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Generating nonisospectral integrable hierarchies via a new scheme


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 [610]. It is well known that many different methods for generating isospectral integrable equations have been proposed [1115]. 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 [2126]. 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 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. (1)

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

  2. (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

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

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 , we introduce the spectral problems

$$\begin{aligned}& \psi _{x}=U\psi ,\quad U=R+u_{1}e_{1}(n)+ \cdots +u_{q}e_{q}(n), \end{aligned}$$
$$\begin{aligned}& \psi _{t}=V\psi ,\quad V=A_{1}e_{1}(n)+ \cdots +A_{p}e_{p}(n), \end{aligned}$$
$$\begin{aligned}& \lambda _{t}=\sum_{i\geq 0}k_{i}(t) \lambda ^{-N_{i}i}, \end{aligned}$$

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

  1. (a)

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

  2. (b)

    R is pseudoregular,

  3. (c)

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

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

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

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

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

Equation (6) can be broken down into

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


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

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

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

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

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



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 \(\widetilde{\operatorname{gl}}(3)\):


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


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

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

that is,

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

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,

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

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:






Based on (12), one has




Hence, (14) can be written as




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

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

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

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

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


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


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




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


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

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


In fact,


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


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 Km=Φm(qxrx).

Proposition 2

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

whereu=(qxrx), H=(0ii0), andIis an identity matrix.

In fact,






We therefore verified that (25) is correct.

Proposition 3

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

whereu=(qxrx), H=(0ii0), and\(K_{1}=\varPhi u\).

In fact,


Then we have


We therefore verified that (26) is correct.

Proposition 4

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

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


Definition 3

([11, 12, 14])

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

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

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


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


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

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

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 \(u_{t}=K_{n}(u)\):

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

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




Moreover, we have



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This work was supported by the National Natural Science Foundation of China (grant No. 11971475).

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Wang, H., Zhang, Y. Generating nonisospectral integrable hierarchies via a new scheme. Adv Differ Equ 2020, 170 (2020).

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