Exact solutions for spectra and Green's functions in random one-dimensional systems

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed.The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.     

Maxwell's equations in anisotropic space

In terms of the covariance of equations under a generalized galilean transformation, a general expression of Maxwell's equations in anistropic space is given here.     

Utilisation des franges d'interference en lumiere diffuse pour l'etude de l'etat de surface d'un diffuseur

A diffuser, illuminated successively by light of two neighbouring wavelengths, produces on a photographic plate placed in a parallel plane two intensity distributions which are homothetic except for certain decorrelation terms. This homothecy (scale variation) can be compensated by a longitudinal displacement of the photographic plate between the two exposures and consequently, the decorrelation depends only on the structure of the diffuser. Under these conditions, if the photographic plate is given an additional displacement in its own plane between the two exposures, the contrast of Young's fringes obtained in the Fourier plane characterises the roughness of the diffuser.     

Electromagnetic waves in dielectrics with memory

We analyse electromagnetic wave propagation in a dielectric with memory of the Maxwell-Hopkinson type. We show that the components of the electric and magnetic fields satisfy two different scalar wave equations and we first look for their harmonic plane wave solutions. Then we prove that dielectrics with memory can also support approximate Courant-Hilbert waves. We discuss the equations to be solved to get all the components of the electromagnetic field from a scalar solution from each wave equation and TE, TM harmonic plane waves are explicitly given.     

Complex symmetries of electrodynamics

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincaré group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincaré transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.     

Electromagnetic fields in general relativity: A constructive procedure

A new procedure for obtaining explicit solutions to Maxwell's equations in curved spaces is presented. The problem is reduced to solving one linear scalar wave equation. The formulation includes astrophysically important cosmological models, neutron star and black hole space-times.     

Solitary wave complexes in two-component condensates

Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intracomponent but varying intercomponent interaction strengths. Several families of solitary wave complexes are found: (1) vortex rings of various radii in each of the components; (2) a vortex ring in one component coupled to a rarefaction solitary wave of the other component; (3) two coupled rarefaction waves; (4) either a vortex ring or a rarefaction pulse coupled to a localized disturbance of a very low momentum. The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative differences between the chemical potentials of two components. Solitary wave formation, their stability, and solitary wave complexes in two dimensions are discussed.     

Perturbation theory for studying the effect of the τ∈ term in lens-like media

In this paper we derive solutions of Maxwell's equations for parabolic-index media, using first-order perturbation theory and taking into account the τ∈ term that is most often neglected as being negligibly small. The theory presented here should find applications in the propagation of electromagnetic waves through Selfoc fibers and p-n junctions.     

On Transverse Electric (TE) wave propagation in a warm moving magnetoplasma with sinusoidal electron-density distribution

It has been shown, using the first three moment equations in conjunction with Maxwell's equations, that the wave equation for the TE mode in a warm moving stratified magnetoplasma transforms, for an assumed sinusoidal distribution of the electron density, to the standard form of the Hill's differential equation.     

Ultrasonic waves in porous ceramics with non-spherical holes

Formulae describing the effective elastic moduli of a porous ceramic medium are derived using Eshelby's solution and Maxwell's approximation. The ceramic medium is considered as an infinite matrix, which has uniform elastic properties and encloses non-spherical pores. A numerical evaluation of the velocity of an ultrasonic wave in the ceramic, as function of the porosity and pore shape, is presented. The theoretical results were combined with those obtained experimentally for different firing temperatures of the ceramic.     

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.     

Nonlinear mixing of oppositely travelling surface plasmons

An analysis of the nonlinear mixing of oppositely travelling surface plasmons on a semi-infinite metal is discussed within the framework of Maxwell's equations. Coupling to free-space radiation fields, and transverse bulk waves in the metal is predicted under certain conditions at the sum frequency of the two surface plasmons.     

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.     

Breathers and solitons on two different backgrounds in a generalized coupled Hirota system with four wave mixing

We study breathers and solitons on different backgrounds in optical fiber system, which is governed by generalized coupled Hirota equations with four wave mixing effect. On plane wave background, a transformation between different types of solitons is discovered. Then, on periodic wave background, we find breather-like nonlinear localized waves of which formation mechanism are related to the energy conversion between two components. The energy conversion results from four wave mixing. Furthermore, we prove that this energy conversion is controlled by amplitude and period of backgrounds. Finally, solitons on periodic wave background are also exhibited. These results would enrich our knowledge of nonlinear localized waves' excitation in coupled system with four wave mixing effect.     

The collision of plane waves in general relativity

A number of exact solutions of Einstein's equations are obtained, which describe the collision and subsequent interaction of two plane parallel waves. Gravitational waves, null electromagnetic fields, and neutrino fields are all considered with collisions between any two types. It is shown that two such waves mutually focus each other with the focus usually appearing as a singularity in space-time. Further conclusions are made regarding the qualitative nature of the interactions, and it is argued that these also apply in more realistic physical situations.     

Analogien zwischen außerordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen

Within the limits of Linear Optics we treat analogies between ordinary and extraordinary waves in uniaxial media which become conspicuous through a nonorthogonal transformation of coordinates. To any ordinary wave solution in unbounded uniaxial media we can construct a corresponding extraordinary wave solution by interchanging electrical and magnetical field components. Boundary conditions for instance for ideal conducting plane surfaces approximately preserve their original form, if the optical axis or the middle wave vector are normal to the surface. The parabolic approximative equations for slowly varying amplitudes are derived, the polarisation of these waves being considered as a slowly varying quantity. Further these approximative equations are expanded to include frequency dispersion. Through the specified transformation we can simplify problems with extraordinary waves.     

Theory of elementary excitations and the metal-insulator transition of semiconductor surfaces

The microscopic theory of density and spin response of surface systems and its application to elementary excitations is discussed. Particular emphasis is placed on semiconductor surfaces, for which the often-used jellium approximation is not valid. The discussion is based on a solution of Maxwell's equations or, formally, of the Bethe-Salpeter equation for the two-particle Green's function of the surface system. This solution is achieved in a local wave function representation and takes density fluctuations on a microscopic scale (surface profile and local-field effects parallel to the surface) into account. Many-body effects of random-phase (RPA) and electron-hole type are included. The resulting spin and density response functions present a practical scheme for a microscopic calculation of surface elementary excitations in conducting as well as non-conducting solids. As examples, the conditions for the appearance of an electronic (charge- and spin-density) instability at the surface and the coupling of the resulting charge-density wave to the lattice are studied in detail.Results of quantitative calculations of the charge- and spin-density-response function of the Si(111) surface establish the importance of including both excitonic (electron-hole) and (RPA) local-field many-body interactions. In particular, they lead to an instability of the ideal paramagnetic surface with respect to spin-density waves (SDW) with wavelength corresponding to the observed (2 × 1) and (7 × 7) superstructures. Another example deals with an a-priori calculation of the phonons and the electron-phonon interaction of the same surface system. Various results of the theory such as phonon softening due to the coupling of the charge-density fluctuations to the lattice are summarized and general aspects of the importance of many-body effects for the a-priori determination of surface structures via elementary excitations are discussed.     

Application of Hamilton's law to forced,damped, stationary systems

The algebraic equations for the forced, damped, periodic, axisymmetric motion of circular plates, solid and annular, are derived directly through the application of Hamilton's law of varying action. The simplicity, for many problems, of direct analytical solutions by means of Hamilton's law has previously been demonstrated. The method is called the Hamilton-Ritz method. In this paper, direct analytical solutions from Hamilton's law are shown to be exactly the same as direct analytical solutions from the ancient and fundamental principle of virtual work. The Hamilton-Ritz formulation is compared to the Galerkin formulation. Results from one- and two-term solutions by direct virtual work (Hamilton-Ritz) are compared to results from the exact solution and to results from the Galerkin method.     

Generalized plane gravitational waves

The definition of plane gravitational waves is generalized to include the case in which rays are not orthogonal to the two-dimensional wave surfaces. All Einstein spaces and some new solutions of the Einstein-Maxwell equations of this type are given.