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.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

Short-wave grazing scattering by periodic inclined half-planes

The problem of scattering of a plane wave by a periodic structure consisting of inclined half-planes is considered. In the expression for the scattered field an exact formula for the excitation coefficients for the plane waves escaping into the upper half-plane is obtained by means of the Wiener-Hopf factorization method. In the case of short-wave grazing scattering of plane waves the leading term of the asymptotics is obtained from this result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 58–66, 1989.The author wishes to express his gratitude to M. M. Popov for useful discussions and constructive remarks.     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

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.     

Critical wave speeds for a family of scalar reaction-diffusion equations

We study the set of traveling waves in a class of reaction-diffusion equations for the family of potentials f_m(U) = 2U^m(1 − U). We use perturbation methods and matched asymptotics to derive expansions for the critical wave speed that separates algebraic and exponential traveling wave front solutions for m → 2 and m → ∞. Also, an integral formulation of the problem shows that nonuniform convergence of the generalized equal area rule occurs at the critical wave speed.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

On the determination of the sound field in a layer : PMM vol. 38, n 6, 1974, pp. 1129–1132

We consider a method for determining the sound field in a two-dimensional layer. The method we present combines the usual method of reflected plane waves with a summation from graphs. It makes it comparatively easy to take into account the complex interference pattern due to the transformation of the various waves at the boundaries of the layer and to obtain integral relations for the sound potentials. When the layer thickness tends to infinity, the problem reduces to one concerning the reflection of sound waves at the interface of two media. We study the potentials of normal waves in the case of a harmonic source in a solid.     

Asymptotics of Maxwell Time in the Plate-Ball Problem

The problem on rolling of a sphere on a plane without slipping or twisting is considered. One should roll the sphere from one contact configuration to another so that the length of the curve traced by the contact point in the plane is the shortest possible. The asymptotics of Maxwell time for rolling of the sphere along small amplitude sinusoids is studied. A two-sided estimate for this asymptotics is obtained.     

Singular Stress Behavior in a Bonded Hereditarily-Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Stress singularity is investigated in a plane problem for a bonded isotropic hereditarily elastic (viscoelastic) aging infinite wedge. The general solution of the operator Lamé equations, which are partial differential equations in space co-ordinates and integral equations in time, respectively, is represented in terms of one-parametric holomorphic functions (the Kolosov–Muskhelishvili complex potentials depending on time) in weighted Hardy-type classes. After application of the Mellin transform with respect to the radial variable, the problem is reduced to a system of linear Volterra integral equations in time. By using the residue theory for the inverse Mellin transform, the stress asymptotics and strain estimates near the singular point are presented here for non-hereditary Dundurs parameters. The general case of the hereditary Dundurs operators is considered in Part II (see [21]). © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

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.     

Mixed Type Multiple Orthogonal Polynomials for Two Nikishin Systems

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.     

Diffraction d'une onde électromagnétique par un dièdre infini absorbant

We study the diffraction of electromagnetic waves by an infinite wedge of dielectric material. For this aim we consider the stationary coupled vacuum-dielectric Maxwell equations with an outgoing condition in the vacuum. We show the equivalence of the latter problem to a Caldéron boundary operator system for particular classes of incoming data. We study the solutions of this system in the case of traces of monochromatic plane waves. In particular, we give the asymptotics of the diffracted signal in high frequency regime at a given point with fixed distance to the boundary and away from some incident directions. To cite this article: J.-M. Caron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Actions of Finitely Generated Groups and Semigroups on a Plane by Means of Isometries

The problem of uniform distribution of orbits on a plane for actions of finitely generated groups of isometries is studied. A criterion for every orbit to be dense is presented. The asymptotics for the frequency with which an orbit hits a compact set is obtained.     

An integro-differential equation arising as a limit of individual cell-based models

In this paper, we study mathematical properties of an integro-differential equation that arises as a particular limit case in the study of individual cell-based model. We obtain global well-posedness for some classes of interaction potentials and finite time blow-up for others. The existence of space homogeneous steady states as well as long-time asymptotics for the solutions of the problem is also discussed.     

Properties of the spectrum in the John problem on a freely floating submerged body in a finite basin

We consider a problem on the interaction of surface waves with a freely floating submerged body, which combines a spectral Steklov problem with a system of algebraic equations. We reduce this spectral problem to a quadratic pencil and then to the standard spectral equation for a self-adjoint operator in a certain Hilbert space. In addition to general properties of the spectrum, we investigate the asymptotics of eigenvalues and eigenvectors with respect to an intrinsic small parameter.     

Asymptotics of the scattered Debye potentials via a generalized convolution

In this paper, we express the scattered Debye potentials via a new generalized convolution related to the Kontorovich-Lebedev integral. The uniform asymptotics of the scattered Debye potentials under very mild conditions on spectral functions are obtained, and an inverse problem of finding spectral functions from given Debye potentials is considered.