Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.We introduce the generalized Lax equation. This enables us to derive the time-evolution of the transition operator. Then, the time-dependent intermediate operator is constructed. The solution of the considered Cauchy problem is expressed in terms of solutions of the fundamental equations in inverse problem. This solution is found uniquely from the given initial condition.     

Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.     

Scattering in a Forked-Shaped Waveguide

We consider wave scattering in a forked-shaped waveguide which consists of two finite and one half-infinite intervals having one common vertex. We describe the spectrum of the direct scattering problem and introduce an analogue of the Jost function. In case of the potential which is identically equal to zero on the half-infinite interval, the problem is reduced to a problem of the Regge type. For this case, using Hermite-Biehler classes, we give sharp results on the asymptotic behavior of resonances, that is, the corresponding eigenvalues of the Regge-type problem. For the inverse problem, we obtain sufficient conditions for a function to be the S-function of the scattering problem on the forked-shaped graph with zero potential on the half-infinite edge, and present an algorithm that allows to recover potentials on the finite edges from the corresponding Jost function. It is shown that the solution of the inverse problem is not unique. Some related general results in the spectral theory of operator pencils are also given. This work was supported by the grant UM1-2567-OD-03 from the Civil Research and Development Foundation (CRDF). YL was partially supported by the NSF grants 0338743, 0354339 and 0754705, by the Research Board and Research Council of the University of Missouri, and by the EU Marie Curie “Transfer of Knowledge” program.     

基于遗传算法重建多个散射体的组合Newton法

本文研究由单个入射声波或电磁波及其远场数据反演多个柔性散射体边界的逆散射问题.通过建立边界到边界总场的非线性算子及其n6chet导数,本文首先给出了基于单层位势的组合Newton法.将组合Newton法转化为泛响优化问题,从而获得了该方法重建单个散射体的收敛性分析.然后,基于遗传算法和正则化参数选取的模型函数方法,给出...     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.     

Scattering Theory for CMV Matrices: Uniqueness,Helson–Szegő and Strong Szegő Theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.     

Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.      

Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

The inverse problem of the scattering theory for Sturm–Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied. Copyright © 2010 John Wiley & Sons, Ltd.     

Aspects of electromagnetic scattering in chiral media

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon‐type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon‐type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far‐field operator that is defined on a unit sphere and contains the far‐field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.     

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.     

An Explicit Example of a Local and a Nonlocal Potential Whose Hamiltonians Are Unitarily Equivalent and Whose Scattering Operators Are Identical

An explicit example is given for the one-dimensional Schrodinger equation in which two unitarily equivalent Hamiltonians, one with a local scattering potential and the other with a nonlocal scattering potential, have the same scattering operator and bound-state measure. The result has obvious implications for the inverse scattering problem. The unitary operator which maps one Hamiltonian to the other is of interest because it is expressed as the product of two operators, neither of which has an inverse.     

Complex solutions of general Korteweg-de Vries equation: Inverse problem method

The inverse method of scattering problem has been applied to find complex solutions of the general Korteweg-de Vries equation. The direct and inverse problem have been considered for nonself-adjoint one-dimensional Schrödinger operator (with complex potential) in L₂(). The used technique of inverse problems for nonself-adjoint operators has been developed by V. É. Lyantse and his disciples.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 223–230, February, 1990.     

Application of the Factorization Method to Recover Cuts with Oblique Derivative Boundary Condition

Direct and inverse problems for the scattering of cracks with mixed oblique derivative boundary conditions from the incident plane wave are considered, which describe the scattering phenomenons such as the scattering of tidal waves by spits or reefs. The solvability of the direct scattering problem is proven by using the boundary integral equation method. In order to show the equivalent boundary integral system is Fredholm of index zero, some relationships concerning the tangential potential operator is used. Due to the mixed oblique derivative boundary conditions, we cannot employ the factorization method in a usual manner to reconstruct the cracks. An alternative technique is used in the theoretical analysis such that the far field operator can be factorized in an appropriate form and fulfills the range identity theorem. Finally, we present some numerical examples to demonstrate the feasibility and effectiveness of the factorization method.     

Analysis of the Newton-Sabatier Scheme for Inverting Fixed-Energy Phase Shifts

Suppose that the inverse scattering problem is understood as follows: given fixed-energy phase shifts, corresponding to an unknown potential q = q ( r ) from a certain class, for example, q ] L 1,1 , recover this potential. Then it is proved that the Newton-Sabatier (NS) procedure does not solve the above problem. It is not a valid inversion method, in the following sense: (1) it is not possible to carry this procedure through for the phase shifts corresponding to a generic potential q ] L 1,1 , where $ L_{1,1} : = \{ q {:}\, q = \overline q, \int ^\infty _0 r |q(r)| dr \lt \infty \} $ and recover the original potential: the basic integral equation, introduced by Newton without derivation, in general, may be not solvable for some r > 0, and if it is solvable for all r > 0, then the resulting potential is not equal to the original generic q ] L 1,1 . Here a generic q is any q which is not a restriction to (0, X ) of an analytic function. (2) the ansatz (*) $ K(r,s) = \sum^\infty _{l = 0} c_l \varphi _l (r) u_l (s) $ , used by Newton, is incorrect: the transformation operator I m K , corresponding to a generic q ] L 1,1 , does not have K of the form (*), and (3) the set of potentials q ] L 1,1 , that can possibly be obtained by NS procedure, is not dense in the set of all L 1,1 potentials in the norm of L 1,1 . Therefore, one cannot justify NS procedure even for approximate solution of the inverse scattering problem with fixed-energy phase shifts as data. Thus, the NS procedure, if considered as a method for solving the above inverse scattering problem, is based on an incorrect ansatz, the basic integral equation of NS procedure is, in general, not solvable for some r > 0, and in this case this procedure breaks down, and NS procedure is not an inversion theory: it cannot recover generic potentials q ] L 1,1 from their fixed-energy phase shifts. Suppose now that one considers another problem: given fixed-energy phase shifts, corresponding to some potential, find a potential which generates the same phase shifts. Then NS procedure does not solve this problem either: the basic integral equation, in general, may be not solvable for some r > 0, and then NS procedure breaks down.     

On an inverse scattering problem for a class of Dirac operators with spectral parameter in the boundary condition

In this work, we consider the inverse scattering problem for a class of one dimensional Dirac operators on the semi-infinite interval with the boundary condition depending polynomially on a spectral parameter. The scattering data of the given problem is defined and its properties are examined. The main equation is derived, its solvability is proved and it is shown that the potential is uniquely recovered in terms of the scattering data. A generalization of the Marchenko method is given for a class of Dirac operator.     

Nonstrictly hyperbolic systems of conservation laws of the conjugate type

We consider an inverse boundary problem for a general second order self-adjoint elliptic differential operator on a compact differential manifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the operator to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical object in the orbit of the group. This object, the canonical Schrödinger operator, is uniquely determined via its incomplete boundary spectral data.     

Spectral analysis of nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrödinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem are given.