scholarly journals GAUGE EQUIVALENCE BETWEEN THE TWO-COMPONENT GENERALIZATION OF THE (2+1)-DIMENSIONAL DAVEY-STEWARTSON I EQUATION AND 𝜞 - SPIN SYSTEM

2020 ◽  
Vol 6 (334) ◽  
pp. 68-73
Author(s):  
N.S. Serikbayev ◽  
◽  
G. N. Nugmanova ◽  
A.A. Meirmanova ◽  
◽  
...  
Keyword(s):  
Spin System ◽  
Linear Equations ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Lax Pair ◽  
Scientific Publications ◽  
Gauge Equivalence ◽  
Zero Curvature ◽  
Two Component ◽  
Component Representation

In recent years, multidimensional nonlinear evolutionary equations have been actively studied within the framework of the theory of solitons. Their relevance is confirmed by numerous scientific publications. In this work the gauge equivalence between the (2+1)-dimensional integrable two-component Davey-Stewartson I (DSI) equation and the Г - spin system is established. Multicomponent generalizations of nonlinear integrable equations are of current interest from both physical and mathematical points of view. In the theory of integrable (soliton) equations, one of the key models is integrable nonlinear Schrodinger-type (NLS) equations. A two-component integrable generalization of the (2+1)-dimensional DSI equation, obtained on the basis of its one-component representation, and its corresponding Lax representation were proposed. A geometric connection between the twolayer spin system and the integrable two-component Manakov system is found. The nonlinear equations are integrated using the inverse scattering problem method by means of a linear system. For each integrable nonlinear equation, as is known, there is a Lax pair of two linear equations, a compatibility condition, that is, a condition of zero curvature, which this equation serves. We have obtained Lax pair whose zero curvature condition gives the Г - spin system.

The 3-D distribution of sources of optical scattering computed from complex-amplitude far-field data

1976 ◽  
Vol 54 (19) ◽  
pp. 1925-1936 ◽  
Author(s):  
D. K. Lam ◽  
H. G. Schmidt-Weinmar ◽  
A. Wouk
Keyword(s):  
Gaussian Elimination ◽  
Singular Systems ◽  
Linear Equations ◽  
Computing Time ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Optical Scattering ◽  
Far Field ◽  
First Approximation ◽  
Systems Of Linear Equations

We present a fast computer algorithm to solve the scalar inverse scattering problem numerically by inverting a linear transformation which maps a 3-D distribution of scattering sources into the angular distribution of the resultant scattered far field. We show how an approximate solution to the problem can be found in discrete form which leads to non-singular systems of linear equations of a type whose matrix can be inverted readily by fast algorithms.The method uses Born's first approximation and is valid for a slowly varying refractive index: the resultant numerical problem can be solved by a fast algorithm which reduces computing time by ~10−7, storage requirement by ~10−5, as compared with Gaussian elimination applied to 125 000 sample points. With this algorithm, computerized 3-D reconstruction becomes feasible.

scholarly journals Nonlinear Electromagnetic Inverse Scattering Imaging Based on IN-LSQR

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Huilin Zhou ◽  
Youwen Liu ◽  
Yuhao Wang ◽  
Liangbing Chen ◽  
Rongxing Duan
Keyword(s):  
Inverse Scattering ◽  
Large Scale ◽  
Linear Equations ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Least Square ◽  
Qr Factorization ◽  
Inexact Newton Method ◽  
Nonlinear Inversion ◽  
Electromagnetic Inverse Scattering

A nonlinear inversion scheme is proposed for electromagnetic inverse scattering imaging. It exploits inexact Newton (IN) and least square QR factorization (LSQR) methods to tackle the nonlinearity and ill-posedness of the electromagnetic inverse scattering problem. A nonlinear model of the inverse scattering in functional form is developed. At every IN iteration, the sparse storage method is adopted to solve the storage and computational bottleneck of Fréchet derivative matrix, a large-scale sparse Jacobian matrix. Moreover, to address the slow convergence problem encountered in the inexact Newton solution via Landweber iterations, an LSQR algorithm is proposed for obtaining a better solution of the internal large-scale sparse linear equations in the IN step. Numerical results demonstrate the applicability of the proposed IN-LSQR method to quantitative inversion of scatterer electric performance parameters. Moreover, compared with the inexact Newton method based on Landweber iterations, the proposed method significantly improves the convergence rate with less computational and storage cost.

scholarly journals Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh-Liem Nguyen ◽  
Trung Truong
Keyword(s):  
Inverse Scattering ◽  
Maxwell Equations ◽  
Field Data ◽  
Periodic Structures ◽  
Near Field ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Factorization Method ◽  
Unique Determination

AbstractThis paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

The inverse elastic scattering by an impenetrable obstacle and a crack

2020 ◽  
Author(s):  
Jianli Xiang ◽  
Guozheng Yan
Keyword(s):  
Mixed Type ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Operator ◽  
Integral Equation Method ◽  
Factorization Method ◽  
Boundary Integral ◽  
Far Field ◽  
Direct Scattering ◽  
Time Harmonic

Abstract This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral matrix operator $A$, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.

Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Leonid L. Frumin
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Numerical Algorithms ◽  
Scattering Problems ◽  
Block Matrices ◽  
Nonlinear Schrödinger ◽  
Direct Scattering ◽  
Vector Case ◽  
Manakov Model ◽  
Vector Variant

AbstractWe introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

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