Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem

Berezanskiĭ, Yu. M.; Gekhtman, Michael; Shmoish, Michael

doi:10.1007/bf01056762

Cited by 33 publications

(45 citation statements)

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“…In case of the Toda equation on the half line a different approach using the spectral measure of H+ is introduced and extended in [28] - [32] (see also [15]). It turns out that the time evolution of the spectral measure can be computed explicitly as the Freud transform of the initial measure.…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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“…In this paper we use the Inverse Spectral Transform approach [7,8,22]. More precisely, we follow the scheme introduced in [38][39][40], see also [41,Ch.…”

Section: )mentioning

confidence: 99%

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Sakhnovich

2015

Journal of Mathematical Analysis and Applications

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“…Inverse Spectral Transform (instead of the Inverse Scattering Transform) was first used for initial-boundary value problems in [15,35]. More precisely, a special kind of initial-boundary value problem for Toda lattice with a linear law of evolution of the spectral and Weyl functions was studied in [15,35].…”

Section: Evolution Of Weyl Functionsmentioning

confidence: 99%

“…More precisely, a special kind of initial-boundary value problem for Toda lattice with a linear law of evolution of the spectral and Weyl functions was studied in [15,35]. In the papers [59,60], an essentially more general case of the initial-boundary value problem for Toda lattice as well as some initial-boundary value problems for continuous integrable systems (including square matrix dNLS) were dealt with, and the law of evolution of the Weyl function was presented in the form of Möbius transformation.…”

Section: Evolution Of Weyl Functionsmentioning

confidence: 99%

Evolution of Weyl Functions and Initial-Boundary Value Problems

Sakhnovich

2016

Math. Model. Nat. Phenom.

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This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well.

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Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem

Cited by 33 publications

References 1 publication

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Evolution of Weyl Functions and Initial-Boundary Value Problems

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