In the paper we first investigate symmetries of isospectral and non-isospectral four-potential Ablowitz-Ladik hierarchies. We express these hierarchies in the form of $u_{n,t}=L^m H^{(0)}$, where $m$ is an arbitrary integer (instead of a nature number) and $L$ is the recursion operator. Then by means of the zero-curvature representations of the isospectral and non-isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. The recursion operator $L$ is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies. The even order members in the four-potential Ablowitz-Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two-potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes $L^2$.
As is well known, many physically interesting problems are modeled by nonlinear differentialdifference equations, such as the Toda lattice, Volterra lattice and discrete nonlinear Schrödinger equation. Since 1970s discrete systems have received considerable attention from variety of aspects (e.g., [1][2][3][4]), such as Inverse Scattering Transform, bilinear method, Sato's approach, symmetry analysis and so on. One of famous discrete spectral problems is given by Ablowitz and Ladik [1,5] which is now referred to as the Ablowitz-Ladik (AL) spectral problem. This spectral problem, coupled with different time evolution parts, has provided Lax integrabilities for many discrete soliton systems, such as integrable discrete nonlinear Schrödinger equation [6], discrete mKdV equation and so forth.
There are two types of the AL spectral problems, which contains two potentials {Q n , R n } and four potentials {Q n , R n , S n , T n }, respectively. The two-potential one is the direct discretization (cf. [6]) of the famous continuous AKNS-ZS spectral problem [7], and besides solutions,
Let us first introduce some basic notations and notions which have been used for discussing symmetries of discrete systems (cf. [11,15,27]).
Assume that u n . = u(t, n) = (u (1) , u (2) , u (3) , u (4) ) T is a four-dimensional vector field, where u (i) = u (i) (t, n), 1 ≤ i ≤ 4, are all functions defined over R × Z and vanish rapidly as |n| → ∞. By V 4 we denote a linear space consisting of all vector fields f = (f (1) , f (2) , f (3) , f (4) ) T , where each f (i) is a function of u(t, n) and its shifts u(t, n + j), j ∈ Z, satisfying f (i) (u(t, n))| un=0 = 0, and each f (i) is C ∞ differentiable w.r.t. t and n, and C ∞ -Gauteaux differentiable w.r.t. u n .
Here the Gateaux (or Fréchet) derivative of f ∈ V 4 (or f an operator on V 4 ) in the direction g ∈ V 4 is defined as
By means of the Gateaux derivative one can define a Lie product for any f, g ∈ V 4 as
We also define a Laurent matrix polynomials space Q 2 (z) composed by all 2 × 2 matrices
, where all the {q ij } are Laurent polynomials of z. Two subspaces of Q 2 (z) we will need are
We note that in a similar way we can define spaces V s , Q m (z) and Q ± m (z) (cf. [11]). In general a discrete evolution equation arises from the compatibility of a pair discrete linear problems
where Φ n is a wave function, U n is a spectral matrix with spectral parameter z and potential vector u(t, n) while V n is a matrix governing time evolution. The compatibility condition, also called discrete zero-curvature equation, reads
Here and in the following E is a shift operator defined as E j f (n) = f (n + j) for j ∈ Z. Suppose that the corresponding nonlinear evolution equation is
Then by means of the Gateaux derivative the flow K(u n ) can be embedded into the zerocurvature equation (2.4) as the following,
which is usually called the zero-curvature representation of the flow K(u n ).
For the nonlinear evolution equation (2.5),
where by ∂σ ∂t we specially denote the derivative of σ w.r.t. t explicitly included in σ, (for example,
Next, let us recall some backgrounds on four-potential AL hierarchy. The four-potential AL spectral problem reads [1,5]
where λ is a spectral parameter and Q n , R n , S n , T n are four potential functions of n and t. When S n = T n = 0 (2.8) reduces to the two-potential AL spectral problem, i.e.,
which is a discrete version of the AKNS-ZS spectral problem (cf. [6]). An alternative (matrix) form of (2.8) is [1]
where Λ n = 1 -S n T n and we have substituted z 2 for λ. This form can be gauge-transformed to [18]
where Φ n and Ψ n are related through
(2.12)
Here on U n we impose a condition
Then the compatibility condition with (2.11) yields
Usually the above discrete zero-curvature equation contributes a discrete nonlinear evolution equation hierarchy with four potentials and their recursion operator, but this is not as easy as in two-potential case (related to (2.9), cf. [9][10][11]15]). However, the spectral matrix U n can be separated into [19] U n = U (2) n U (1) n , U (1)
Then (2.15) holds if [19] U
where
Thus, one can consider the two auxiliary systems given in (2.17), where each of U
n contains two potentials. Recently, starting from (2.17) Geng and Dai [19] derived a four-potential AL hierarchy and their recursion relation and considered Hamiltonian structures and nonlinearization of Lax pair.
Noting that in the AL spectral problem (2.8) the spectral parameter λ appears symmetrically w.r.t. positive and negative powers, it is then understood that the recursion operator and its inverse can be derived symmetrically. So are the negative order and positive order hierarchies. In [15], starting from (2.9) we have expressed isospectral and non-isospectral two-potential AL hierarchies in the form of u n,t = L m H (0) , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. This is also true for four-potent
This content is AI-processed based on open access ArXiv data.
