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

Kako, Fujio; Mugibayashi, Nobumichi

doi:10.1143/ptp.60.975

Cited by 22 publications

(17 citation statements)

References 2 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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“…We shall keep the same front-index at all quantities characterizing the primary system (1.1)-(1.6) in the framework of Hamiltonian treatment. The experience given by other semi-discrete integrable systems of nonlinear Schrödinger type [26,31,46,58] prompts us to seek the Hamiltonian density of the primary system (1.1)-(1.6) as some superposition of second local densities entering the local conservation laws (see formulas (7.21), (7.22) and (7.29), (7.30) of Section 7 accompanied by the reduction formulas (2.20)-(2.25) of Section 2). This observation gives rise to the following candidate [66,67]…”

Section: Hamiltonian Representation Of Primary Nonlinear Schrödinger System With Thementioning

confidence: 99%

Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling

Vakhnenko¹

2021

JNMP

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We summarize the most featured items characterizing the semi-discrete nonlinear Schrödinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized dressing procedure based on the Darboux transformation. On the other hand the system integrability inspires an infinite hierarchy of local conservation laws some of which were found explicitly in the framework of generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem and it permits the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of inter-site resonant coupling and as a consequence it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. Adjusting the background parameter we are able to switch over the system dynamics between two essentially different regimes separated by the critical point. The system criticality against the background parameter is manifested both indirectly by the auxiliary linear spectral problem and directly by the nonlinear dynamical equations themselves. The physical implications of system criticality become evident after the rather sophisticated canonization procedure of basic field variables. There are two variants of system standardization equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus in the under-critical region the system support the bright excitations in both subsystems, while in the over-critical region one of subsystems converts into the subsystem of dark excitations.

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“…In subsection 3.1, we present its discrete analogue, that is, the (2 + 1)-dimensional Ablowitz-Ladik lattice (2.2) is a linear combination of four commuting flows. This is a quite natural result because (i) each of the two Davey-Stewartson flows provides an asymmetric (2 + 1)-dimensional generalization of the NLS system and (ii) the Ablowitz-Ladik discretization of the NLS system is actually a sum of two elementary flows (and one trivial flow) in the same hierarchy [3,7,[27][28][29]].…”

Section: Continuum Limitmentioning

confidence: 99%

On a (2 + 1)-dimensional generalization of the Ablowitz–Ladik lattice and a discrete Davey–Stewartson system

Tsuchida¹,

Dimakis²

2011

J. Phys. A: Math. Theor.

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We propose a natural (2 + 1)-dimensional generalization of the Ablowitz-Ladik lattice that is an integrable space discretization of the cubic nonlinear Schrödinger (NLS) system in 1 + 1 dimensions. By further requiring rotational symmetry of order 2 in the two-dimensional lattice, we identify an appropriate change of dependent variables, which translates the (2 + 1)-dimensional Ablowitz-Ladik lattice into a suitable space discretization of the Davey-Stewartson system. The space-discrete Davey-Stewartson system has a Lax pair and allows the complex conjugation reduction between two dependent variables as in the continuous case. Moreover, it is ideally symmetric with respect to space reflections. Using the Hirota bilinear method, we construct some exact solutions such as multidromion solutions.

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

Cited by 22 publications

References 2 publications

Poisson brackets of orthogonal polynomials

Poisson brackets of orthogonal polynomials

Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling

On a (2 + 1)-dimensional generalization of the Ablowitz–Ladik lattice and a discrete Davey–Stewartson system

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