On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

Journal of Mathematical Physics

2017

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Using Gelfand-Levitan theory on a half line, we construct a vessel for the class of potentials, whose spectral functions satisfy a certain regularity assumption. When the singular part of the spectral measure is absent, we construct a canonical model of the vessel. Finally, evolving the constructed vessel, we solve the Korteweg de Vries equation on the half line, coinciding with the given potential for t = 0. It is shown that the initial value for x = 0 is prescribed by this construction, but can be perturbed using an "orthogonal" to the problem measure.The results, presented in this work 1. include formulas for the ingredients of the GelfandLevitan equation, 2. are shown to be general in the sense that NLS, Canonical systems and many more equations can be solved using theory of vessels, analogously to Zacharov-Shabath scheme, 3. present a generalized inverse scattering theory on a line for potentials with singularities using pre-vessels, 4. present the tau function and its role.

Section: Such Functions Are Called Transfer Functions Of Vessels Defmentioning

confidence: 65%

Section: Vesselsmentioning

confidence: 99%

Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. On solution of the Korteweg-de Vries equation in the first quadrant

Journal of Mathematical Physics

2017

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“…31. As a result, the ideas presented in this work can be used to prove a similar to Main Theorem 5.6 result for the evolutionary NLS equation.…”

Section: Main Theorem 56 Suppose That Q(x) Is An Analytic Function mentioning

confidence: 68%

Solution of the Korteweg-de Vries equation on the line with analytic initial potential

Journal of Mathematical Physics

2014

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Articles you may be interested inGeneralized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transform Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methodsWe present a theory of Sturm-Liouville non-symmetric vessels, realizing an inverse scattering theory for the Sturm-Liouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the Sturm-Liouville operator, defined with an analytic potential on the line. Evolving such vessels we generate Korteweg-de Vries (KdV) vessels, realizing solutions of the KdV equation. As a consequence, we prove the theorem as follows: Suppose that q(x) is an analytic function on R. Then there exists a closed subset ⊆ R 2 and a KdV vessel, defined on . For each x ∈ R one can find TThe potential q(x) is realized by the vessel for t = 0. Since we also show that if q(x, t) is a solution of the KdV equation on R × [0, t 0 ), then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary nonlinear Schrödinger and Boussinesq equations, since both of these equations possess vessel constructions. C 2014 AIP Publishing LLC.differentiable potential by Gelfand-Levitan theory, 19 but not always it can be used to solve the KdV equation (1). For this situation in case var[dρ] < ∞ a solution of KdV in the first quadrant (x, t ≥ 0) is presented in Ref. 28.Although there is a good scattering theory of the SL equation (2) on the line (see Ref. 27, Chap. 2, p. 128, and Ref. 26, Chap. 2), even with arbitrary singularities, 13 the solutions of KdV, corresponding to them are not developed. In fact, the classes of initial potentials, for which solutions of (1) were presented using inverse scattering are as follows:Suppose that dρ = dρ + − dρ − for two positive measures dρ + , dρ − , creating two Hilbert spaces of column-functions H + , H − as above. We make the following definition. Definition 2.2. A Krein space Kρ, associated to the finite measure dρ is

“…For example, the famous Korteweg-de Vries (KdV) equation [KdV95,GGKM67], which was first considered by the author in [Melb], was further significantly improved in [Mele] and finally culminated in a scattering theory of analytic parameters in [Mel14c], and KdV hierarchy [Melc]. 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].…”

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

Complex Anal. Oper. Theory

2014

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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).