Complete Integrability of General Nonlinear Differential-Difference Equations Solvable by the Inverse Method. II

Kako, Fujio; Mugibayashi, Nobumichi

doi:10.1143/ptp.61.776

Cited by 39 publications

(19 citation statements)

References 5 publications

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“…Indeed, a key to our progress was learning of this work and the earlier paper of Faybusovich-Gekhtman [14]. We also mention not unrelated earlier papers of Kako-Mugibayashi [29,30] and Kulish [35]. During the refereeing of this paper, we learned of a preprint of Tsiganov [53], clearly independent of ours, that also computes Poisson brackets of transfer matrices which include some of our OP Poisson brackets.…”

Section: Introductionmentioning

confidence: 86%

Poisson brackets of orthogonal polynomials

Cantero

Simon

2009

Journal of Approximation Theory

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Section: Introductionmentioning

confidence: 86%

Poisson brackets of orthogonal polynomials

Cantero

Simon

2009

Journal of Approximation Theory

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“…The experience given by other semidiscrete integrable systems of nonlinear Schrödinger type 18,24,25 prompts us to seek the Hamiltonian density as some superposition of second local densities entering the local conservation laws. This observation gives rise to the following candidate:…”

Section: Hamiltonian Representation Of Nonlinear Schrödinger Systmentioning

confidence: 99%

Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice

Vakhnenko

2015

Journal of Mathematical Physics

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A new integrable two-component system with cubic nonlinearityFollowing the considerable progress in nanoribbon technology, we propose to model the nonlinear Frenkel-like excitations on a triangular-lattice ribbon by the integrable nonlinear ladder system with the background-controlled intersite resonant coupling. The system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy. The most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach. The obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy. Using the recovered second densities, the Hamiltonian formulation of integrable nonlinear ladder system with background-controlled intersite resonant coupling is presented. In doing so, the relevant Poisson structure turns out to be essentially nontrivial. The Darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of Bäcklund transformation in justification of its self-consistency is pointed out. The spectral properties of Darboux matrix allow to restore the whole Darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system. The power of Darboux-dressing method is explicitly demonstrated in generating the multicomponent one-soliton solution to the integrable nonlinear ladder system with background-controlled intersite resonant coupling. C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4914510] under the proper choice of spectral L(n|z) and evolution A(n|z) operators. Here, the integer n standing for the discrete spatial coordinate runs from minus to plus infinity, the letter z marks the time-independent complex-valued spectral parameter, while the overdot assumes the differentiation over the time variable τ:ḟ (n) ≡ d f (n)/dτ. In its original formulation, 1 the system of our interest has been derived relying upon certain Laurent-type form of 4 × 4 square matrices L(n|z) and A(n|z). In doing so, the analysis of a limiting spectral problem dictated by the auxiliary spectral operator a) vakhnenko@bitp.kiev.ua

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“…For example, its inverse scattering transform, soliton solutions, the Hamiltonian structures, Darboux transformation, and others have been discussed in Refs. [23][24][25][26][27]. In what follows, we first give solutions of the stationary discrete zero-curvature equation.…”

Section: The Hierarchy Of Differential-difference Equationsmentioning

confidence: 99%

“…In particular, letting R n = ∓Q n = u n , S n = ∓T n = v n , and α 0 = −β 0 = 1 2 , (2.33) is reduced to the self-dual network [23][24][25][26][27][28] …”

Section: )mentioning

confidence: 99%

Nonlinearization of the Lax pairs for discrete Ablowitz–Ladik hierarchy

Geng

Dai

2007

Journal of Mathematical Analysis and Applications

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The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finitedimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.

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Complete Integrability of General Nonlinear Differential-Difference Equations Solvable by the Inverse Method. II

Cited by 39 publications

References 5 publications

Poisson brackets of orthogonal polynomials

Poisson brackets of orthogonal polynomials

Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice

Nonlinearization of the Lax pairs for discrete Ablowitz–Ladik hierarchy

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