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 gl(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{gl}(3)$\end{document} 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.


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][7][8][9][10]. It is well known that many different methods for generating isospectral integrable equations have been proposed [11][12][13][14][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 λ t = λ n (n > 0). Qiao [18] adopted the Lenard series method to obtain some nonisospectral integrable hierarchies under the case λ t = λ 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 λ t = n j=0 k j (t)λ n-j . Obviously, this case is a generalized expression for the case λ t = λ n [19,20]. Under obtaining nonisospectral integrable systems, some of their properties, including Darboux transformations, exact solutions, and so on, could be studied [21][22][23][24][25][26]. We first recall some fundamental facts. Definition 1 One basis element R ∈ G is called pseudoregular if the following conditions hold: (1) G = Ker ad R ⊕ Im ad R, Definition 2 For any basis element e i (n) (i = 1, 2, . . . , p), we define its gradation by Obviously, for ∀g ∈ G, g can be expressed by g = n k n e i (n) =: n g n , k n are constants. We can decompose g into two parts as follows: and call g + the positive part of g, μ ∈ Z is some chosen integer.
In the following, the steps for generating nonisospectral integrable hierarchies of evolution equations are presented.
Step 2: Solving the following stationary zero curvature equation for A i , i = 1, 2, . . . , 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 n ∈ G so that where B i (i = 1, 2, . . . , q) ∈ 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].

A nonisospectral integrable hierarchy of evolution equations
A basis of the Lie algebras gl(3) is given by And the corresponding loop algebra is taken by where h(n) = hλ 2n , e(n) = eλ 2n-1 , f (n) = f λ 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 gl (3): where It follows that we obtain Furthermore, the following equation can be derived by taking λ t = i≥0 k i (t)λ 1-2i with Eq. (6): that is, In terms of Eq. (12), we take the initial values and then one has where β 0 (t) = 0 is an integral constant. From (12), we deduce that where β 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: In order to obtain the nonisospectral integrable hierarchies, we take the modified term 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 A, B = 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 γ = 0 via substituting the initial values of (12) into (22), and then we obtain b n c n = δH n δu = 0 1 1 0 where H n = 2ia nqb nrc n 2n -2 , 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).

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 Thus, Φ is the hereditary symmetry of (17). In what follows we can also prove that the following relation holds.
In fact, 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 K m = Φ m q x r x .

Proposition 2
where u = q x r x , H = 0 i∂ -i∂ 0 , and I is an identity matrix.
In fact, We therefore verified that (25) is correct.
The proofs of Proposition 4 and Proposition 5 were presented in [20].
is called the conserved covariance, where K is the linearized operator of K , and K * denotes a conjugate operator of K .

Proposition 6 ([14])
If σ is a symmetry of Eq. u t = K n (u), v is its conserved covariance, then we have which is independent of time t, that is, d dt v, σ = 0. Definition 4 ([11, 12, 14]) If F f = v, f for ∀f ∈ S, then v is called the gradient of the functional F, which is denoted by v = δF δu .

Proposition 7 ([14])
If v = v * , then v is the gradient of the following functional: According to the symbols above, we can deduce the following.