On a nonlinear scattering model

We obtain conditions for the existence and blow-up of global solutions of systems of nonlinear wave equations with compactly supported initial data and critical nonlinearities arising from the scattering theory of electromagnetic waves.     

Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

In the present paper we introduce the notion of dilation of a multiparametric linear stationary dynamical system (systems of this type, in particular dissipative, and conservative scattering ones were first introduced in [6]). We establish the criterion for existence of a conservative dilation of a multiparametric dissipative scattering system. This allows to distinguish the class of so-calledN-dissipative systems preserving the most important properties of one-parametric dissipative scattering systems.Research supported in part by the Ukrainian-Israeli project of scientific co-operation (contract no. 2M/1516-97).     

Factorization of transfer functions. II. The minimality of passive scattering systems under a step-by-step combination

By the method of functional models of operators and systems one proves necessary and sufficient conditions in order that a step-by-step combination of n minimal passive scattering systems be a minimal system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 488–494, April, 1990.     

Inverse scattering problem for nonstationary Dirac-type systems on the plane

In this paper the forward and inverse scattering problems for the nonstationary Dirac-type systems on the plane are considered. The scattering data for the inverse scattering problem (ISP) is defined and a unique restoration of the potential from the scattering data is proved.     

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Given a normally hyperbolic invariant manifold Λ for a map f, whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future.We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions.We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type.We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64-188] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.     

Landscape of wave focusing and localization at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.     

Multipole methods for boundary-value problems involving a sphere in a tube

It is shown how the same mathematical technique can be usedto solve a number of different problems all concerned with thesame geometry. The multipole method, which can be thought ofas an extension of separation of variables to problems wherethe boundaries lie on the coordinate surfaces of two differentcoordinate systems, is used to solve the problems of potentialflow, Stokes flow, and acoustic scattering for the case of asphere situated on the axis of a circular cylindrical tube.     

Reflection and Refraction of Normal Waves in a Composite Rectangular Orthotropic Waveguide

A numerical-analytic method employing series over basis systems of propagating and edge standing normal waves is used to study dynamic edge effects in the scattering of the lowest order travelling wave at the contact surface between two semi-infinite components of a prismatic waveguide with a rectangular cross section made of single crystals of Rochelle salt and sodium ammonium selenate dihydrate.     

Criterion of unitary similarity of minimal passive scattering systems with a given transfer function

We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.     

On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition

On the positive semi‐infinite interval, we obtained a generalization of the Marchenko method for a Dirac equation system with a discontinuous coefficient and a quadratic polynomial on a spectral parameter in the boundary condition. In this connection, we use an new integral representation of the Jost solution of equation systems, which does not have a ‘triangular’ form. The scattering function of the problem is defined, and its properties are examined. The Marchenko‐type main equation is obtained, and it is shown that the potential is uniquely recovered in terms the scattering function. Copyright © 2012 John Wiley & Sons, Ltd.     

The determinant of the scattering matrix and its relation to the number of eigenvalues

For a large class of scattering systems we study the behavior of the determinant of the scattering matrix as a function on the spectrum of the unperturbed operator. The variation of this determinant is related to the number of eigenvalues due to the perturbation. This relation generalizes results of Levinson and others. The range of physical systems to which these results apply is thus considerably extended.     

Fast iteration method in the problem of waves interacting with a set of thin screens

A new iteration method is proposed for the wave equation describing the scattering of a harmonic wave from an arbitrary configuration in the form of an array of thin straight barriers. The problem is reduced to a system of boundary integral equations, which are discretized by applying the Belotserkovskii-Lifanov method. In discrete form, a finite number of systems with Toeplitz matrices (the number of systems is equal to the number of barriers) are solved at each iteration step by applying special fast methods. The algorithm is tested on several geometries, and its convergence in these cases is analyzed.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problems

Bitangential input scattering problems are formulated and analyzed for canonical integral systems. Special attention is paid to the case when the input scattering matrix is ap×q matrix valued function of Wiener class. Formulas for the solution of the inverse input scattering problem are obtained by reproducing kernel Hilbert space methods. A number of illustrative examples are presented. Additional examples for the case when the input scattering matrix is of Wiener class/rational will be presented in a future publication.     

Spectral and scattering theory for symmetric systems of nonconstant deficit

We study the spectral and scattering problems for symmetric systems of nonconstant deficit. We prove (i) discreteness of eigenvalues, (ii) absolute continuity of continuous spectrum. (iii) principle of limiting absorption. As an application to the scattering problems, the decay of local energy and the completeness of wave operators are obtained. The class of systems we consider here includes the linearized equation of magnetohydrodynamics as a typical example.     

Scattering of Fermions on a Superfluid Bose Condensate

The influence of a fermionic impurity on the spectrum of overcondensate excitations of a superfluid Bose system is investigated. The scattering of neutrons on the Bose condensate of a superfluid ⁴He and a mixture of the liquid ³He and ⁴He is considered. An analytic expression for the spectrum of overcondensate excitations for these two systems is found. Bibliography: 10 titles.     

Augmented conjugate gradient. Application in an iterative process for the solution of scattering problems

We discuss the application of an augmented conjugate gradient to the solution of a sequence of linear systems of the same matrix appearing in an iterative process for the solution of scattering problems. The conjugate gradient method applied to the first system generates a Krylov subspace, then for the following systems, a modified conjugate gradient is applied using orthogonal projections on this subspace to compute an initial guess and modified descent directions leading to a better convergence. The scattering problem is treated via an Exact Controllability formulation and a preconditioned conjugate gradient algorithm is introduced. The set of linear systems to be solved are associated to this preconditioning. The efficiency of the method is tested on different 3D acoustic problems. This revised version was published online in August 2006 with corrections to the Cover Date.     

On elements of the Lax-Phillips scattering scheme for ρ-perturbations of an abstract wave equation

We give the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition includes perturbations with compact support for the classical wave equation. By using the Lax-Phillips method, we study scattering of “ρ-perturbed” systems and establish some properties of corresponding scattering matrices.