The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrdinger Equations

This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.

A Method of Wide Convergence Region for Solving Inverse Problems of Wave Equations

Based on the embedding thought, a method of wide convergence region for solving the coefficient inverse problem of wave equations in the space-time variable domain is presented. The numerical simulation shows that the method is feasible and effective.

A NEW APPROACH TO INVERSE PROBLEMS OF WAVE EQUATIONS

We introduce a multi-cost-functional method for solving inverse problems of wave equations. This method has its simplicity, efficiency and good physical interpretation. It has the advantage of being programmed for two- or three- (space) dimensional problems as well as for one-dimensional problems.

A Differential Continuation－Regularization Method for Solving Inverse Problems of Acoustic Wave Equations

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THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE SMOOTH MIXED BOUNDARY CONDITIONS

The spectral distribution exp( ), where {} are the eigenvalues of the negative Laplacian -△=- in the (x^1,x^2)-plane, is studied for a variety of domains, where -∞< t <∞ and i=(1/2)(-1) . The dependence of (t)on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain Ω in R^2 with a smooth boundary Ω, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts Γj(j = 1,……,n) of Ω are considered such that Some geometrical properties of Ω(e.g., the area of Ω, the total lengths of the boundary, the curvature of its boundary, etc.) are determined, from the asymptotic expansions of (t) for |t| → 0.

Difference inversion model of wave equation

A numerical iterative model was derived from the difference method and a perturbation assumption to calculate the coefficient function of a wave equation. The method was used to solve the disaccord problem of numerical precision between the direct problem model and inverse problem model, and its serial problems using the old method. Numerical simulation calculation shows that the method is feasible and effective.

Some Universal Properties of the Green’s Functions Associated with the Wave Equation in Bounded Partially-Homogeneous Domains and Their Use in Acoustic Tomography

Direct and inverse scattering problems connected with the wave equation in non-homogeneous bounded domains constitute challenging actual subjects for both mathematicians and engineers. Among them one can mention, for example, inverse source problems in seismology, nondestructive archeological probing, mine prospecting, inverse initial-value problems in acoustic tomography, etc. In spite of its crucial importance, almost all of the available rigorous investigations concern the case of unbounded simple domains such as layered planar or cylindrical or spherical structures. The main reason for the lack of the works related to non-homogeneous bounded structures is the extreme complexity of the explicit expressions of the Green’s functions. The aim of the present work consists in discovering some universal properties of the Green’s functions in question, which reduce enormously the difficulties arising in various applications. The universality mentioned here means that the properties are not depend on the geometrical and physical properties of the configuration. To this end one considers first the case when the domain is partially-homogeneous. Then the results are generalized to the most general case. To show the importance of the universal properties in question, they are applied to an inverse initial-value problem connected with photo-acoustic tomography.

Reconstruction of density and wave velocity from reflection and transmission data

Consider an inverse problem of reconstructing the coefficient in a linearwave equation on an inhomogeneous slab with density ρ(z) and wave velocity c(z). The inversioninput information is the reflection and transmission data corresponding to a point source. Byapplying the characteristic theory for hyperbolic equations, we establish an integral system fromwhich ρ(z) and c(z) can be recovered simultaneously. In contrast to some known results, our inverseapproach is carried out for depth variable, rather than for travel-time variable. Thereforeinversion results in this paper are more appropriate for the physical interpretation of a mediumslab.

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

An Alternative Method to Study Wave Scattering by Semi-infinite Inertial Surfaces

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green＇s second identity to the potential functions and appropriate Green＇s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.

基于弹性反射波正演表达的转换波逆时偏移与反偏移

针对线性化弹性反射波方程无法直接用于弹性转换波反偏移模拟的问题,本文提出基于高频近似的一次反射波模拟新方法.该方法基于地震波传播的散射理论,先对地下速度扰动的空间变化进行高频近似,再基于散射波传播方程推导得到线性化一次反射波传播方程.基于弹性反射地震数据表达,本文采用线性算子的伴随估计反射系数,实现反射波逆时偏移.根据该线性表达,联合背景速度和成像值来模拟一次反射横波传播,实现了弹性波角度域反偏移,并进一步导出成像值关于波速的梯度.数值实验表明,基于弹性反射地震数据表达的偏移成像方法,既克服了极性反转效应,又使成像结果包含了更为丰富的反射角信息;一次反射波模拟方法获得的弹性地震数据具有较高的精度和稳定性,得到的弹性波场的振幅和相位信息与理论解析解吻合较好,有效保证了成像域层析速度反演梯度的稳定求解.

The Sasa-Satsuma equation with high-order discrete spectra in space-time solitonic regions:soliton resolution via the mixed■-Riemann-Hilbert problem

In this paper,we investigate the Cauchy problem of the Sasa-Satsuma(SS)equation with initial data belonging to the Schwartz space.The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3×3 Lax representation.With the aid of the■nonlinear steepest descent method of the mixed■-Riemann-Hilbert problem,we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.

声波反散射问题的正则化集合卡尔曼方法

集合卡尔曼滤波在处理反问题时存在稳定性不够、对初值集合依赖过大等不足,文章将集合卡尔曼滤波算法与正则化思想相结合处理声波反散射问题。在集合卡尔曼滤波框架中加入正则化思想,根据最大后验估计给出滤波过程的具体计算步骤,并进行仿真实验。仿真结果表明,该正则化集合卡尔曼滤波在处理声波反散射问题时滤波精度、稳定性都优于传统的集合卡尔曼滤波。

Uniqueness and local stability for the inverse scattering problem of determining the cavity

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.

ON AN INVERSE PROBLEM FOR 1-DIMENSIONAL WAVE EQUATION

The inverse problem for the 1-dimensional acoustic wave equation is discussed to deter-mine propagation velocity from impulse response. A relation between the propagation velocityand the wavefield can be established from the analysis of propagation of discontinuities forhyperbolic equations. As a result, the inverse problem discussed in this paper is reduced to aparticular initial value problem of a semilinear system of P. D. E.. The Picard iteration forsolving this initial value problem is constructed and the convergence of iteration is proved.The main results are the following: (i) the propagation velocity can always be recovered fromthe impulse response, unless the inverse problem contains a singular point, where the propa-gation velocity is infinite or zero, or its total variation in the neighborhood of the singularpoint is infinite; (ii) the stability behaviour of the solutions of this inverse problem is es-sentially dependent on the total variation of logarithm of propagation velocity.

The inverse problem based on a full dispersive wave equation

The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.

Estimation of interfacial damage in composites with reinforced particles based on ultrasonic data

The scattering of elastic waves by a spherical particle with imperfect interface and the nondestructive detection of interfacial damage were studied. First, the scattering of elastic waves by a spherical particle with imperfect interface, i.e. spring interface model, was studied. Then, multiple scattering by random distributed particles was investigated and the equations to evaluate the velocity and attenuation of effective waves defined by statistic averaging were given. Furthermore, on the basis of the established relation between the velocity and interfacial constants, a method to evaluate the interfacial damage nondestructively from the ultrasonic data was pro- posed. Numerical simulation was performed for the SiC-Al composites. The velocities of the effective waves were computed to show the influence of the interface constants. Using the genetic algorithm, the interfacial damage was evaluated from the synthetic experi- mental data with various noise levels. The numerical results showed the feasibility of the method proposed.

DISCRETENESS OF THE EXTERIOR TRANSMISSION EIGENVALUES

In this paper we consider a kind of exterior transmission problem in which the refractive index n（x） is a piecewise positive constant. Through establishing an equivalent boundary integral system, we obtain that the set of exterior transmission eigenvalues is a discrete set. Furthermore, we prove that there does not exist a transmission eigenvalue under some conditions.

ON THE SECOND ORDER WAVE DIFFRACTION IN TWO LAYER FLUIDS

With assumption that the stratified ocean consists of two layer fluids with different densities, the problem of the second order wave diffraction by three dimensional bodies in the stratified ocean was investigated. The boundary value problem of the second order multi-chromatic wave scattering potential was firstly formulated based on a weakly radiation condition and with the use of regular perturbation method, and a formal solution was then found. Using the Green theorem and introducing an assisting potential, the integral expressions, which do not explicitly connect with the second order scattering potential of the second order wave loads were also derived. The analysis indicates that the effects of the stratification upon the second order difference frequency wave loads on the structures may be significant.

Inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrodinger equation with zero/nonzero boundary conditions

The present work studies the inverse scattering transforms(IST)of the inhomogeneous fifth-order nonlinear Schrodinger(NLS)equation with zero boundary conditions(ZBCs)and nonzero boundary conditions(NZBCs).Firstly,the bound-state solitons of the inhomogeneous fifth-order NLS equation with ZBCs are derived by the residue theorem and the Laurent's series for the first time.Then,by combining with the robust IST,the Riemann-Hilbert(RH)problem of the inhomogeneous fifth-order NLS equation with NZBCs is revealed.Furthermore,based on the resulting RH problem,some new rogue wave solutions of the inhomogeneous fifth-order NLS equation are found by the Darboux transformation.Finally,some corresponding graphs are given by selecting appropriate parameters to further analyze the unreported dynamic characteristics of the corresponding solutions.