The radial Marchenko-Blažek integral equation for complete spectra

The same simple and standard way, by means of which an inverse problem for scattering of spinless particles by central potentials is solved in the Gelfand-Levitan method, is applied to the Marchenko method for general angular momenta including the bound states. We first derive an integral equation for the kernel with a triangularity property, which relates it to a potential and then the other one, which connects the kernel with spectral data. A solution corresponding to the general Yukawa potential is found and some formulae are checked by solving the problem of the restrained phase equivalent potentials.     

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.     

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.     

Practical pulse synthesis via the discrete inverse scattering transform

This paper provides the practical details required to use the inverse scattering (IST) approach to design selective RF-pulses. As in the Shinnar-Le Roux (SLR) approach, we use a hard pulse approximation to actually design the pulse. Unlike SLR, the pulse is designed using the full inverse scattering data (the reflection coefficient and the bound states) rather than the flip angle profile. We explain how to approximate the reflection coefficient to obtain a pulse with a prescribed rephasing time. In contrast to the SLR approach, we retain direct control on the phase of the magnetization profile throughout the design process. We give explicit recursive algorithms for computing the hard pulse from the inverse scattering data. These algorithms are quite different from the SLR recursion, being essentially discretizations of the Marchenko equations. We call our approach the discrete inverse scattering transform or DIST. Overall, it is as fast as the SLR approach. When bound states are present, we use both the left and right Marchenko equations to improve the numerical stability of the algorithm. We compute a variety of examples and consider the effect of amplitude errors on the magnetization profile.     

A new approach to the inverse scattering and spectral problems for the Sturm-Liouville equation

New proofs of the known uniqueness theorems for the one-dimensional inverse spectral and scattering problems are given. Proof of the invertibility of all of the steps in the inversion procedures of Gelfand-Levitan and Marchenko is given. The proposed method of investigation yields some new results, for example, a Marchenko-type equation at x = 0 which holds on the whole axis, rather than on a half-axis, as usual for the scattering theory on half-axis. It also yields a new method, shorter and simpler than earlier published, for proving that the potential in the class L_1,1, obtained by the Marchenko reconstruction procedure, generates the scattering data from which it was reconstructed.     

COMPLETENESS OF THE JOST SOLUTIONS IN THE CASE OF THE NLS+ EQUATION WITH NONVANISHING BOUNDARY CONDITION

In the case of the NLS⁺ equation with nonvanishing boundary condition, a complete set of the Jost solutions is chosen, and its completeness is shown by means of the Marchenko inverse scattering equation.     

Sound propagation and scattering in media with random inhomogeneities of sound speed, density and medium velocity

This review presents both classical and new results of the theory of sound propagation in media with random inhomogeneities of sound speed, density and medium velocity (mainly in the atmosphere and ocean). An equation for a sound wave in a moving inhomogeneous medium is presented, which has a wider range of applicability than those used before. Starting from this equation, the statistical characteristics of the sound field in a moving random medium are calculated using Born-approximation, ray, Rytov and parabolic-equation methods, and the theory of multiple scattering. The results obtained show, in particular, that certain equations previously widely used in the theory of sound propagation in moving random media must now be revised. The theory presented can be used not only to calculate the statistical characteristics of sound waves in the turbulent atmosphere or ocean but also to solve inverse problems and develop new remote-sensing methods. A number of practical problems of sound propagation in moving random media are listed and the further development of this field of acoustics is considered.     

Extension of the Marchenko equation to non-hermitian differential systems

The results of Marchenko and Ljance are unified and extended. We prove the existence of a fundamental equation for an l ≠ 0 non-Hermitian system and identify which the scattering data are. However we have included neither threshold energies nor Coulomb interactions. Conventional methods are used to determine which necessary conditions must be imposed upon the scattering data so that they can be used in the inverse problem for definite angular momentum. Indications are given on how to overcome our restrictions for the proof of the existence of the fundamental equation and the analysis of its properties.     

Sound propagation and scattering in media with random inhomogeneities of sound speed,density and medium velocity

Abstract

This review presents both classical and new results of the theory of sound propagation in media with random inhomogeneities of sound speed, density and medium velocity (mainly in the atmosphere and ocean). An equation for a sound wave in a moving inhomogeneous medium is presented, which has a wider range of applicability than those used before. Starting from this equation, the statistical characteristics of the sound field in a moving random medium are calculated using Born-approximation, ray, Rytov and parabolic-equation methods, and the theory of multiple scattering. The results obtained show, in particular, that certain equations previously widely used in the theory of sound propagation in moving random media must now be revised. The theory presented can be used not only to calculate the statistical characteristics of sound waves in the turbulent atmosphere or ocean but also to solve inverse problems and develop new remote-sensing methods. A number of practical problems of sound propagation in moving random media are listed and the further development of this field of acoustics is considered.     

Zakharov-Shabat Equations for Dark Soliton Solutions to the NLS Equation

For dark soliton solutions of the NLS equation, an inverse scattering transform is redeveloped. Deductions are essentially simplified in terms of an auxiliary spectral parameter from the beginning. Equations of inverse scattering transform in the form of Zakharov-Shabat are found to be simpler than those in the form of Marchenko. An explicate expression for the dark N-soliton solution and its asymptotic behaviors in the limits as t →±∞ are simply derived.     

COMPLETENESS OF THE JOST SOLUTIONS IN THE CASE OF THE NLS+ EQUATION WITH NONVANISHING BOUNDARY CONDITION

In the case of the NLS⁺ equation with nonvanishing boundary condition, a complete set of the Jost solutions is chosen, and its completeness is shown by means of the Marchenko inverse scattering equation.     

A relationship between Gel'fand-Levitan and Marchenko kernels

A new integral equation which relates the output kernels of the Gel'fand-Levitan and Marchenko inverse scattering equations in a continuous range of their variables is specified. Structural details of this integral equation are studied when theS-matrix is a rational function, and the output kernels are separable in terms of Bessel, Hankel and Jost solutions.     

Three-dimensional radiative transfer in an anisotropically scattering, semi-infinite medium: generalized reflection function

Three-dimensional radiative transfer in an anisotropic scattering medium exposed to spatially varying, collimated radiation is studied. The generalized reflection function for a semi-infinite medium with a very general scattering phase function is the focus of this investigation. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The integration is over both the polar and azimuthal angles; hence, the integral equation is said to be in the double-integral form. The double-integral, reflection function formulation can handle a variety of anisotropic phase functions and does not require an expansion of the phase function in a Legendre polynomial series. Complicated kernel transformations of previous single-integral studies are eliminated. Single and double scattering approximations are developed. Numerical results are presented for a Rayleigh phase function to illustrate the computational characteristics of the method and are compared to results obtained with the single-integral method. Agreement between the two approaches is excellent; however, as the transform variable increases beyond five the number of quadrature points required for the double-integral method to produce accurate solutions significantly increases. A new interpolation scheme produces accurate results when the transform variable is large.     

Fredholm determinants and inverse scattering problems

The Gel'fand-Levitan and Marchenko formalisms for solving the inverse scattering problem are applied together to a single set of scattering phase-shifts. The result is an identity relating two different types of Fredholm determinant. As an application of the method, an asymptotic formula of high accuracy is derived for a particular Fredholm determinant that determines the level-spacing distribution-function in the theory of random matrices.Research sponsored by the National Science Foundation, Grant No. GP-40768X.     

Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material

This paper provides a temporal model of the direct and inverse scattering problem for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material having a rigid frame. This new time domain model of wave propagation takes into account the viscous and thermal losses of the medium as described by the model of Johnson et al. [D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid. Mech. 176, 379 (1987)] and Allard [J. F. Allard (Chapman and Hall, London, 1993)] modified by a fractional calculus based method applied in the time domain. This paper is devoted to the analytical calculus of acoustic field in a slab of porous material. The main result is the derivation of the expression of the scattering operators (reflection and transmission) which are the responses of the medium to an incident acoustic pulse. In this model the reflection operator is the sum of two contributions: the first interface and the bulk of the medium. Experimental and numerical results are given as a validation of our model.     

Inverse scattering for the one-dimensional Helmholtz equation: fast numerical method

The inverse scattering problem for the one-dimensional Helmholtz wave equation is studied. The equation is reduced to a Fresnel set that describes multiple bulk reflection and is similar to the coupled-wave equations. The inverse scattering problem is equivalent to coupled Gel'fand-Levitan-Marchenko integral equations. In the discrete representation its matrix has T?plitz symmetry, and the fast inner bordering method can be applied for its inversion. Previously the method was developed for the design of fiber Bragg gratings. The testing example of a short Bragg reflector with deep modulation demonstrates the high efficiency of refractive-index reconstruction.     

An Inverse Scattering Transformation for the Modified Nonlinear Schrijdinger Equation

A new Lax pair of the modified nonlinear Schrödinger equation is introduced in terme of the variable of the Fourier transform λ. The Lax pair has no usual symmetries between 12 and 21 elements and avoids the factor λ^1/2. The basic equation of inverse scattering transformation is deduced in the Zakharov-Shabat form as well as in the Marchenko form.     

平面介质波导折射率剖面重建的欠松弛迭代方法

本文研究存在导(行)模时Gel'fand-Levitan-Marchenko方程的数值解法。由于反射系数在复数k平面正虚数轴上有极点,其特征函数中出现指数增长项,应用数值迭代求解到一定距离后便产生发散。为了克服这一困难,我们在迭代中采用了松弛方法,通过引入欠松弛因子延伸了势函数重建的有效距离。应用Schrodinger方程下的比例变换关系,逆散射所重建势函数可以直接用于介质波导折射率剖面设计。     

Finite difference computation of acoustic scattering by small surface inhomogeneities and discontinuities

The use of finite difference schemes to compute the scattering of acoustic waves by surfaces made up of different materials with sharp surface discontinuities at the joints would, invariably, result in the generations of spurious reflected waves of numerical origin. Spurious scattered waves are produced even if a high-order scheme capable of resolving and supporting the propagation of the incident wave is used. This problem is of practical importance in jet engine duct acoustic computation. In this work, the basic reason for the generation of spurious numerical waves is first examined. It is known that when the governing partial differential equations of acoustics are discretized, one should only use the long waves of the computational scheme to represent or simulate the physical waves. The short waves of the computational scheme have entirely different propagation characteristics. They are the spurious numerical waves. A method by which high wave number components (short waves) in the wave scattering process is intentionally removed so as to minimize the scattering of spurious numerical waves is proposed. This method is implemented in several examples from computational aeroacoustics to illustrate its effectiveness, accuracy and efficiency. This method is also employed to compute the scattering of acoustic waves by scatterers, such as rigid wall acoustic liner splices, with width smaller than the computational mesh size. Good results are obtained when comparing with computed results using much smaller mesh size. The method is further extended for applications to computations of acoustic wave reflection and scattering by very small surface inhomogeneities with simple geometries.     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.