Mapping Properties of the Nonlinear Fourier Transform in Dimension Two

A class of compactly supported Schrödinger potentials in dimension two is given for which the inverse scattering method related to the Novikov–Veselov evolution equation is well-defined. There is no smallness assumption on the initial potential. Regularity results are proven for the direct and inverse scattering transforms, also called nonlinear Fourier transforms.     

On Nonlinear Kelvin-Helmholtz Waves Near Direct Resonance

The evolution equation for the nonlinear Kelvin-Helmholtz wave envelope with its carrier wavenumber near direct resonance is formulated directly by using the nonlinear dispersion relation. The stability of a wavetrain is examined, and the long-time evolution for an arbitrary initial condition is studied through inverse scattering transforms.     

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.     

The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.     

A multiwave range test for obstacle reconstructions with unknown physical properties

We develop a new multiwave version of the range test for shape reconstruction in inverse scattering theory. The range test [R. Potthast, et al., A ‘range test’ for determining scatterers with unknown physical properties, Inverse Problems 19(3) (2003) 533–547] has originally been proposed to obtain knowledge about an unknown scatterer when the far field pattern for only one plane wave is given. Here, we extend the method to the case of multiple waves and show that the full shape of the unknown scatterer can be reconstructed. We further will clarify the relation between the range test methods, the potential method [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Inverse Problems (Oberwolfach, 1986), Internationale Schriftenreihe zur Numerischen Mathematik, vol. 77, Birkhäuser, Basel, 1986, pp. 93–102] and the singular sources method [R. Potthast, Point sources and multipoles in inverse scattering theory, Habilitation Thesis, Göttingen, 1999]. In particular, we propose a new version of the Kirsch–Kress method using the range test and a new approach to the singular sources method based on the range test and potential method. Numerical examples of reconstructions for all four methods are provided.     

Generating operators for integrable nonlinear evolution equations

For linear problems which are associated with known, exactly integrable nonlinear evolution equations, one gives the corresponding integrodifferential Λ-operators. Relative to the expansions with respect to the elgenfunctions of Λ-operators, the method of the inverse scattering problem can be considered as the analog of the Fourier transform of linear problems, while the Λ-operators are the analogues of the differentiation operator. One considers the equations: Koteweg-de Vries, the nonlinear Schrödinger equations, the nonlinear Schrödinger equations with a derivative, the system of three waves, the matricial analog of the KdV equation, the Toda chain equation.     

Composition of Fourier Integral Operators with Fold and Blowdown Singularities

ABSTRACT

The purpose of this work is to present results about the composition of Fourier integral operators with certain singularities, for which the composition is not again a Fourier integral operator. The singularities considered here are folds and blowdowns. We prove that for such operators, the Schwartz kernel of F*F belongs to a class of distributions associated to two cleanly intersection Lagrangians. Such Fourier integral operators appear in integral geometry, inverse acoustic scattering theory and Synthetic Aperture Radar imaging, where the composition calculus can be used as a tool for finding approximate inversion formulas and for recovering images.     

A Nonlinear Difference Scheme and Inverse Scattering

A nonlinear partial difference equation is obtained and solved by the method of inverse scattering. In a certain continuum limit it is shown how this equation approximates the nonlinear Schrodinger equation and a related nonlinear differential-difference equation. At all times the solutions can be compared, and the scheme is shown to be convergent. These ideas apply to other nonlinear evolution equations as well.     

Closure of the squared Zakharov-Shabat eigenstates

By solving the inverse scattering problem for a third-order (degenerate) eigenvalue problem, we can find the closure of the squared eigenfunctions of the Zakharov-Shabat equations. The question of the completeness of squared eigenstates occurs in many aspects of “inverse scattering transforms” (solving nonlinear evolution equations exactly by inverse scattering techniques) as well as in various aspects of the inverse scattering problem. The method we use is quite suggestive as to how one might find the closure of the squared eigenfunctions of other eigenvalue equations, and we point the strong analogy between our results and the problem of finding the closure of the eigenvectors of a nonself-adjoint matrix.     

Locating an obstacle in a 3D finite depth ocean using the convex scattering support

We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525–1548]. A few examples are presented to show the feasibility of the method.     

Simple Harmonic Generation: an Exact Method of Solution

The general solution for simple harmonic generation is obtained by inverse scattering techniques. The time evolution of the scattering data is found to be much more complicated than in all other cases which are solvable by inverse scattering. It is also pointed out that the instability of the real solutions of simple harmonic generation is related to the singular nature of the scattering data about ζ = 0.     

Estimation of interface damage of fiber-reinforced composites

The estimation of interface damage of fiber-reinforced composites based on the propagation constants of coherent waves is studied in this paper. First, the relation between the interface damage and the propagation constants is investigated by using the theory of multiple scattering. Next, single and multiple scatterings in the composites in the case of incident P, SV, and SH waves are considered. The propagation constants of coherent waves are computed numerically, and the influence of interface damage on them is discussed. Then, based on the relation between the propagation constants and the interface damage, an inverse method to estimate the interface damage is proposed. Finally, a numerical example is given. The numerical results are obtained by using synthetic experimental data and the genetic algorithm. It is shown by the numerical example that the interface damage can be approximately estimated from the wave propagation constants measured with various degrees of accuracy.     

Dressing the boundary: On soliton solutions of the nonlinear Schrödinger equation on the half‐line

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.     

Toward a General Solution of the Three‐Wave Partial Differential Equations

The three‐wave, resonant interaction equations appear in many physical applications. These partial differential equations (PDEs) are known to be completely integrable, and have been solved with initial data that decay rapidly in space, using inverse scattering theory. We present a new way to solve these equations, which makes no use of inverse scattering theory, and which can be used with a wide variety of boundary conditions. A “general solution” of these PDEs would involve six free, real‐valued functions of space. At this time, our “nearly general solution” accepts five free, real‐valued functions of space, and embeds them in convergent series in a deleted neighborhood of a pole.     

On the inverse problem at fixed energy in the "Iterative" range

The scattering amplitude by a spherically symmetric potential at fixed energy is given in the Born approximation by a filtered Fourier transform, whose inverse is not unique. It is well known that matrix methods enable one to study exactly the problem at fixed energy in classes of potentials parametrised by sequences of numbers. In the range of potentials (or of phase shifts) where these methods can be managed by iteration, Born case is a limit. This article is a brief survey of the inverse problem (scattering amplitude?→?potential?) recalling how the nonuniqueness predicted in the Born approximation appears in these exact methods, showing henceforth that the inverse problem ill-posedness corresponds to physical features of the potential on which experiments at finite energy are unable to give information.     

High‐Order Soliton Solution of Landau–Lifshitz Equation

The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well‐posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high‐order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.     

Connections between three-dimensional inverse scattering and linear least-squares estimation of random fields

The three-dimensional Schrödinger equation inverse scattering problem with a nonspherically-symmetric potential is related to the filtering problem of computing the linear leastsquares estimate of the three-dimensional random field on the surface of a sphere from noisy observations inside the sphere. The relation consists of associating an estimation problem with the inverse scattering problem, and vice-versa. This association allows equations and quantities for one problem to be given interpretations in terms of the other problem. A new fast algorithm is obtained for the estimation of random fields using this association. The present work is an extension of the connections between estimation and inverse scattering already known to exist for stationary random processes and one-dimensional scattering potentials, and isotropic random fields and radial scattering protentials.     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

INVERSE SCATTERING BY AN INHOMOGENEOUS PENETRABLE OBSTACLE IN A PIECEWISE HOMOGENEOUS MEDIUM

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.