Explicitly recovering asymptotics of short range potentials

Inverse scattering for real-valued short range potentials on Rⁿ is studied. It is shown that the scattering matrix at fixed energy is the pull-back of a pseudo-differential operator and that the symbol of the operator determines the asymptotics of the potential. This is done by an explicit construction of the Poisson operator for the scattering problem as an oscillatory integral     

Recovering asymptotics of metrics from fixed energy scattering data

The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on Rⁿ from fixed energy scattering data is studied. It is shown that if two such metrics, g₁,g₂, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism N 'fixing infinity' such that N^*g₁-g₂ is rapidly decreasing. Given the scattering matrix at two energies, it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously. These results also hold for a wide class of scattering manifolds.     

The Augmented Scattering Matrix and Exponentially Decaying Solutions of an Elliptic Problem in a Cylindrical Domain

The self-adjoint elliptic boundary-value problem in a domain with cylindrical outlets to infinity is considered. The notion of an augmented scattering matrix is introduced on the basis of artificial radiation conditions. Properties of the augmented scattering matrix are studied, and the relationship with the classical scattering matrix is demonstrated. The central point is the possibility of calculating the number of linearly independent solutions of a homogeneous problem with fixed rate of decrease at infinity by analyzing the spectrum of the augmented scattering matrix. This property is applied to the problem on diffraction on a periodic boundary as an example. Bibliography: 21 titles.     

Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations

The possibility of remotely sensing the optical properties of scattering particulates from the variations in either the angular or the spectral characteristics, or both, of the radiation they transmit or scatter, is a problem of fundamental importance in the monitoring of environmental particulate pollution. It is shown that the corresponding problem is (or can be brought to) one of inverting first kind Fredholm integral equations. The solution to this problem would also enable one to follow the dynamical evolution of the polluted environment if it can be obtained in a time scale that is comparable to, or shorter than, the time constant of the physical measurements.The direct problem of how given physical parameters of such particles affect the transmission and scattering of incident radiation is first analyzed on the basis of the corresponding radiative transfer problem, including single and multiple scattering, and polarization induced on scattering. The various available methods for reconstructing the size distribution from the observed directly transmitted or scattered light are reviewed, particularly with regard to their main advantages and shortcomings. For direct light transmission, these include: library, iterative, and least-squares methods; the trial-and-error method; the matrix inversion method with smoothing constraint (MIM); the resolution accuracy trade-off method; and the analytical method. A minimization search method with smoothing constraint, an essential modification to the MIM, is also proposed. The corresponding methods for singly and multiply scattered light are likewise reviewed. The proposed forward-scattering method is shown to provide excellent reconstructions from either angular or spectral light measurements under proper experimental conditions. It can also be coupled with a minimization search in order to provide simultaneously the complex refractive index of the particles. The potentialities of other methods for the complete multiple scattering problem—the minimization search method, quasilinearization method, and small-angle Gaussian approximation method—are also studied.     

Radiation conditions for the linear water‐wave problem in periodic channels

We study the well‐posedness of the linearized water‐wave problem in a periodic channel with fixed or freely floating compact bodies. Floquet–Bloch–Gelfand‐transform techniques lead to a generalized spectral problem with quadratic dependence on a complex parameter, and the asymptotics of the solutions at infinity can be described using Floquet waves. These are constructed from Jordan chains, which are related with the eigenvalues of the quadratic pencil and which are calculated in the paper in some typical cases. Posing proper radiation conditions requires a careful study of spaces of incoming and outgoing waves, especially in the threshold situation. This is done with the help of a certain skew‐Hermitian form q , which is closely related to the Umov–Poynting vector of energy transportation. Our radiation conditions make the problem operator into a Fredholm operator of index zero and provides natural (energy) classification of outgoing/incoming waves. They also lead to a novel, most natural properties and interpretation of the scattering matrix, which becomes unitary and symmetric even at threshold.     

Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges

We analyze the Sommerfeld solution to the stationary diffraction by a half‐plane. We prove that this solution is the limiting amplitude for time‐dependent scattering of incident plane waves with a broad class of the profile functions. We also show that this solution is the asymptotics of the limiting amplitudes of solutions to time‐dependent scattering problem with narrow wedges when the angle of the wedge tends to zero.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Description of pair potentials for which the scattering in a quantum system of three one-dimensional particles is free of diffraction effects

The coordinate asymptotics of the solution of the scattering problem for a system of three one-dimensional particles contains, besides plane and spherical waves, also Fresnel waves which arise also in the two-dimensional problem of a plane wave on a semiinfinite screen. One describes explicitly the class of potentials for which the Fresnel waves do not occur in the coordinate asymptotics. This class is somewhat wider than the class of nonrefleeting potentials.     

Long-Time Asymptotics of Complex mKdV Equation with Weighted Sobolev Initial Data

In this paper, we apply $\bar{\partial}$-steepest descent method to analyze the long-time asymptotics of complex mKdV equation with the initial value belonging to weighted Sobolev spaces. Firstly, the Cauchy problem of the complex mKdV equation is transformed into the corresponding Riemann-Hilbert problem on the basis of the Lax pair and the scattering data. Then the long-time asymptotics of complex mKdV equation is obtained by studying the solution of the Riemann-Hilbert problem.     

The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition

We examine the Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition. The question is addressed of existence of a solution with the same asymptotics at infinity as the initial condition. We demonstrate that the method of the inverse scattering problem is applicable to this problem.     

Convergence of Monte Carlo algorithms for reconstructing the scattering phase function with polarization

The problem of reconstructing the atmospheric scattering phase function from groundbased solar almucantar sky brightness observations is considered.A newiterative algorithm for solving this problem was developed as a combination of existing additive and multiplicative methods of refining the single-scattering contribution to the observed brightness with polarization of scattered radiation in the atmosphere. Also, some modifications of these methods were proposed. The objective of this paper is to numerically substantiate the convergence of these methods. For this purpose, an algorithm of Jacobi matrix calculation for the iteration operators of the methods was developed, and calculations were carried out for various parameters of the atmosphere.     

Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements

The paper investigates the asymptotic behavior of solutions to the 2 × 2 matrix factorization (Riemann-Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein’s theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrödinger equation at large times.     

On scattering by a cylinder with a narrow slit and walls of finite thickness

A two-dimensional analog of the Helmholtz resonator with walls of finite thickness and Neumann's boundary conditions is considered. The method of matching asymptotic expansions is used to find the asymptotics for the poles of the analytic continuation of the Green's function. For the corresponding eigendistributions and the solutions of the scattering problem, the leading terms in the asymptotics of the poles are obtained directly.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 24–43, January, 1996.     

Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons

The two-component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.     

The Inverse Scattering Transform for the Benjamin–Ono Equation

We extend the inverse scattering transform (IST) for the Benjamin–Ono (BO) equation, given by A. S. Fokas and M. J. Ablowitz ( Stud. Appl. Math. 68:1, 1983), in two important ways. First, we restrict the IST to purely real potentials, in which case the scattering data and the inverse scattering equations simplify. Second, we extend the analysis of the asymptotics of the Jost functions and the scattering data to include the nongeneric classes of potentials, which include, but may not be limited to, all N -soliton solutions. In the process, we also study the adjoint equation of the eigenvalue problem for the BO equation, from which, for real potentials, we find a very simple relation between the two reflection coefficients (the functions β(λ) and f (λ)) introduced by Fokas and Ablowitz. Furthermore, we show that the reflection coefficient also defines a phase shift, which can be interpreted as the phase shift between the left Jost function and the right Jost function. This phase shift leads to an analogy of Levinson's theorem, as well as a condition on the number of possible bound states that can be contained in the initial data. For both generic and nongeneric potentials, we detail the asymptotics of the Jost functions and the scattering data. In particular, we are able to give improved asymptotics for nongeneric potentials in the limit of a vanishing spectral parameter. We also study the structure of the scattering data and the Jost functions for pure soliton solutions, which are examples of nongeneric potentials. We obtain remarkably simple solutions for these Jost functions, and they demonstrate the different asymptotics that nongeneric potentials possess. Last, we show how to obtain the infinity of conserved quantities from one of the Jost functions of the BO equation and how to obtain these conserved quantities in terms of the various moments of the scattering data.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

Scattering by a Cylindrical Trap in the Critical Case

We study a two-dimensional analog of the Helmholtz resonator with walls of finite thickness in the critical case, for which there exists a frequency which is simultaneously the limit of poles generated both by the bounded component of the resonator and by a narrow communication channel. Under the assumption that the limit frequency is a simple frequency for the bounded component, by using the method of matched asymptotic expansions, we construct asymptotics for the two sequences of poles converging to this frequency. We obtain explicit formulas for the leading terms of the asymptotics of poles and for the solution of the scattering problem.     

Asymptotics of the optimal time in a time-optimal problem with two small parameters

A time-optimal problem of control of a small-mass point by a force of bounded magnitude in an unresisting medium is considered. An asymptotic expansion of the optimal time and optimal control is constructed with respect to two independent small parameters: the mass of the point and the perturbation of the initial conditions. It is shown that the asymptotics of the optimal time in this problem is complicated even for cases of general position.     

On the soliton-free structure of scattering data on perturbation of a two-dimensional soliton for the Davey-Stewartson equation II

The asymptotics of scattering data for the Davey-Stewartson equation II is obtained in the case of perturbation of a solition. It is shown that the scattering data of the perturbed problem have a soliton-free character.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 200–208, February, 1996.