508 research outputs found
Volterra integral equations with highly oscillatory kernels: A new numerical method with applications
No full textThe aim of this paper is to present a Nyström-type method for the numerical approximation of the solution of Volterra integral equations of the second kind having highly oscillatory kernels. The method is based on a mixed quadrature scheme which combines the classical product rule with a dilation quadrature formula. The convergence and the stability of the method are investigated and the accuracy of the presented approach is assessed by some numerical tests. The proposed procedure is also applied to the computation of initial scattering data related to the initial value problem associated to the Korteweg-de Vries equation
A Matrix Schrodinger Approach to Focusing Nonlinear Schrodinger Equations with Nonvanishing Boundary Conditions
No full textWe relate the scattering theory of the focusing AKNS system with equally sized nonvanishing boundary conditions to that of the matrix Schrodinger equation. This (shifted) Miura transformation converts the focusing matrix nonlinear Schrodinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method of solving the Marchenko integral equations by separation of variables to derive the multisoliton solutions of this nonlocal equation, thus proposing a method to solve the reflectionless matrix NLS equation
Reflectionless Solutions for Square Matrix NLS with Vanishing Boundary Conditions
No full textIn this article we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the matrix triplet method of representing the Marchenko integral kernel in separated form. Apart from using the Marchenko method, these solutions are also verified by direct substitution in the 2 + 2 NLS equation
Weil pairing and the Drinfeld modular curve
No full textIn 1974 verscheen er van de hand van de Oekraïnse wiskundige Vladimir Gershonovich Drinfeld een baanbrekend artikel getiteld 'Elliptic modules'. Met deze elliptische modulen, die tegenwoordig Drinfeld modulen worden genoemd, voegde de toen 19-jarige Drinfeld een nieuwe en belangwekkend onderwerp toe aan de arithmetische theorie van functielichamen. Eén van de mooiste resultaten binnen deze theorie is voor een groot deel verkregen dankzij Drinfelds werk. Samen met de resultaten verkregen in zijn artikel uit 1974 weet hij in 1977 een bewijs te geven van een speciaal geval van het zogenaamde Langlands' vermoeden. ...
Zie: Samenvatting
The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions
No full textThe inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schr ̈odinger (NLS) equation with non-zero boundary values as x →∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with and .
The direct problem is shown to be well-defined for NLS solutions q(x, t) such that with respect to x for all t ≥ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λ_{l/r} =\sqrt{k^2 + A^2_{l/r} , where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x →±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the longtime asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations
Additive and Multiplicative Perturbations of Exponentially Dichotomous Operators on General Banach Spaces
No full textNonlocal integrable PDEs from hierarchies of symmetry laws: The example of Pohlmeyer–Lund–Regge equation and its reflectionless potential solutions
No full textBy following the ideas presented by Fukumoto and Miyajima in Fukumoto and Miyajima (1996) we derive a generalized method for constructing integrable nonlocal equations starting from any bi-Hamiltonian hierarchy supplied with a recursion operator. This construction provides the right framework for the application of the full machinery of the inverse scattering transform. We pay attention to the Pohlmeyer–Lund–Regge equation coming from the nonlinear Schrödinger hierarchy and construct the formula for the reflectionless potential solutions which are generalizations of multi-solitons. Some explicit examples are discussed
One-dimensional Photonic Crystal Design
No full textIn this article we present a method to determine the band spectrum, band gaps, and discrete energy levels, of a one-dimensional photonic crystal with localized impurities. For one-dimensional crystals with piecewise constant refractive indices we develop an algorithm to recover the refractive index distribution from the period map. Finally, we derive the relationship between the period map and the scattering matrix containing the information on the localized modes
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