Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries

An approach to express an incident shaped beam with respect to an arbitrarily oriented spheroidal particle having layers with non-confocal boundaries is presented. To overcome the difficulty of non-confocal boundary conditions connected with different spheroidal coordinate systems, a theoretical procedure is developed to deal with the non-confocal boundary conditions by virtue of a transformation for vector wave functions. The unknown coefficients of scattered and internal electromagnetic fields are determined by solving a system of linear equations derived from the boundary conditions and relations between the spheroidal vector wave functions and spherical ones. Numerical results of the normalized scattering cross section for a two-layered non-confocal prolate spheroid are evaluated. PACS 42.25.Fx; 42.25.Bs     

Comparison between the generalized tanh–coth and the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ^′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.     

Gaussian beam scattering by a chiral sphere

Based on the generalized Lorenz–Mie theory that provides the general framework, an analytic solution to Gaussian beam scattering by a chiral sphere is constructed, by expanding the incident Gaussian beam, scattered fields and internal fields in terms of spherical vector wave functions. The unknown expansion coefficients are determined by a system of equations derived from the boundary conditions. For a localized beam model, numerical results of the normalized differential scattering cross section are presented.     

Magnetooptical properties of a ferromagnetic superlattice

Transmission and reflection of a normally incident wave from a magnetic superlattice consisting of 2N ferromagnetic layers with alternating orientation of the magnetization vector are considered. The characteristic matrix of a superlattice relating wave amplitudes at the entrance to the system and at the exit from it is calculated in the closed form and Jones matrices determining all the basic magnetooptical characteristics of the structure (transmission and reflection coefficients, the degree of polarization of transmitted and reflected waves, and so on) are constructed. A significant dependence of these characteristics on the number of layers is demonstrated.     

Regularization of boundary integral equations in the problems of wave field diffraction by curved boundaries

A regularization of the exact Fredholm integral equations for the field or its derivative on a scattering surface is proposed. This approach allows one to calculate the scattering or diffraction of pulsed wave fields by curved surfaces of arbitrary geometry. Mathematically, the method is based on the replacement of the exact Fredholm integral equations by their truncated analogs, in which the contributions of the geometrically shadowed regions are cancelled. This approach has a clear physical meaning and provides stable solutions even when the direct numerical solution of mathematically exact initial integral equations leads to unstable results. The method is mathematically substantiated and tested using the problem of plane-wave scattering by a cylinder as an example.     

Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

柱面非线性麦克斯韦方程组的行波解

柱面电磁波在各种非均匀非线性介质中的传播问题具有非常重要的研究价值.对描述该问题的柱面非线性麦克斯韦方程组进行精确求解,则是最近几年新兴的研究热点.但由于非线性偏微分方程组的极端复杂性,针对任意初边值条件的精确求解在客观上具有极高的难度,已有工作仅解决了柱面电磁波在指数非线性因子的非色散介质中的传播情况.因此,针对更为确定的物理场景,寻求能够精确描述其中更为广泛的物理性质的解,是一种更为有效的处理方法.本文讨论了具有任意非线性因子与幂律非均匀因子的非色散介质中柱面麦克斯韦方程组的行波精确解,理论分析表明这种情况下柱面电磁波的电场分量E已不存在通常形如E=g(r-kt)的平面行波解;继而通过适当的变量替换与求解相应的非线性常微分方程,给出电场分量E=g(lnr-kt)形式的广义行波解,并以例子展示所得到的解中蕴含的类似于自陡效应的物理现象.     

Exact solution to the problem of nonlinear pulse propagation through random layered media and its connection with number triangles

An energy pulse refers to a spatially compact energy bundle. In nonlinear pulse propagation, the nonlinearity of the relevant dynamical equations could lead to pulse propagation that is nondispersive or weakly dispersive in space and time. Nonlinear pulse propagation through layered media with widely varying pulse transmission properties is not wave-like and a problem of broad interest in many areas such as optics, geophysics, atmospheric physics and ocean sciences. We study nonlinear pulse propagation through a semi-infinite sequence of layers where the layers can have arbitrary energy transmission properties. By assuming that the layers are rigid, we are able to develop exact expressions for the backscattered energy received at the surface layer. The present study is likely to be relevant in the context of energy transport through soil and similar complex media. Our study reveals a surprising connection between the problem of pulse propagation and the number patterns in the well known Pascal’s and Catalan’s triangles and hence provides an analytic benchmark in a challenging problem of broad interest. We close with comments on the relationship between this study and the vast body of literature on the problem of wave localization in disordered systems.     

Solution of the three-dimensional problem of wave diffraction by an ensemble of objects

The pattern equations method is extended to solving three-dimensional problems of wave diffraction by an ensemble of bodies. The method is based on the reduction of the initial problem to a system of N (N is the number of scatterers in the ensemble) integro-operator equations of the second kind for the scattering patterns of scatterers. With the use of the series expansions of the scattering patterns in angular spherical harmonics, the problem is reduced to an algebraic system of equations in the expansion coefficients. An explicit (asymptotic) solution to the problems is obtained in the case when the scattering bodies are separated by sufficiently long distances. It is shown that the method can be used to model the characteristics of wave scattering by complex-shaped bodies.     

A simple reduction procedure for the three- and N-body scattering equations

A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful unitary reduction of the three- and N-body scattering equations. Employing generalized half-off-shell functions that satisfy of-sshell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is simple example of how the scattering problem can be split into an internal and an external part.     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

On accounting for the effect of particles of a condensed dispersed phase on radiant energy transfer in gaseous media

Questions concerning the formation of the optical properties of dense gaseous and plasma media in relation to the specific features of radiant energy transfer are considered. The integral equations describing the radiation trapping are investigated as a new class of generalized wave equations of Schrödinger type. Starting from the methods of quantum mechanics, original analytical and numerical approaches are suggested for solving problems of the radiative kinetics of both spatially homogeneous and inhomogeneous absorbing media containing dispersed particles. In terms of the quasi-classical approximation, two classes of reference problems for determination of phase factors are formulated. Solutions for a number of model problems are presented that demonstrate the efficiency of the methods developed.     

Microscopic Theory of the Two-Dimensional Quantum Antiferromagnet in a Paramagnetic Phase

We have developed a consistent theory of the Heisenberg quantum antiferromagnet in the disordered phase with a short range antiferromagnetic order on the basis of the path integral for spin coherent states. In the framework of our approach we have obtained the response function for the spin fluctuations for all values of the frequency ω and the wave vector k and have calculated the free energy of the system. We have also reproduced the known results for the spin correlation length in the lowest order in 1/N. We have presented the Lagrangian of the theory in a form which is explicitly invariant under rotations and found natural variables in terms of which one can construct a natural perturbation theory. The short wave spin fluctuations are similar to those in the spin wave theory and they are on the order of the smallness parameter 1/2s where s is the spin magnitude. The long-wave spin fluctuations are governed by the nonlinear sigma model and are on the order of the smallness parameter 1/N, where N is the number of field components. We also have shown that the short wave spin fluctuations must be evaluated accurately and the continuum limit in time of the path integral must be performed after the summation over the frequencies ω.     

Boundary diffraction wave theory of resistive surfaces with edge discontinuities

The line integral of the boundary diffraction wave theory is derived by considering the exact diffracted fields of a resistive half-plane. The line integral is generalized for arbitrary resistive surface with edge discontinuity. The method is applied to the diffraction problem of waves by a convex resistive spherical reflector and the resultant field expressions are investigated numerically.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Reduced Order Podolsky Model

We perform the canonical and path integral quantizations of a lower-order derivatives model describing Podolsky’s generalized electrodynamics. The physical content of the model shows an auxiliary massive vector field coupled to the usual electromagnetic field. The equivalence with Podolsky’s original model is studied at classical and quantum levels. Concerning the dynamical time evolution, we obtain a theory with two first-class and two second-class constraints in phase space. We calculate explicitly the corresponding Dirac brackets involving both vector fields. We use the Senjanovic procedure to implement the second-class constraints and the Batalin-Fradkin-Vilkovisky path integral quantization scheme to deal with the symmetries generated by the first-class constraints. The physical interpretation of the results turns out to be simpler due to the reduced derivatives order permeating the equations of motion, Dirac brackets and effective action.     

Viscoelastic micropolar continuum and viscoelastic waves

Generalized Kelvin model is applied to isotropic viscoelastic micropolar continuum. Constitutive equations in integral and differential form, generalized Lamé equations, and wave equations for displacement and microrotation are derived for this model. For damped viscoelastic waves, which are realized in the continuum under examination, the wave vectors and vectors of decay are explicitly given as functions of elastic moduli, viscosity coefficients, angular frequency, density, and microinertia coefficient. Analogous relations are derived for further eleven simpler mass models. Results are compared for three models in current use. For two of them (classical elastic and micropolar elastic medium) are the derived results in agreement with the ones usually used, for the third model (viscoelastic continuum) are the usual formulas a limiting case of the relations derived in this paper.     

Vector Diffraction Integrals for Solving Inverse Problems of Radio-Holographic Sensing of the Earth's Surface and Atmosphere

Vector relationships between the fields on a certain surface confining an inhomogeneous three-dimensional volume and the fields inside this volume are obtained by the Stratton–Chu method developed for the case of homogeneous media. The vector relationships allow us to solve the direct and inverse problems of determining the fields inside an inhomogeneous medium given the field on its boundary. The vector equations take into acount the polarization changes of direct and inverse waves propagated in an inhomogeneous medium. In the case of a two-dimensional homogeneous medium, the vector equations reduce to the previously obtained scalar equations used in the approximation of spherical symmetry to describe the process of backward wave propagation during the atmospheric and ionospheric radio-occultation monitoring. It is shown that the Green's function of the scalar wave equation in an inhomogeneous medium should be used as the reference signal for solving the inverse problem of radio-occultation monitoring. This validates the method of focused synthetic aperture previously used for high-accuracy retrieval of the vertical refractive-index profiles in the ionosphere and atmosphere. In this method, the reference-signal phase was determined from a model which describes with sufficient accuracy the radiophysical parameters of a refracting medium in the region of radio-occultation sensing. The obtained equations can be used for the high-accuracy solving of inverse problems of radio-holographic sensing of the Earth's atmosphere and surface by precision signals from radio-navigation satellites.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations

A simple method is developed for constructing the solutions of the short-wave model equations associated with the Camassa–Holm (CH) and Degasperis–Procesi (DP) shallow-water wave equations. Taking an appropriate scaling limit of the N-soliton solution of the CH equation, we obtain the N-cusp soliton solution for the CH short-wave model. The similar procedure also leads to the N-loop soliton solution for the DP short-wave model. We describe the property of the solutions. In particular, we derive the large-time asymptotics of the solutions as well as the formulas for the phase shift.