Inverse Scattering of Canonical Systems and Their Evolution

Melnikov, Andrey

doi:10.1007/s11785-014-0391-1

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…Another choice of these parameters generates solutions of the Sturm-Liuoville differential equation and the Kortweg-de-Vries equation (see [2,9] for details). A similar result was presented for the setting canonical systems and their evolution [10].…”

Section: Definition Of a Regular Nls Vessel And Its Propertiessupporting

confidence: 80%

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

Melnikov

2014

International Journal of Partial Differential Equations

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In this work we present an application of a theory of vessels to a solution of the evolutionary nonlinear Schrödinger (NLS) equation. The classes of functions for which the initial value problem is solvable rely on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of Zakharov-Shabath for solving evolutionary NLS equation but has an advantage of simpler formulas and new techniques and notions to understand the solutions.

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Section: Definition Of a Regular Nls Vessel And Its Propertiessupporting

confidence: 80%

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

Melnikov

2014

International Journal of Partial Differential Equations

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“…The theory of operator colligations in Hilbert spaces is deeply connected with the problem of wave dispersions, collective motions of spatio-temporal systems, classical scattering theory ( [16], [17], [25], [32], and etc.). In series of papers (for example, [25], [27], [28], [29], [1], and etc.) many common points of operator colligations (vessels) with the classical scattering theory of Sturm-Liouville operator, inverse scattering of linear differential equations are presented.…”

Section: Introductionmentioning

confidence: 99%

Solitonic combinations, commuting nonselfadjoint operators, and applications

Borisova¹

2019

Preprint

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In this paper, applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by M.S. Livšic and Y. Avishai, are considered. An approach to the inverse scattering problem and to the wave equations is presented, based on the Livšic operator colligation theory (or vessel theory) in the case of commuting bounded nonselfadjoint operators in a Hilbert space, when one of the operators belongs to a larger class of nondissipative operators with asymptotics of the corresponding nondissipative curves. The generalized Gelfand-Levitan-Marchenko equation of the cases of different differential equations (the Korteweg-de Vries equation, the Schrödinger equation, the Sine-Gordon equation, the Davey-Stewartson equation) are derived. Relations between the wave equations of the input and the output of the generalized open systems, corresponding to the Schrödinger equation and the Korteweg-de Vries equation, are obtained. In these two cases, differential equations (the Sturm-Liouville equation and the 3dimensional differential equation), satisfied by the components of the input and the output of the corresponding generalized open systems, are derived.

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“…Second example is a notable Evolutionary Non Linear Schrödinger (ENLS) equation [Kat89], which was inserted into the setting of vessels in [Mel14b]. Third example, developed by the author addressed the scattering theory and a corresponding completely integrable PDE for so called canonical (or Dirac) systems [Mel14a]. Worth noticing that a different from these three examples type of vessels was used in [Meld] to solve a shallow waters (Boussinesq) equation, originated in [Bou72].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we would like to mention some works of the author, originating the theory of vessels [Mel11,Mel14a,Mel14c,Mel14b] and joint works with collaborators [AMV09, AMV12, MV14].…”

Section: Introductionmentioning

confidence: 99%

Classification of KdV Vessels with Constant Parameters and Two Dimensional Outer Space

Melnikov

2014

Complex Anal. Oper. Theory

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In this article we classify vessels producing solutions of some completely integrable PDEs, presenting a unified approach for them. The classification includes such important examples as Korteweg-de Vries (KdV) and evolutionary Non Linear Schrödingier (ENLS) equations. In fact, employing basic matrix algebra techniques it is shown that there are exactly two canonical forms of such vessels, so that each canonical form generalize either KdV or ENLS equations. Particularly, Dirac canonical systems, whose evolution was recently inserted into the vessel theory, are shown to be equivalent to the ENLS equation in the sense of vessels. This work is important as a first step to classification of completely integrable PDEs, which are solvable by the theory of vessels. We note that a recent paper of the author, published in Journal of Mathematical Physics, showed that initial value problem with analytic initial potential for the KdV equation has at least a "narrowing" in time solution. The presented classification, inherits this idea and a similar theorem can be easily proved for the presented PDEs. Finally, the the resuts of the work serve as a basis for the investigation of the following problems: 1. hierarchy of the generalized KdV, ENLS equations (by generalizing the vessel equations), 2. new completely integrable PDEs (by changing the dimension of the outer space), 3. addressing the question of integrability of a given arbitrary PDE (the future classification will create a list of solvable by vessels equations, which may eventually include many existing classes of PDEs).

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Inverse Scattering of Canonical Systems and Their Evolution

Cited by 6 publications

References 22 publications

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

Solitonic combinations, commuting nonselfadjoint operators, and applications

Classification of KdV Vessels with Constant Parameters and Two Dimensional Outer Space

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