The inverse scattering problem for partially penetrable obstacles

We consider the direct and inverse problems for the scattering of a partially penetrable obstacle. Here ‘partially penetrable obstacle’ means that the waves transmit into the obstacle just from partial boundary of the obstacle with the rest of the boundary touching a known perfect and thin scatterer. The solvability of the direct scattering problem is presented using the classical boundary integral equation method. An interesting interior transmission problem is investigated for the purpose of solving the inverse obstacle scattering problem. Then the linear sampling method is proposed to reconstruct the shape and location of the obstacle from near field measurements. We note that the inversion algorithm can be implemented by avoiding the use of background Green function as a test function due to a mixed reciprocal principle.     

The DBAR Problem and the Thirring Model

We study the Thirring model by means of a DBAR-Riemann-Hilbert problem. We undertake a rigorous study for both the direct and the inverse problem on the class of rapidly decaying potentials. This study includes a discussion of the existence and uniqueness of solutions of the direct and inverse problems, a characterization of the solutions, and an explicit expression for the scattering data in terms of initial scattering functions.     

Inverse scattering from an orthotropic medium

We consider the inverse scattering problem of determining the shape of an orthotropic homogeneous medium from a knowledge of the far field pattern corresponding to incident time harmonic plane waves. We solve the direct problem by potential theory, show the uniqueness of the inverse problem and then use a ‘simple’ method recently developed by Colton and Kirsch (1996) and Colton and Monk (1997) to solve the inverse problem. Numerical examples are given showing the practicality of our method for solving the inverse problem.     

The propagation problem and far-field patterns in a stratified finite-depth ocean

In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves. A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].     

The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line

The initial value problem for the Kadomstev–Petviashvili II (KPII) equation is considered with given data that are nondecaying along a line. The associated direct and inverse scattering of the two-dimensional heat equation is constructed. The direct problem is formulated in terms of a bounded Green's function. The inverse data are decomposed into scattering data along the line and     data from the decaying portion of the potential. The solution of the KPII equation is then obtained via coupled linear integral equations.     

On the direct and inverse problems in fluid-structural interaction

This paper is concerned with the direct and inverse scattering problems in fluid-structure interaction. The scattering problem in the fluid-structure interaction can be simply described as follows: an acoustic wave propagates in the fluid domain of infinite extent where a bounded elastic body is immersed. The direct problem is to determine the scattered pressure and velocity fields in the fluid domain as well as the displacement fields in the elastic body, while the inverse problem is to reconstruct the shape of the elastic scatterer from a knowledge of the far field pattern of the fluid pressure or from the measured scattered fluid pressure field. As is well known, the inverse problems are generally nonlinear and highly ill-posed. For treating inverse problem of this kind, we reformulate the problem as a nonlinear optimization problem including special regularization terms. The precise formulation of the nonlinear objective functional will depend on the approaches of the direct problem. In this paper, the direct problem is reformulated by introducing an artificial boundary and the corresponding inverse problem will be analyzed. Some of the basic results are summarized without proofs. The latter are available in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems

Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

Inverse Scattering Transform for the Focusing Ablowitz‐Ladik System with Nonzero Boundary Conditions

The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz‐Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann‐Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz‐Ladik potential.     

Uniqueness For An Inverse Boundary Value Problem For Dirac Operators

The uniqueness of both the inverse boundary value problem and inverse scattering problem for Dirac equation with a magnetic potential and an electrical potential are proved. Also, a relation between the Dirichlet to Dirichlet map for the inverse boundary value problem and the scattering amplitude for the inverse scattering problem is given     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

Dense sets and the projection theorem for acoustic harmonic waves in a homogeneous finite depth ocean

The scattering of a ‘plane wave’ off a submerged body situated in an ocean of finite depth is investigated. The index of refraction is considered to be depth-independent. It is shown that the far field is not unique; hence, the problem of determining the shape of an object from its far field is not well-posed. If solutions are sought among a restricted class of problems the ‘dense set’ property implies that the problem can be made well-posed.     

An inverse spectral problem for complex Jacobi matrices

We introduce the concept of generalized spectral function for finite order complex Jacobi matrices and solve the inverse problem with respect to the generalized spectral function. The results obtained can be used for solving of initial-boundary value problems for finite nonlinear Toda lattices with the complex-valued initial conditions by means of the inverse spectral problem method.     

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.     

The Inverse Scattering Transform for the Benjamin-Ono Equation—A Pivot to Multidimensional Problems

The initial value problem associated with the Benjamin-Ono equation is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation and solution of an associated nonlocal Riemann-Hilbert problem in terms of initial scattering data. Solitons are given a definitive spectral characterization. Pure soliton solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on [INLINEEQUATION], [INLINEEQUATION].     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

Inverse Scattering Problem For The Schrödinger Equation With An Additional Quadratic Potential On The Entire Axis

We consider the Schrödinger equation with an additional quadratic potential on the entire axis and use the transformation operator method to study the direct and inverse problems of the scattering theory. We obtain the main integral equations of the inverse problem and prove that the basic equations are uniquely solvable.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

Explicit solutions of an integrable boundary value problem for the two-dimensional toda lattice

We obtain an explicit solution of the integrable boundary value problem for the two-dimensional Toda lattice using the inverse scattering method. We interpret the integrability property in terms of the corresponding linear problem, the Gel’fand-Levitan-Marchenko equation, and the dressing procedure. The simplest initial solutions of the boundary value problem become new nontrivial solutions after the dressing procedure is applied.