Using the connection between closed solution curves of the Vortex Filament Equation and the periodic problem for the nonlinear Schrr odinger equation (NLS), we investigate the possibility of relating the knot types of nite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply-covered circles.

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs forced version of this equation. The forcing consists terms involving quadratic products certain eigenfunctions the associated Lax pair. Remarkably, some these equations arise modelling important physical phenomena. initial value problem can be formulated as Riemann-Hilbe...

The one-dimensional Schrodinger equation is considered when the potential and its rst moment are absolutely integrable. The potential is uniquely constructed in terms of the scattering data consisting of the re ection coe cient from the right (left) and the knowledge of the potential on the right (left) half line of the real axis. Hence, neither the bound state energies nor the bound state nor...

The equation of motion a microparticle in various force fields is the Schr¨odinger wave equation.Many questions quantum mechanics, particular thermal radiation electromagneticwaves, lead to problem separability singular differential operators. One such operator isthe above Schr?dinger operator. In this paper, named studied by methods offunctional analysis. Found sufficient conditions for existe...

The one-dimensional Schrodinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coe cient vanishes when the potential is nontrivial, but in the exceptional case this coe cie...

The ared horn is modeled according to Webster's equation. A change of variables transforms the equation into the form of the one dimensional Schrodinger wave equation. The Schrodinger form facilitates speci cation of arbitrary axisymmetric wavefronts for the acoustic disturbance within the horn. To provide a physically motivated choice of wavefront shape, Poisson's equation is solved inside t...

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a prior...

In a previous article 1] we have shown how one can employ Artiicial Neu-ral Networks (ANNs) in order to solve non-homogeneous ordinary and partial diierential equations. In the present work we consider the solution of eigen-value problems for diierential and integrodiierential operators, using ANNs. We start by considering the Schrr odinger equation for the Morse potential that has an analytica...

This work is devoted to the study of a family of almost periodic one-dimensional Schrr odinger equations. We deene a monodromy matrix for this family. We study the asymptotic behavior of this matrix in the adiabatic case. Therefore, we develop a complex WKB method for adiabatic perturbations of periodic Schrr odinger equations. At last, the study of the monodromy matrix enables us to get some s...