations

A basis of the Lie algebras $\operatorname{gl}(3)$ is given by

$$ \operatorname{gl}(3)=\operatorname{span}\{h, e, f\} $$

with

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

And the corresponding loop algebra is taken by

$$ \widetilde{\operatorname{gl}}(3)=\operatorname{span}\bigl\{ h(n), e(n), f(n)\bigr\} , $$

where $h(n)=h\lambda ^{2n}$, $e(n)=e\lambda ^{2n-1}$, $f(n)=f\lambda ^{2n-1}$.

After simple calculations, one can find

$$ \begin{gathered} \bigl[h(n),e(m)\bigr]=f\lambda ^{2n+2m-1}=f(m+n),\qquad \bigl[h(n),f(m) \bigr]=-e(m+n), \\ \bigl[e(n),f(m)\bigr]=h(m+n-1), \quad m,n\in Z, \end{gathered} $$

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

$$ \deg h(n)=2n, \qquad \deg e(n)=2n-1,\qquad \deg f(n)=2n-1, \quad n\in Z. $$

We consider the following nonisospectral problems based on $\widetilde{\operatorname{gl}}(3)$:

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

(9)

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

(10)

where $\overline{i}^{2}=-1$, $a=\sum_{i\geq 0}a_{i}\lambda ^{-2i}$, $b=\sum_{i\geq 0}b_{i}\lambda ^{-2i}$, $c=\sum_{i\geq 0}c_{i}\lambda ^{-2i}$.

It follows that we obtain

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

Furthermore, the following equation can be derived by taking $\lambda _{t}=\sum_{i\geq 0}k_{i}(t)\lambda ^{1-2i}$ with Eq. (6):

$$ \textstyle\begin{cases} a_{ix}=qc_{i+1}-rb_{i+1}-2\overline{i}k_{i+1}(t), \\ b_{ix}=\overline{i}c_{i+1}+ra_{i}+k_{i}(t)q, \\ c_{ix}=-\overline{i}b_{i+1}-qa_{i}+k_{i}(t)r, \end{cases} $$

(11)

that is,

$$ \textstyle\begin{cases} a_{ix}=-\overline{i}(qb_{ix}+rc_{ix}-q^{2}k_{i}(t)-r^{2}k_{i}(t)+2k_{i+1}(t)), \\ c_{i+1}=\overline{i}(-b_{ix}+ra_{i}+qk_{i}(t)), \\ b_{i+1}=\overline{i}(c_{ix}+qa_{i}-rk_{i}(t)). \end{cases} $$

(12)

In terms of Eq. (12), we take the initial values

$$ b_{0}=k_{0}\partial ^{-1}q, \qquad c_{0}=k_{0}\partial ^{-1}r, $$

and then one has

$$ a_{0}=-2\overline{i}k_{1}(t)x+\beta _{0}(t), $$

where $\beta _{0}(t)=0$ is an integral constant. From (12), we deduce that

$$\begin{aligned}& b_{1}=2k_{1}(t)qx,\qquad c_{1}=2k_{1}(t)rx, \\& a_{1}=-\overline{i}k_{1}(t)x\bigl(q^{2}+r^{2} \bigr)-2\overline{i}k_{2}(t)x+ \beta _{1}(t), \\& b_{2}=\overline{i}k_{1}(t) (r+2xr_{x})+qx \bigl(k_{1}(t)q^{2}+k_{1}(t)r^{2}+2k_{2}(t) \bigr), \\& c_{2}=-\overline{i}k_{1}(t) (q+2xq_{x})+rx \bigl(k_{1}(t)q^{2}+k_{1}(t)r^{2}+2k_{2}(t) \bigr), \\& \cdots, \end{aligned}$$

where $\beta _{1}(t)=0$ is an integral constant. Denote that

$$\begin{aligned}& V_{+}^{(n)}=\sum_{i=0}^{n} \bigl(a_{i}h(n-i)+b_{i}e(n+1-i)+c_{i}f(n+1-i) \bigr), \\& V_{-}^{(n)}=\sum_{i=n+1}^{\infty } \bigl(a_{i}h(n-i)+b_{i}e(n+1-i)+c_{i}f(n+1-i) \bigr), \\& \lambda _{t,+}^{(n)}=\sum_{i=0}^{n}K_{i}(t) \lambda ^{2n-2i+1}, \qquad \lambda _{t,-}^{(n)}=\sum _{i=n+1}^{\infty }K_{i}(t) \lambda ^{2n-2i+1}. \end{aligned}$$

In what follows, the gradations of the left-hand side of (7) can be obtained by using (1), (9), and (10)

$$\begin{aligned}& \deg V_{+}^{(n)}=:(0,1,1)\geq 0, \qquad \deg \frac{\partial U}{\partial \lambda }\lambda _{t,+}^{(n)}=:(2,1,1) \geq 1, \\& \deg \bigl(\bigl[U,V_{+}^{(n)}\bigr]\bigr)=:(2,1,1;0,1,1)\geq 1, \end{aligned}$$

which indicates that the minimum gradation of the left-hand side of (7) is zero. Additionally, we also obtain the gradations of the right-hand side of (7) as follows:

$$\begin{aligned}& \deg V_{-}^{(n)}=:(-2,-1,-1)\leq -1, \qquad \deg \frac{\partial U}{\partial \lambda }\lambda _{t,-}^{(n)}=:(0,-1,-1) \leq 0, \\& \deg \bigl(\bigl[U,V_{-}^{(n)}\bigr]\bigr)=:(2,1,1;-2,-1,-1) \leq 1, \end{aligned}$$

which means the maximum gradation of the right-hand side of (7) is 1. Thus, we further infer the following equation by taking these terms which have the gradations 0 and 1:

$$\begin{aligned}& V_{-,x}^{(n)}-\frac{\partial U}{\partial \lambda } \lambda _{t,-}^{(n)}-\bigl[U,V_{-}^{(n)} \bigr] \\& \quad =\overline{i}b_{n+1}f(1)- \overline{i}c_{n+1}e(1)-qc_{n+1}h(0) +rb_{n+1}h(0)+2\overline{i}K_{n+1}(t)h(0), \end{aligned}$$

that is,

$$ \begin{aligned}[b] & -V_{+,x}^{(n)}+ \frac{\partial U}{\partial \lambda } \lambda _{t,+}^{(n)}+\bigl[U,V_{+}^{(n)} \bigr] \\ &\quad =\overline{i}b_{n+1}f(1)- \overline{i}c_{n+1}e(1)-qc_{n+1}h(0) +rb_{n+1}h(0)+2\overline{i}K_{n+1}(t)h(0). \end{aligned} $$

(13)

In order to obtain the nonisospectral integrable hierarchies, we take the modified term $\triangle _{n}=-a_{n}h(0)$ so that for $V^{(n)}=V_{+}^{(n)}-a_{n}h(0)$, we have from (13) that

$$ \begin{aligned} -V_{x}^{(n)}+\frac{\partial U}{\partial \lambda } \lambda _{t,+}^{(n)}+\bigl[U,V^{(n)}\bigr]=(- \overline{i}c_{n+1}-ra_{n})e(1)+( \overline{i}b_{n+1}+qa_{n})f(1). \end{aligned} $$

Therefore, the nonisospectral integrable hierarchy is derived by Eq. (8) as follows:

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

(14)

or

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

(15)

where

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

Based on (12), one has

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

(16)

where

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

Hence, (14) can be written as

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

(17)

where

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

(18)

When $n=1$, the nonisospectral integrable hierarchy (17) becomes

$$ \textstyle\begin{cases} q_{t}=2K_{1}(qx)_{x}+K_{1}q, \\ r_{t}=2K_{1}(rx)_{x}+K_{1}r. \end{cases} $$

(19)

When $n=2$, the nonisospectral integrable hierarchy (17) reduces to

$$ \textstyle\begin{cases} q_{t}=K_{1}(q^{3}x+qr^{2}x+\overline{i}r+2\overline{i}r_{x}x)_{x}+2K_{2}(qx)_{x}-K_{2}q, \\ r_{t}=K_{1}(r^{3}x+rq^{2}x-\overline{i}q-2\overline{i}q_{x}x)_{x}+2K_{2}(rx)_{x}-K_{2}r. \end{cases} $$

(20)

Additionally, we focus on a format of Hamiltonian construction of hierarchy (17) via the trace identity proposed by Tu [4]. Denote the trace of the square matrices A and B by $\langle A,B\rangle =\operatorname{tr}(AB)$.

Equation (9) and Eq. (10) admit that

$$ \biggl\langle V,\frac{\partial U}{\partial q}\biggr\rangle =-2b\lambda ^{2}, \qquad \biggl\langle V, \frac{\partial U}{\partial r}\biggr\rangle =-2c\lambda ^{2},\qquad \biggl\langle V, \frac{\partial U}{\partial \lambda }\biggr\rangle =-2cr \lambda +4\overline{i}a \lambda -2bq\lambda , $$

which can be substituted into the trace identity to get

$$\begin{array}{c}\frac{\delta }{\delta u}\left(〈V,\frac{\partial U}{\partial \lambda }〉\right)={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }{\lambda }^{\gamma }\left(\begin{array}{c}〈V,\frac{\partial U}{\partial q}〉\\ 〈V,\frac{\partial U}{\partial r}〉\end{array}\right),\\ \frac{\delta }{\delta u}(-2cr\lambda +4\bar{i}a\lambda -2bq\lambda )={\lambda }^{-\gamma }\frac{\partial }{\partial \lambda }\left(\begin{array}{c}-2b{\lambda }^{2+\gamma }\\ -2c{\lambda }^{2+\gamma }\end{array}\right).\end{array}$$

(21)

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

$$\frac{\delta }{\delta u}(4\bar{i}{a}_n-2q{b}_n-2r{c}_n)=-2(2-2n+\gamma )\left(\begin{array}{c}{b}_n\\ c_n\end{array}\right).$$

(22)

One can find $\gamma =0$ via substituting the initial values of (12) into (22), and then we obtain

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

where

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

Hence, hierarchies (14) and (15) can be written as

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

(23)

It is remarkable that when $K_{n}(t)=K_{n+1}(t)=0$, (23) is the Hamiltonian structure of the corresponding isospectral integrable hierarchy of (17).

3 Discussion on symmetries and conserved quantities

In [8], the authors applied the isospectral and nonisospectral integrable AKNS hierarchy to construct K symmetries and τ symmetries, which constitute an infinite-dimensional Lie algebra. Thus, we also study the K symmetries and τ symmetries of hierarchy (17) in this section. Moreover, some conserved qualities of hierarchy (17) can be found based on the obtained symmetries. After simple calculations, one can find that Φ presented in (18) satisfies

$$ \varPhi '[\varPhi f]g-\varPhi '[\varPhi g]f=\varPhi \bigl\{ \varPhi '[f]g-\varPhi '[g]f\bigr\} $$

for $\forall f,g\in S$. Thus, Φ is the hereditary symmetry of (17). In what follows we can also prove that the following relation holds.

Proposition 1

$$ \varPhi '[K_{0}]=\bigl[K_{0}', \varPhi \bigr], $$

(24)

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

In fact,

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

for $\forall f=(f_{1},f_{2})^{T}\in S$, we have

$$\begin{array}{c}{\Phi }^{\prime }[K_0]f=\left(\begin{array}{c}{q}_{xx}{\partial }^{-1}q{f}_1+{(q^2)}_x{f}_1+q_x{\partial }^{-1}{q}_x{f}_1+q_{xx}{\partial }^{-1}r{f}_2+{(qr)}_x{f}_2+q_x{\partial }^{-1}{r}_x{f}_2\\ r_{xx}{\partial }^{-1}q{f}_1+{(qr)}_x{f}_1+r_x{\partial }^{-1}{q}_x{f}_1+r_{xx}{\partial }^{-1}r{f}_2+{(r^2)}_x{f}_2+r_x{\partial }^{-1}{r}_x{f}_2\end{array}\right),\hfill \\ [K_0^{\prime },\Phi ]\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)\hfill \\ =K_0^{\prime }\Phi \left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)-\Phi K_0^{\prime }\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)\hfill \\ =\left(\begin{array}{cc}\partial & 0\\ 0& \partial \end{array}\right)\partial \left(\begin{array}{cc}q{\partial }^{-1}q& i+q{\partial }^{-1}r\\ -i+r{\partial }^{-1}q& r{\partial }^{-1}r\end{array}\right)\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)-\Phi \left(\begin{array}{c}{f}_{1x}\\ f_{2x}\end{array}\right)\hfill \\ =\left(\begin{array}{c}{q}_{xx}{\partial }^{-1}q{f}_1+3q{q}_x{f}_1-q_x{\partial }^{-1}q\partial f_1+q_{xx}{\partial }^{-1}r{f}_2+q_{x}r{f}_2+{(qr)}_x{f}_2-q_x{\partial }^{-1}r{f}_{2x}\\ r_{xx}{\partial }^{-1}q{f}_1+r_{x}q{f}_1+q_{x}r{f}_1-r_x{\partial }^{-1}q\partial f_1+q{r}_x{f}_1+r_{xx}{\partial }^{-1}r{f}_2+3r{r}_x{f}_2-r_x{\partial }^{-1}r{f}_{2x}\end{array}\right).\hfill \end{array}$$

We therefore verified that (24) is correct. It follows that we can get the following equation because Φ is a hereditary symmetry:

$$ \varPhi '[K_{m}]=\bigl[K_{m}', \varPhi \bigr], $$

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

Proposition 2

$$ \varPhi '[xu]+\varPhi (xu)'-(xu)' \varPhi =HI, $$

(25)

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

In fact,

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

where

$$\begin{array}{c}\{\begin{array}{c}A=q_x{\partial }^{-1}q+x{q}_{xx}{\partial }^{-1}q+2x{q}_{x}q+q_x{\partial }^{-1}x{q}_x,\hfill \\ B=q_x{\partial }^{-1}r+x{q}_{xx}{\partial }^{-1}r+x{q}_{x}r+xq{r}_x+q_x{\partial }^{-1}x{r}_x,\hfill \\ C=r_x{\partial }^{-1}q+x{r}_{xx}{\partial }^{-1}q+x{r}_{x}q+xr{q}_x+r_x{\partial }^{-1}x{q}_x,\hfill \\ D=r_x{\partial }^{-1}r+x{r}_{xx}{\partial }^{-1}r+2x{r}_{x}r+r_x{\partial }^{-1}x{r}_x.\hfill \end{array}\hfill \\ \Phi {(xu)}^{\prime }=\left(\begin{array}{cc}x{q}^2\partial +xq{q}_x-q_x{\partial }^{-1}(q+x{q}_x)& xqr\partial +\bar{i}\partial +\bar{i}x{\partial }^2+xr{q}_x-q_x{\partial }^{-1}(r+x{r}_x)\\ xqr\partial -\bar{i}\partial -\bar{i}x{\partial }^2+xq{r}_x-r_x{\partial }^{-1}(q+x{q}_x)& x{r}^2\partial +xr{r}_x-r_x{\partial }^{-1}(r+x{r}_x)\end{array}\right),\hfill \\ {(xu)}^{\prime }\Phi =\left(\begin{array}{cc}x{q}_{xx}{\partial }^{-1}q+3xq{q}_x+x{q}^2\partial & \bar{i}x{\partial }^2+x{q}_{xx}{\partial }^{-1}r+2xr{q}_x+xq{r}_x+xqr\partial \\ -\bar{i}x{\partial }^2+x{r}_{xx}{\partial }^{-1}q+2xq{r}_x+xr{q}_x+xqr\partial & x{r}_{xx}{\partial }^{-1}r+3xr{r}_x+x{r}^2\partial \end{array}\right),\hfill \end{array}$$

where

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

We therefore verified that (25) is correct.

Proposition 3

$$ [K_{1},xu]=[\varPhi u,xu]=Hu+K_{1}, $$

(26)

where$u=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)$, $H=\left(\begin{array}{cc}0& \bar{i}\partial \\ -\bar{i}\partial & 0\end{array}\right)$, and$K_{1}=\varPhi u$.

In fact,

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

Then we have

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

We therefore verified that (26) is correct.

Proposition 4

$$ [K_{m},K_{n}]=0, \quad m,n=0,1,2,\ldots, $$

(27)

where$K_{m}=\varPhi ^{m}u$, $K_{n}=\varPhi ^{n}u$.

Proposition 5

$$ \bigl[\varPhi ^{m}xu,xu\bigr]=m\varPhi ^{m-1}(xu). $$

The proofs of Proposition 4 and Proposition 5 were presented in [20].

From the above results we can get

$$ \bigl[\varPhi ^{m}xu,\varPhi ^{n}xu\bigr]=(m-n)\varPhi ^{m+n-1}(xu),\quad m=0,1,2,\ldots ; n=0,1,2,\ldots . $$

From (26), one can find that $\{\varPhi ^{n}u, \varPhi ^{m}xu\}$ cannot constitute a Lie algebra. However, $\{\varPhi ^{n}u, n=0,1,2,\ldots \}$ and $\{\varPhi ^{n}xu, n=0,1,2,\ldots \}$ constitute the infinite-dimensional Lie algebra, respectively based on the above analysis.

Next we derive some conserved qualities of Tu isospectral hierarchy

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

(28)

Definition 3

([11, 12, 14])

If we have known the integrable hierarchy $u_{t}=K_{n}(u)$, then v satisfying the following equation

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

(29)

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

Proposition 6

([14])

Ifσis a symmetry of Eq. $u_{t}=K_{n}(u)$, vis its conserved covariance, then we have

$$ \int _{-\infty }^{\infty }v\sigma \,dx=\langle v,\sigma \rangle , $$

which is independent of timet, that is, $\frac{d}{dt}\langle v,\sigma \rangle =0$.

Definition 4

([11, 12, 14])

If $F'f=\langle v,f\rangle$ 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

([14])

If$v'=v^{\prime \ast }$, thenvis the gradient of the following functional:

$$ F= \int _{0}^{1}\bigl\langle v(\lambda u),u\bigr\rangle \,d\lambda . $$

(30)

According to the symbols above, we can deduce the following.

Proposition 8

([11, 12])

IfIis a conserved quality of the hierarchy$u_{t}=K_{n}(u)$, and the conserved covariancevsatisfies

$$ I'K_{n}=\langle v,K_{n}\rangle , $$

then one obtains

$$ \frac{\partial I}{\partial t}+\langle v,K_{n}\rangle =0, $$

that is,

$$ \frac{\partial v}{\partial t}+K_{n}^{\prime \ast }v+v'K_{n}=0. $$

Hence, we derive the following conserved qualities related to the integrable hierarchy $u_{t}=K_{n}(u)$:

$$ I_{m}= \int _{0}^{1}\bigl\langle \partial _{x}^{-1}K_{m}(\lambda u),u\bigr\rangle \,d\lambda . $$

(31)

In addition, a few conserved qualities are also derived for the integrable hierarchy (28) as follows:

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

where

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

Moreover, we have

$$\begin{array}{c}\begin{array}{rl}{K}_1& =\Phi u=\left(\begin{array}{c}\bar{i}{r}_{xx}+\frac{1}{2}{q}_x(q^2+r^2)+qr{r}_x+q^2{q}_x\\ -\bar{i}{q}_{xx}+\frac{1}{2}{r}_x(q^2+r^2)+qr{q}_x+r^2{r}_x\end{array}\right)\\ & =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}\bar{i}{q}_{xx}-\frac{1}{2}{r}_x(q^2+r^2)-qr{q}_x-r^2{r}_x\\ \bar{i}{r}_{xx}+\frac{1}{2}{q}_x(q^2+r^2)+qr{r}_x+q^2{q}_x\end{array}\right),\end{array}\hfill \\ \begin{array}{rl}{I}_1=& {\int }_0^1〈{\left[\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}\bar{i}{r}_{xx}\lambda +\frac{1}{2}{q}_x(q^2+r^2){\lambda }^3+qr{r}_x{\lambda }^3+q^2{q}_x{\lambda }^3\\ -\bar{i}{q}_{xx}\lambda +\frac{1}{2}{r}_x(q^2+r^2){\lambda }^3+qr{q}_x{\lambda }^3+r^2{r}_x{\lambda }^3\end{array}\right)\right]}^T,\left(\begin{array}{c}q\\ r\end{array}\right)〉d\lambda \\ =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[\frac{\bar{i}}{2}(q{q}_{xx}+r{r}_{xx})+\frac{1}{8}(q^2+r^2)(q_{x}r-r_{x}q)]dx,\end{array}\hfill \\ ⋮\hfill \\ I_k={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}〈\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)K_k(\lambda u),\left(\begin{array}{c}q\\ r\end{array}\right)〉d\lambda .\hfill \end{array}$$