Nonlinear Schr�dinger equation on a semi-axis and an inverse problem associated with it

Sakhnovich, Alexander

doi:10.1007/bf01057016

Cited by 44 publications

(87 citation statements)

References 2 publications

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“…There is a unique Weyl function of system (5.39) with a bounded on [0, ∞) potential [53,69]. If (5.41) holds, we can put in (5.42) M 1 = M .…”

Section: Skew-self-adjoint Dirac Systemmentioning

confidence: 99%

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On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.

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“…There is a unique Weyl function of system (5.39) with a bounded on [0, ∞) potential [53,69]. If (5.41) holds, we can put in (5.42) M 1 = M .…”

Section: Skew-self-adjoint Dirac Systemmentioning

confidence: 99%

“…In this subsubsection we consider system Analogously to Definition 5.2 one can define Weyl functions of the skew-self-adjoint Dirac system [16,29,53,69]. …”

Section: Skew-self-adjoint Dirac Systemmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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“…The term "canonical differential equations" for the system (1.1) has been used by MelikAdamyan [32][33][34], L. A. Sakhnovieh [39,40], and A. L. Sakhnovich [38], who have studied the direct and inverse scattering problems for (1.1) on the half line. Under minor restrictions on the given so-called reflection function, a characterization of the scattering data corresponding to an Ll-potential on the half line was given by Melik-Adamyan [34], who also supplied a method to reduce the inverse spectral problem on the full line for a canonical equation of order 2n to an inverse spectral problem on the half line for a canonical equation of order 4n [32] (see also [41]).…”

Section: F At()) = Hn + Ij2~ Aye -~J~oy V(y) Ft(y )) Oomentioning

confidence: 99%

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

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A direct and inverse scattering theory on the full line is developed for a class of firstorder selfadjoint 2n • 2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

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“…Weyl theory of skew-selfadjoint Dirac systems was studied in the papers [22,28,46,48]. Some further references as well as the results of this subsection are contained in [58,Ch.3].…”

Section: Theorem 26 [56]mentioning

confidence: 99%

“…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. See further results and references, for instance, in [46,48,58].…”

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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Nonlinear Schr�dinger equation on a semi-axis and an inverse problem associated with it

Cited by 44 publications

References 2 publications

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Evolution of Weyl Functions and Initial-Boundary Value Problems

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