Restricted flows of the Ablowitz-Ladik hierarchy and their continuous limits

Zeng, Yunbo; Rauch-Wojciechowski, Stefan

doi:10.1088/0305-4470/28/1/016

Cited by 39 publications

(35 citation statements)

References 18 publications

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“…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.…”

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confidence: 99%

New symmetries for the Ablowitz–Ladik hierarchies

Zhang¹,

Ning²,

Bi³

et al. 2006

Physics Letters A

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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

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confidence: 99%

New symmetries for the Ablowitz–Ladik hierarchies

Zhang¹,

Ning²,

Bi³

et al. 2006

Physics Letters A

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“…Firstly, we study the integrable discrete GNLS' remarkable properties: recursion operator, infinite sets of conservation laws, infinite symmetries. The recursion operator (3.6) what we obtain is more concise than the recursion operator for the corresponding AKNS equation in [28,29]. The recursion operator is not only the generating operator for the family of equations connected with a given equation but also the generating operator for the family of Hamiltonian structures.…”

Section: Introductionmentioning

confidence: 78%

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

Li¹,

Li²,

Chen³

2014

Commun. Theor. Phys.

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A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrödinger (GNLS) equation for which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. To show the complete integrability of the discrete GNLS equation, the recursion operator, symmetries and conservation quantities are obtained. Furthermore, all of reductions for the discrete GNLS equation are given and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Chen

2010

Studies in Applied Mathematics

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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 .

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Restricted flows of the Ablowitz-Ladik hierarchy and their continuous limits

Cited by 39 publications

References 18 publications

New symmetries for the Ablowitz–Ladik hierarchies

New symmetries for the Ablowitz–Ladik hierarchies

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

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

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