The Atkinson-Wilcox expansion theorem for electromagnetic chiral waves

Consider the problem of scattering of a time-harmonic electromagnetic wave by a three-dimensional bounded and smooth obstacle. The infinite space outside the obstacle is filled by a homogeneous isotropic chiral medium. In the region exterior to a sphere that includes the scatterer, any solution of the generalized Helmholtz's equation that satisfies the Silver-Müller radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. The coefficients of the expansion can be determined from the leading coefficient, “the radiation pattern”, by a recurrence relation.     

四个对激波与涡旋有高分辨率的计算格式的比较*

近十年来,计算非定常无粘可压气体力学Euler方程组的高分辨率差分格式有显着进展.本文选择四个近年受到重视的格式,用一较复杂的二维不定常问题作进一步的考验.所选算例为平面激波遇矩形障碍物初始阶段的绕射与反射.在挡板头部有两个尖角点,角点附近流场参量变化剧烈,会有中心稀疏波和集中涡的出现,要模拟好它们,就要求格式有较好的适应性.本文选择特殊的激波马赫数M_s=2.068,使静止坐标系下激波后流速恰为声速,并沿中心稀疏波区从角点发出的一条曲线也有这一现象,以考察各格式在方程组某一特征值恰为零时的计算特点,因零特征值可以使某些格式局部受损.计算结果的图形显示可表明四个格式在激波分辨率,格式粘性、膨胀波的计算、模拟非定常集中涡产生过程的能力等方面的性质.     

On the connection of the superposition and homogeneous-solutions methods in wave propagation problems in a semi-infinite rigidly-clamped elastic layer

The process of harmonic wave propagation is investigated in a semi-infinite rigidly-clamped elastic layer. An analytic solution of the problem is obtained by the superposition method. The wave field expansion in the form of a normal mode series for a corresponding infinite waveguide is established. According to residue theory, the explicit form of the expansion coefficients is established with physical requirements of the radiation conditions taken into account.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 3–10, 1988.     

“Electron ping-pong” on a one-dimensional lattice: Multiple reflections of the wave packet and capture of the wave function by an acceptor

Considering the problem of multiple reflections of a wave function from the ends of a lattice, we observe an interesting phenomenon: the wave function amplitude is concentrated on the impurity center after reflections. The solution obtained by expanding the total wave function a(t) on the impurity center in the partial amplitudes a_k(t), whose contributions become essential only after the kth reflection from the lattice end, seems to agree very well with the results of numerical modeling. We solve the problem of the capture of the wave function by an acceptor. The obtained results can be used to explain experimental data on charge transfer in artificial oligonucleotides and polypeptides. We find expressions for the electron capture probability in some limit cases, which can be considered estimates of the quantum output of the charge transport.     

Short-wave asymptotic behavior of space-time creeping waves in elasticity theory

We consider the elastic space-time (ST) wave on an unstressed convex surface in a deep shadow zone. The uniform high-frequency asymptotic expansion of the wave field is constructed as the sum of the caustic expansion for the longitudinal (transverse) wave containing the Airy function and the space-time ray series for the transverse (longitudinal) wave. The contribution of the ray expansion with the transverse eikonal is comparable to the contribution of the longitudinal creeping wave to the wave field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 176–189, 1985.I would like to thank V. M. Babich for suggesting the topic and for discussion of results.     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

Stability of the High Frequency Fast Multipole Method for Helmholtz’ Equation in Three Dimensions

Stability limits for the high frequency plane wave expansion, which approximates the free space Greens function in Helmholtz equation, are derived. This expansion is often used in the Fast Multipole Method for scattering problems in electromagnetics and acoustics. It is shown that while the original approximation of the Greens function, based on Gegenbauers addition theorem, is stable except for overflows, the plane wave expansion becomes unstable due to errors from roundoff, interpolation, choice of quadrature rule and approximation of the translation operator. Numerical experiments validate the theoretical estimates. AMS subject classification (2000) 65B10, 65G99.Received November 2003. Revised July 2004. Communicated by Anders Szepessy.Martin Nilsson: Financial support has been obtained from Parallel and Scientific Computing Institute (PSCI), which is a competence center financed by Vinnova, The Swedish Agency for Innovation Systems, and the Swedish National Aeronautical Research Program, NFFP.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain

An inverse problem for the wave equation outside an obstacle with a dissipative boundary condition is considered. The observed data are given by a single solution of the wave equation generated by an initial data supported on an open ball. An explicit analytical formula for the computation of the coefficient at a point on the surface of the obstacle, which is nearest to the center of the support of the initial data, is given. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of the three‐dimension wave equation by using wave polynomials

The paper demonstrates a specific power series expansion technique to solve the three‐dimensional wave equation. As solving functions, so‐called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry without strength restrictions. The wave polynomials are defined. Recurrent formulas for the wave polynomials and their derivatives are obtained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Wave Field near the Point of Incidence of the Limiting Ray

The plane scalar problem on the refraction of a high-frequency wave, given by its ray expansion, from a curvilinear interface of two media is considered. It is assumed that the velocity in the medium where the refracted wave propagates is larger than the velocity in the medium where the incident wave propagates. It is also assumed that, on the interface, there is a point on one side of which the ordinary refraction of the wave holds and on the other side of which the complete internal reflection of the wave occurs. An analytic expression of the wave field near this limiting point is found. Bibliography: 8 titles.     

On the Fourier transform of SO(d)-finite measures on the unit sphere

The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.Received: 7 May 2004; revised: 11 October 2004     

Focusing of WKB solution of the equation [Δ+K2n2(x)]u=0, K→∞

For a given incoming wave converging to a point, a focal solution and ray expansion for the outgoing wave are constructed. These nonuniform expansions are matched to all orders.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematichskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 123–128, 1975.The author is grateful to V. M. Babich for useful discussions during the course of the work.     

Davey StewartsonⅠ的周期波解

利用新近提出的F展开法，导出了Davey StewartsonⅠ方程的由Jacobi椭圆函数表示的周期波解；并且在极限的情况下，得到了Davey StewartsonⅠ方程的孤波解以及其他形式解.     

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.     

Exact solutions of nonlinear Schrödinger equation by using symbolic computation

The (G′/G,1/G)‐expansion method and (1/G′)‐expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

The direction of the secondary wave motion in the Ekman boundary layer

The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show, that the waves move away from the center of rotation.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.     

Asymptotic behavior of the wave function of three particles in a continuum

We study the wave function of a system of three particles in a continuum. The Faddeev equations are used to explicitly identify the singularities of the wave function in the momentum space. We obtain the asymptotic behavior of the wave function in the configuration space by calculating the asymptotic behavior of the Fourier transform of the wave function in the momentum space. Our attention is focused on configurations in which two particles are at a relatively small distance from each other while the third particle is significantly remote from the center of mass of the pair. We show that the coordinate asymptotic form of the wave function for such a configuration contains scattered waves of a new type in addition to the standard terms. We use the obtained exact data concerning the coordinate asymptotic form of the wave function to critically analyze the multiplicative ansatz used in several works to describe systems of three particles in a continuum.     

Asymptotic Solution of the Problem on an Accelerated Source of High-Frequency Oscillations

The problem of the wave field of a source moving in an inhomogeneous medium is considered. It is assumed that the velocity of the source is less than the velocity of sound for t < 0 and is greater than this velocity for t > 0 (subsonicsupersonic transition). An asymptotic expansion for the wave field in a neighborhood of the source is constructed on the basis of the well-known Hadamard ansatz. The expansion derived is uniform with respect to the velocity of the source and contains new special functions. These functions are generalizations of the Hankel and Bessel functions and possess some remarkable properties. Bibliography: 7 titles.