V. S. Serov

RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering. The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W 1,p -functions due to P. Hajlasz, and a modification of the classical Triebels Maximal Inequality.

Inverse eigenvalue problems for a singular Sturm-Liouville operator on (0,1)

In this paper we investigate two boundary value problems on the finite interval with potentials that may have non-integrable singularity at the origin. The first problem allows unique determination of the potential by only one spectrum for certain values of the parameter in one of the boundary conditions. The second problem deals with the Dirichlet problem for potentials which are the sum of Bessels main part with constant bounded from below and non-integrable potential. Uniqueness results are proved for both problems.

Reflection and transmission of a plane TE-wave at a lossless nonlinear dielectric film

Abstract A general analytical solution of the Helmholtz equation describing the scattering of a plane, monochromatic, TE-polarized wave with a film exhibiting a local Kerr-like nonlinearity is presented. The film is situated between two semi-infinite media. All media are assumed to be non-absorbing, non-magnetic isotropic and homogeneous. The results derived contain conditions for unbounded field intensities expressed in terms of the imaginary half-period of Weierstrass’ elliptic function ℘ . The reflectivity R is calculated as a function of the film thickness and the transmitted intensity. The critical values of R are determined. The results are a generalization of the linear optics results.

Recovery of Singularities of a Multidimensional Scattering Potential

We prove that in multidimensional potential scattering the leading order singularities of the unknown potential are obtained exactly from the scattering amplitude by the linearized inversion method. The proof is based on the appropriate mapping properties of the fundamental solution in weighted L p-spaces and on a homogeneity argument concerning the bilinear term.

Solutions to the Helmholtz equation for TE-guided waves in a three-layer structure with Kerr-type nonlinearity

We study certain solutions (TE-polarized electromagnetic waves) of the Helmholtz equation on the line describing waves propagating in a nonlinear three-layer structure consisting of a film surrounded by semi-infinite media. All three media are assumed to be lossless, nonmagnetic, isotropic and exhibiting a local Kerr-type dielectric nonlinearity. The linear component of the permittivity is modelled by a continuous real-valued function of the transverse coordinate. We show that the solution of the Helmholtz equation in the form of a TE-polarized electromagnetic wave exists and can be obtained by iterating the equivalent Volterra equation. The associated dispersion equation has a simple root (if the semi-infinite media are linear and if the nonlinearity parameter of the film is sufficiently small) that uniquely determines this solution.

Integral equation approach to reflection and transmission of a plane TE-wave at a (linear/nonlinear) dielectric film with spatially varying permittivity

A method is proposed for obtaining certain solutions (TE-polarized electromagnetic waves) of the Helmholtz equation, describing the reflection and transmission of a plane monochromatic wave at a (linear or nonlinear) dielectric film situated between two linear semi-infinite media. All three media are assumed to be lossless, nonmagnetic and isotropic. The permittivity of the film is modelled by (i) a continuously differentiable real-valued function of the transverse coordinate, and by (ii) a Kerr-nonlinearity. It is shown that the solution of the Helmholtz equation exists in the form of a uniformly convergent series (in case (i)) and in the form of a uniformly convergent sequence (in case (ii)) of iterations of the equivalent Volterra integral equation. Numerical results of the approach are presented.

Some Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation

An approach is proposed to obtain some ex art explicit solutions in terms of elliptic functions to the Novikov-Veselov equation (NTVE[psi(x,y,t)] = 0). An expansion ansatz psi rarr g = Sigma 2 j=0ajfj is used to reduce the NVE to the ordinary differential equation (f)2 = R(f), where R(f) is a fourth degree polynomial in f. The well-known solutions of (f)2 = R(f) lead to periodic and solitary wave like solutions psi. Subject, to certain conditions containing the parameters of the NVE and of the ansatz psi rarr g the periodic solutions psi can be used as start solutions to apply the (linear) superposition principle proposed by Khare and Sukhatme

An inverse Born approximation for the general nonlinear Schrödinger operator on the line

This work deals with the inverse scattering problems for the one-dimensional Schrodinger equation of the most general form with general nonlinearity F and with a real parameter k. Under some assumptions on F we prove that all singularities and jumps of F(x, u0), where u0(x, k) = eikx, can be recovered by the reflection coefficient with arbitrary large k.

On the theory of TM- electromagnetic guided waves in a film with nonlinear permittivity

TM- electromagnetic guided waves propagating in a lossless nonmagnetic structure, consisting of a film with Kerr-type nonlinear permittivity, bounded by two semi-infinite media, are investigated. The problem is reduced to an exact differential equation leading to a first integral that relates the electric field components. Eliminating one field component by means of the first integral an exact analytical relation for the remaining field component is derived. Combination with the boundary conditions yields the exact analytical dispersion relation expressed in terms of elliptic integrals. Numerical results for the propagation constant are presented.

Some elliptic traveling wave solutions to the Novikov-Veselov equation

An approach is proposed to obtain some ex art explicit solutions in terms of elliptic functions to the Novikov-Veselov equation (NTVE[psi(x,y,t)] = 0). An expansion ansatz psi rarr g = Sigma 2 j=0ajfj is used to reduce the NVE to the ordinary differential equation (f)2 = R(f), where R(f) is a fourth degree polynomial in f. The well-known solutions of (f)2 = R(f) lead to periodic and solitary wave like solutions psi. Subject, to certain conditions containing the parameters of the NVE and of the ansatz psi rarr g the periodic solutions psi can be used as start solutions to apply the (linear) superposition principle proposed by Khare and Sukhatme