Method Of Inverse Scattering Problem Research Articles

Integration of the mKdV equation with time-dependent coefficients, with an additional term and with an integral source in the class of rapidly decreasing functions

The work is devoted to the integration of the modified Korteweg–de Vries equation with time-dependent coefficients, an additional term and an integral source in the class of rapidly decreasing functions using the inverse scattering problem method. In this paper, we consider the case when the Dirac operator included in the Lax pairs is not self-adjoint, therefore the eigenvalues of the Dirac operator can be multiples. The evolution of scattering data is obtained for the non-self-adjoint Dirac operator, the potential of which is a solution of the modified Korteweg–de Vries equation with time-dependent coefficients, with an additional term and with an integral source of a class of rapidly decreasing functions. An example is given to illustrate the application of the results obtained.

Deep Neural Network-Oriented Indicator Method for Inverse Scattering Problems Using Partial Data

We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial data inverse problems.

Integration of the Loaded Sine-Gordon Equation by the Inverse Scattering Problem Method

Integration of the Loaded Sine-Gordon Equation by the Inverse Scattering Problem Method

Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model

This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge.

Numerical Verification of the Convexification Method for a Frequency-Dependent Inverse Scattering Problem with Experimental Data

Numerical Verification of the Convexification Method for a Frequency-Dependent Inverse Scattering Problem with Experimental Data

Hybrid Method for Inverse Elastic Obstacle Scattering Problems

The problem of determining the shape of an object from knowledge of the far-field of a single incident wave in two-dimensional elasticity was considered. We applied an iterative hybrid method to tackle this problem. An advantage of this method is that it does not need a forward solver, and therefore, the exact boundary condition is not essential. By deriving the Fréchet derivatives of two boundary operators, we established reconstruction algorithms for objects with Dirichlet, Neumann, and Robin boundary conditions; by introducing a general boundary condition, we also established the reconstruction algorithm for objects with unknown physical properties. Numerical experiments showed the effectiveness of the proposed method.

Bayesian Method for Inverse Scattering Problem of Interior Cavity with Phaseless Data

This paper is concerned with the inverse acoustic scattering problem of a sound soft cavity based on phaseless near-field data. By combining the reference circle technique with the Bayesian method, this paper presents a numerical algorithm to retrieve the shape of the interior cavity from phaseless data. In this paper, the well-posedness of the Bayesian method is discussed. The posterior distribution of the shape parameters is obtained by DRAM algorithm and the reconstruction of the cavity boundary is acquired by taking the average value of the posterior distribution.

A Modified Major Current Coefficients Method for Inverse Scattering Problems Using Multiple-Frequency Data

In this letter, a modified major current coefficients method (M-MCC), which combines the major current coefficients method (MCC), the hierarchical scheme, generalized Tikhonov regularization, and frequency rejection mechanism is proposed to solve inverse scattering problems using multiple-frequency data. The M-MCC utilizes the reconstruction results obtained at lower frequencies as prior information to improve the higher frequency result. The hierarchical scheme utilizes the reconstruction results of the lower frequency to refine the area of objects by using a threshold to judge if the area includes the objects. The inversion at higher frequencies will only be done for the refined area instead of the domain of interests. The generalized Tikhonov regularization is used at higher frequencies, which makes use of the reconstruction results at lower frequencies. In addition, the frequency rejection mechanism also utilizes the result of the lower frequency to exclude the frequencies that have invalid data. The proposed method outperforms the MCC method, which is verified through simulations.

Time Domain Linear Sampling Method for Inverse Scattering Problems with Cracks

Time Domain Linear Sampling Method for Inverse Scattering Problems with Cracks

The generalized nonlinear Schrödinger-like equation of cosmogonical body forming: Justification and determination of its particular solutions

This work justifies the generalized Schrödinger-like equation with logarithmic nonlinearity in the statistical theory of cosmogonical body formation. Within the framework of this theory, the models and evolution equations of the statistical mechanics have been proposed, while well-known problems of gravitational condensation of infinite distributed cosmic substances have been solved on the basis of the proposed statistical model of spheroidal bodies. Equation relative to the complete time-derivative of the common (hydrodynamic plus anti-diffusion) velocity is obtained. Taking into account equations obtained the generalized nonlinear time-dependent Schrödinger-like equation of cosmogonical body formation is derived. This paper investigates different particular forms of the generalized nonlinear time-dependent Schrödinger-like equation corresponding to the respective dynamical states of a forming spheroidal body. The cubic nonlinear time-dependent Schrödinger equation describing cosmogonical body formation in the state of nonlinear wave disturbances is derived. The soliton solution of the one-dimensional cubic nonlinear Schrödinger equation is obtained her e. This paper considers the determination of nonlinear wave solutions from the point of view of the inverse scattering problem method as well as the modified (G/G′)-expansion method.

Integral Equation Method for Inverse Scattering Problem from the Far-Field Data

Integral Equation Method for Inverse Scattering Problem from the Far-Field Data

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

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.

An Improved Subspace-Regularized DBIM-MLGFIM Method for Three-Dimensional Inverse Scattering Problems

The repetitive computation of the forward problem in distorted Born iterative method (DBIM)-type inversion method is time-consuming and memory-costly, especially in dealing with 3-D inverse scattering problem. An improved subspace-regularized DBIM is proposed for the efficient solution of the 3-D inverse problem, which introduces the multilevel Green’s function interpolation method (MLGFIM) to accelerate the computation of the forward problem. In order to stabilize the inversion, an improved subspace-regularized DBIM is used with one new subspace-regularized induced current term in the cost function in virtue of truncated singular value decomposition (TSVD). To further reduce the computation burden, the incident field induced with updated inhomogeneous background medium is used approximately as the total field in the cost function, which saves one forward solver calculation in each iteration. Besides, the resource-intensive multiplication of inhomogeneous Green’s function can be avoided, which further improves the efficiency. Three numerical and two Fresnel experimental examples are given to demonstrate both the efficiency of the forward solver and the accuracy of the inversion algorithm.

Multiplicatively Regularized Iterative Updated Background Inversion Method for Inverse Scattering Problems

A multiplicatively regularized iterative updated background method (MR-IUBM) based on difference integral equations is proposed to solve the electromagnetic inverse scattering problems (ISPs) involving both inhomogeneous and homogeneous background media. This aims to improve the ability for solving highly nonlinear ISPs with source-typed inversion solver by consistently updating the inhomogeneous background media from the updated unknown scatterers via the use of the Green's function with homogeneous medium. In virtue of MR into the cost function, the stability and the capability of tackling highly nonlinear ISPs can be further greatly improved. Synthetic and experimental results are presented to illustrate the efficiency of this method.

Extended Sampling Method for Inverse Elastic Scattering Problems Using One Incident Wave

We consider the inverse elastic scattering problems using the far field data due to one incident plane wave. A simple method is proposed to reconstruct the location and size of the obstacle using different components of the far field pattern. The method sets up linear ill-posed integral equations for sampling points in the domain of interrogation and uses the (approximate) solutions to compute indicators. Using the far field patterns of rigid disks as the kernels of the integral equations and moving the measured data to the right-hand side, the method has the ability to process limited aperture data. Numerical examples show that the method can effectively determine the location and approximate the support of the obstacle with little a priori information.

A resonant soliton in a waveguide in the presence of a graphene monolayer

A theory of an optical self-induced transparency soliton propagating in a waveguide in the presence of a graphene monolayer is constructed. The perturbation theory based on the inverse scattering problem method is used for obtaining explicit analytical expressions for parameters of the (2π pulse) soliton. It is demonstrated that taking into consideration the effects of conductivity in the graphene monolayer leads to exponential variation of the amplitude (width) of the waveguide-mode soliton during propagation.

Multiplicative-Regularized FFT Twofold Subspace-Based Optimization Method for Inverse Scattering Problems

In this paper, we combine two techniques together, i.e., the fast Fourier transform-twofold subspace-based optimiza- tion method (FFT-TSOM) and multiplicative regularization (MR) to solve inverse scattering problems. When applying MR to the objective function in the FFT-TSOM, the new method is referred to as MR-FFT-TSOM. In MR-FFT-TSOM, a new stable and effec- tive strategy of regularization has been proposed. MR-FFT-TSOM inherits not only the advantages of the FFT-TSOM, i.e., lower computational complexity than the TSOM, better stability of the inversion procedure, and better robustness against noise compared with the SOM, but also the edge-preserving ability from the MR. In addition, a more relaxed condition of choosing the number of current bases being used in the optimization can be obtained compared with the FFT-TSOM. Particularly, MR-FFT-TSOM has even better robustness against noise compared with the FFT- TSOM and multiplicative regularized contrast source inversion (MR-CSI). Numerical simulations including both inversion of synthetic data and experimental data from the Fresnel data set validate the efficacy of the proposed algorithm.

“Smart Design” of Quantum Wells and Double-Quantum Wells Structures

In the work, we propose an approach to “smart design” of heterostructures (quantum wells and superlattices) based on the combination of Inverse Scattering Problem Method and the direct solution of the eigenvalue problem for the Schr?dinger equation with reconstructed potentials. Potential shape reconstructed in this way can be substituted then by some approximation, so that the output spectrum obtained by solving the Schr?dinger equation with such approximated potential, differs only slightly from the input one. In our opinion, the approach can be used in many applications, for instance, for developing the new electronic devices such as tunable THz detectors.

Fast Methods for Three-dimensional Inverse Obstacle Scattering Problems

We study the inverse problem to reconstruct the shape of a three– dimensional sound–soft obstacle from measurements of scattered acoustic waves. To solve the forward problem we use a wavelet based boundary element method and prove 4th order accuracy both for the evaluation of the forward solution operator and its Frechet derivative. Moreover, we discuss the characterization and implementation of the adjoint of the Frechet derivative. For the solution of the inverse problem we use a regularized Newton method. The boundaries are represented by a class of parametrizations, which include non star–shaped domains and which are not uniquely determined by the obstacle. To prevent degeneration of the parametrizations during the Newton iteration, we introduce an additional penalty term. Numerical examples illustrate the performance of our method.

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.