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.
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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
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:
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
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 \(A_{i}\), \(i=1,2,\ldots,p\):
It follows that one can get the compatibility condition of (2) and (3)
Equation (6) can be broken down into
where
Step 3: Choose \(\triangle _{n}\in \widetilde{G}\) so that
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
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
with
And the corresponding loop algebra is taken by
where \(h(n)=h\lambda ^{2n}\), \(e(n)=e\lambda ^{2n-1}\), \(f(n)=f\lambda ^{2n-1}\).
After simple calculations, one can find
where the gradations of \(h(n)\), \(e(n)\), and \(f(n)\) are given by
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):
that is,
In terms of Eq. (12), we take the initial values
and then one has
where \(\beta _{0}(t)=0\) is an integral constant. From (12), we deduce that
where \(\beta _{1}(t)=0\) is an integral constant. Denote that
In what follows, the gradations of the left-hand side of (7) can be obtained by using (1), (9), and (10)
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:
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:
that is,
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
Therefore, the nonisospectral integrable hierarchy is derived by Eq. (8) as follows:
or
where
Based on (12), one has
where
Hence, (14) can be written as
where
When \(n=1\), the nonisospectral integrable hierarchy (17) becomes
When \(n=2\), the nonisospectral integrable hierarchy (17) reduces to
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
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
where
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
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
where.
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:
which means that Φ is a strong symmetry, where .
Proposition 2
where, , andIis an identity matrix.
In fact,
where
where
We therefore verified that (25) is correct.
Proposition 3
where, , and\(K_{1}=\varPhi u\).
In fact,
Then we have
We therefore verified that (26) is correct.
Proposition 4
where\(K_{m}=\varPhi ^{m}u\), \(K_{n}=\varPhi ^{n}u\).
Proposition 5
The proofs of Proposition 4 and Proposition 5 were presented in [20].
From the above results we can get
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
If we have known the integrable hierarchy \(u_{t}=K_{n}(u)\), then v satisfying the following equation
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
which is independent of timet, that is, \(\frac{d}{dt}\langle v,\sigma \rangle =0\).
Definition 4
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:
According to the symbols above, we can deduce the following.
Proposition 8
IfIis a conserved quality of the hierarchy\(u_{t}=K_{n}(u)\), and the conserved covariancevsatisfies
then one obtains
that is,
Hence, we derive the following conserved qualities related to the integrable hierarchy \(u_{t}=K_{n}(u)\):
In addition, a few conserved qualities are also derived for the integrable hierarchy (28) as follows:
where
Moreover, we have
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Wang, H., Zhang, Y. Generating nonisospectral integrable hierarchies via a new scheme. Adv Differ Equ 2020, 170 (2020). https://doi.org/10.1186/s13662-020-02600-5
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DOI: https://doi.org/10.1186/s13662-020-02600-5