New symmetries for the Ablowitz–Ladik hierarchies

Abstract

In the letter we give new symmetries for the isospectral and non-isospectral Ablowitz-Ladik hierarchies by means of the zero curvature representations of evolution equations related to the Ablowitz-Ladik spectral problem. Lie algebras constructed by symmetries are further obtained. We also discuss the relations between the recursion operator and isospectral and non-isospectral flows. Our method can be generalized to other systems to construct symmetries for non-isospectral equations. IntroductionIt is well-known that infinitely many symmetries and their Lie algebra serve as one of mathematical structures of integrability for evolution equations [1]. In general, a Lax integrable isospectral evolution equation can have two sets of symmetries, isospectral and non-isospectral symmetries, or called K-and τ -symmetries, respectively. One efficient way to construct τ -symmetries was proposed by Fuchssteiner[2] by using the master symmetry. This method was later developed to many continuous (1+1)-dimensional Lax integrable systems [3,4], (1+2)-dimensional systems [5] and further to some differential-difference cases [6,7]. This letter will discuss K-and τ -symmetries for the isospectral Ablowitz-Ladik(AL) hierarchy, which is a well-known discrete hierarchy[8]- [11]. We will also construct new infinitely many symmetries for the non-isospectral AL hierarchy. The AL spectral problem can have two sets of isospectral hierarchies [12] which respectively correspond to positive and negative powers of the spectral parameter λ in the time-evolution part in Lax pair. The same results hold for the nonisospectral hierarchies as well, as shown in [7], where the algebraic relations between isospectral and non-isospectral flows related to positive powers of λ and the algebraic relations between isospectral and non-isospectral flows related to negative powers of λ were discussed, respectively.Our method to construct K-and τ -symmetries for the isospectral AL hierarchy is essentially the same as used in Ref. [7,6], and as well as a direct generalization of its continuous version [4]. Recently, we uniformed the two sets of isospectral flows (positive order and negative order) to one hierarchy with a uniform recursion operator [13]. This motivates us to do the same thing for the two sets of non-isospectral flows. Then we investigate the algebraic relations of the

Symmetries for the Ablowitz-Ladik Hierarchy: Part I. Four-Potential Case

“…Let us first introduce some basic notations and notions which have been used for discussing symmetries of discrete systems (cf., [11,15,27]).…”

“…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, the related Hamiltonian structures, constraint flows, nonlinearization, Darboux transformation, conservation laws, symmetries and Lie algebra structures have been studied (cf., [8][9][10][11][12][13][14][15][16][17]). The four-potential AL spectral problem is more complicated than the two-potential case because of containing two more potentials and its unsymmetrical matrix form.…”

“…Then we can imbed these flows into their zero-curvature equations by means of functional derivatives. The resulting expressions, which we refer to as zero-curvature representations, have been shown to be powerful in constructing symmetries for Lax integrable systems (cf., [10,15,[23][24][25][26][27]). We will derive algebraic relations for isospectral and non-isospectral flows and then derive symmetries and their Lie algebras for not only isospectral AL hierarchy but also non-isospectral hierarchy.…”

The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform

“…Usually, one eigenvalue problem admits a pair of linear problem, for which the hierarchy of soliton equation is just the compatible condition. For more details, one can see for example [31][32][33][34][35] and references therein. Recently, the discrete mKdV hierarchy with self-consistent sources is investigated by constructing non-auto-Bäcklund transformations [19], where the self-consistent sources are expressed by both of the eigenfunctions and adjoint eigenfunctions corresponding to the A-L spectral problem and its adjoint problem, respectively.…”

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

“…Inspired by [16][17][18][19][20], in the next sections, we will focus on discussing the algebraic structure of zero curvature equation of integrable couplings (1.13), i.e., the enlarged triple (Ū ,V ,K) satisfying (1.14). Further, we illustrate our approach by an application example for the Volterra lattice hierarchy and a τ -symmetry algebra is engendered for this coupling system.…”