Theory of Vessels was started by M. Livšic in late 70's as a part of more general theory, developed for n dimensional systems, defined by non self-adjoint commuting operators. In the Phd thesis of the author this theory is developed for the case n = 2 and there arise overdetermined time invariant systems and corresponding Vessels. A key idea of the construction is that transfer function of the system intertwines solutions of Linear Differential Equations (LDEs) with a spectral parameter. In this manner there are constructed solutions of Sturm Liouville differential equationswith the spectral parameter λ and coefficient q(x), called potential. On the one hand this work can be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm Liouville differential equations on a half axis (0, ∞) with singularity at 0. On the other hand, there is developed a rich and interesting theory of Vessels which has connections to the notion of τ function, arising in non linear differential equations and to the Galois differential theory for linear ODEs.The transfer function of a Vessel plays a key role in this research work. From a realization formula for the transfer function one can construct ,,tau" function τ , which is a determinant of a self-adjoint matrix function, corresponding to the SL Vessel, and can prove that there is a differential ring R * generated by τ τ , eq , to which all the relevant objects belong. Further, using R * one can evaluate the Picard-Vessiot ring of the output LDE and so connections to the Differential Galois theory are obtained.It seems that finite dimensional SL Vessels as the most convenient environment to handle the ,,deformation theory" of Sturm Liouville differential equations. In order to precisely understand this deformation, there is studied the dependence of Vessels on spectral parameters.

In this paper we extend and solve in the class of functions RSI mentioned in the title, a number of problems originally set for the class RS of rational functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned self-adjoint matrix. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on the one-to-one correspondence between elements in a given subclass of RSI and elements in RS. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.Contents 1991 Mathematics Subject Classification. Primary:47A48, 93C15, 93C35; Secondary: 16E45, 46E22.

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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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

In this work we present an analogue of the inverse scattering for Canonical systems using theory of vessels and associated to them completely integrable systems. Analytic coefficients fits into this setting, significantly expanding the class of functions for which the inverse scattering exists. We also derive an evolutionary equation, arising from canonical systems, which describes an evolution of the logarithmic derivative of the tau function, associated to these systems.Main contribution of this paper is that also analytic initial potentials (when p(x), q(x) are analytic functions) can serve as initial values for (2) and there is a prescribed way how to solve it, using vessels. In order to do it, a more general setting of evolutionary nodes (prevessels) is presented. We show in Section 6.5 how the formulas are incorporated to create this. Moreover, when served as an initial condition, we show that a solution of (1) is created, which coincides for t = 0 with the given value, which is created from the scattering of L (3).The notion of KdV vessel comes from an analogous solution of the Korteweg-de Vries (KdV) and evolutionary Non Linear Schrödinger (NLS) equations. Consider the following collection of operators

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The oocyte maturity is potentially related to the mitochondria activity of cumulus cells.