scholarly journals Integration of the Harry Dym equation with an integral type source

2021 ◽  
Vol 31 (2) ◽  
pp. 285-295
Author(s):  
G.U. Urazboev ◽  
A.K. Babadjanova ◽  
D.R. Saparbaeva
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Evolutionary Equation ◽  
Integral Type ◽  
Self Consistent ◽  
The Cauchy Problem ◽  
Nonlinear Evolutionary Equation ◽  
Dym Equation ◽  
Harry Dym Equation

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

AN INVERSE SCATTERING PROBLEM FOR NONLOCAL POTENTIALS I: THE FAMILY OF PHASE EQUIVALENT WAVEFUNCTIONS

1986 ◽  
Vol 01 (07) ◽  
pp. 449-454 ◽  
Author(s):  
V.M. MUZAFAROV
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
The Family ◽  
Consistent Approach ◽  
Nonlocal Potentials

We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.

Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures

2015 ◽  
Vol 23 (2) ◽  
Author(s):  
Michael V. Klibanov ◽  
Vladimir G. Romanov
Keyword(s):  
Born Approximation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattered Wave ◽  
Scattering Data ◽  
Unknown Coefficient ◽  
X Rays ◽  
Reconstruction Formula ◽  
Nano Structures ◽  
Complex Valued

AbstractImaging of nano structures is necessary for the quality control in their manufacturing. In the case when X-rays probe the medium, only the modulus of the complex valued scattered wave field can be measured. The phase cannot be measured. In the case of the Born approximation, we obtain an explicit reconstruction formula for the unknown coefficient from the phaseless scattering data.

A characterisation of the scattering data in the 3D inverse scattering problem

1987 ◽  
Vol 3 (3) ◽  
pp. L49-L52 ◽  
Author(s):  
A G Ramm ◽  
O L Weaver
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data

scholarly journals A robust expectation-maximization method for the interpretation of small-angle scattering data from dense nanoparticle samples

2019 ◽  
Vol 52 (5) ◽  
pp. 926-936
Author(s):  
M. Bakry ◽  
H. Haddar ◽  
O. Bunău
Keyword(s):  
Small Angle ◽  
Expectation Maximization ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Small Angle Scattering ◽  
Fine Tuning ◽  
Scattering Data ◽  
Model Parameters ◽  
Dry Powders ◽  
Angle Scattering

The local monodisperse approximation (LMA) is a two-parameter model commonly employed for the retrieval of size distributions from the small-angle scattering (SAS) patterns obtained from dense nanoparticle samples (e.g. dry powders and concentrated solutions). This work features a novel implementation of the LMA model resolution for the inverse scattering problem. The method is based on the expectation-maximization iterative algorithm and is free of any fine-tuning of model parameters. The application of this method to SAS data acquired under laboratory conditions from dense nanoparticle samples is shown to provide good results.

scholarly journals Two-Step Extended Sampling Method for the Inverse Acoustic Source Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jiyu Sun ◽  
Yuhui Han
Keyword(s):  
Inverse Scattering ◽  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Indicator Function ◽  
Scattering Data ◽  
Acoustic Source ◽  
Scattering Problems ◽  
Linear Sampling Method ◽  
Ill Posed

Recently, a new method, called the extended sampling method (ESM), was proposed for the inverse scattering problems. Similar to the classical linear sampling method (LSM), the ESM is simple to implement and fast. Compared to the LSM which uses full-aperture scattering data, the ESM only uses the scattering data of one incident wave. In this paper, we generalize the ESM for the inverse acoustic source problems. We show that the indicator function of ESM, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. This behavior is similar to the ESM for the inverse scattering problem. Numerical examples are presented to show the effectiveness of the method.

The inverse scattering problem for a discrete dirac system on the whole axis

2017 ◽  
Vol 25 (6) ◽  
Author(s):  
Hidayat M. Huseynov ◽  
Agil K. Khanmamedov ◽  
Rza I. Aleskerov
Keyword(s):  
Inverse Problem ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Sufficient Condition ◽  
Dirac System ◽  
Entire Line

AbstractThis paper investigates the inverse scattering problem for a discrete Dirac system on the entire line with coefficients that stabilize to zero in one direction. We develop an algorithm for solving the inverse problem of reconstruction of coefficients. We derive a necessary and a sufficient condition on the scattering data so that the inverse problem is uniquely solvable.

scholarly journals Integration of the Matrix Nonlinear Schr¨odinger Equation with a Source

2021 ◽  
Vol 37 ◽  
pp. 63-76
Author(s):  
G.U. Urazboev ◽  
◽  
A.A. Reyimberganov ◽  
A.K. Babadjanova ◽  
◽  
...  
Keyword(s):  
Inverse Scattering Method ◽  
The Self ◽  
Scattering Data ◽  
Scattering Method ◽  
Self Consistent ◽  
The Matrix ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Step Algorithm ◽  
Decreasing Functions

This paper is concerned with studying the matrix nonlinear Schr¨odinger equation with a self-consistent source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the matrix Zakharov-Shabat system which has not spectral singularities. The theorem about the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system which potential is a solution of the matrix nonlinear Schr¨odinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schr¨odinger equation with a self-consistent source in the class of the rapidly decreasing functions via the inverse scattering method. A one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data provide the uniqueness of the solution of the considering problem. A step-by-step algorithm for finding a solution to the problem under consideration is presented.

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