Maxwell’s Equations

Two-dimensional wavelets collocation method for electromagnetic waves in dielectric media

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

The generation and detection of scalar or pseudoscalar high-frequency gravitational waves in some dielectrics, excited by ultrashort intense laser pulses are proposed. Hidden photons and axion particles are elemental excitations of such gravitational waves. The properties of longitudinal electromagnetic waves in dielectric media have been described from the point of view of photon-hidden photon conversion processes in these crystals. Scalar bound photonic states in dielectric media are analyzed. The experimental schemes for photon-hidden photon conversion with taking part in longitudinal electromagnetic excitations in media have been proposed. Hertz gravitational laboratory experiments with the illumination of the dielectric media by intense ultrashort laser pulses and the “Light Sching Trough Wall” scheme using are analyzed.

The propagation of electromagnetic waves in dielectric medium with decreasing perturbations (defects) is studied. We use the Maxwell operator M=∇×p(x)∇× and the perturbed operator Mλ=∇×(p(x)+λq(x))∇× to describe the “background” medium and perturbed medium, respectively, where λ>0 is the coupling constant. Under the assumption that M has a spectral gap G, we prove that a defect can create an eigenvalue of any given fixed value τ∊G for some λ>0, provided that the support of the defect is large enough. We also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches.

Local nonlinear electrodynamics

A plane-wave-based approach for complex photonic band structure is developed and applied to analyse the behaviour of electromagnetic (EM) waves at the surface of semi-infinite and finite photonic crystals (PCs). The complex photonic band structures of two-dimensional and three-dimensional PCs have been calculated by the present approach and verified by conventional plane wave expansion method and plane-wave-based transfer matrix method. Based on the eigen modes of present approach, equations in matrix form are given to describe the boundary conditions of PCs, and consequently the behaviour of EM waves at the surface of semi-infinite and finite PCs is discussed. These results have also verified the validity and efficiency of the current approach.

Carathèodory’s classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton–Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.

This paper presents a canonical boundary value problem to understand the behavior of electromagnetic surface waves propagated on the planar interface of rugate filter and hyperbolic material. The electromagnetic surface waves are generated on the planar interface of two different materials. The hyperbolic materials constructed by making a dyadic negative in columnar thin films. Multiple electromagnetic surface waves were investigated by varying phase of rugate filter form 0 to 180° and wavelength of incident light from 400 to 700 nm. Material’s manufacturing fault is also introduced by adding 0.001 i in refractive index of hyperbolic medium.Propagation of multiple electromagnetic waves were observed on the planar interface of rugate filter and hyperbolic material. In order to verify the existence of surface electromagnetic waves at the interface of hyperbolic medium and rugate filter, electric and, magnetic fields are plotted for different material specifications.

In this paper, we investigate the behavior of electromagnetic localized waves [LW's], which are concentrated moving pulses of energy, at an interlace between two electrically different media. We employ a previously given Fourier transform domain representation of LW's that is geometric in nature; the Fourier transform of an LW consists of a support line on a particular surface together with a weighting associated with this line. The behavior of electromagnetic LW's at an interface is somewhat involved owing to their broadband nature, but this behavior has a lucid pictorial representation in terms of the transform domain geometry and, in this paper, we exploit this representation. If an LW strikes the interface obliquely then a different LW is reflected, but the pulse transmitted into the second medium is not a LW. We give analytical and graphical evidence of this behavior. Moreover, we show that if one desires a particular LW to be propagated into the second medium, one can in some cases design the necessary non-LW pulse one needs to launch at the interface to achieve this goal.

A method for constructing a solution for some systems of first-order quasilinear partial differential equations is presented. The type of equations can either be hyperbolic or elliptic. The method is based on the application of the generalized hodograph method, which allows us to write the solution in an implicit form. There is a system of first-order linear partial differential equations that is used for commuting flows in the generalized hodograph method. We discover an analogy between the commuting flows and divided differences for the Hermite polynomial. This analogy allows us to obtain an explicit representation for commuting flows. The introduction of new (Lagrangian) variables, which are conserved on the characteristics of the original system, suggests a way to transform the solution of the Cauchy problem for first-order quasilinear partial differential equations to the solution of the Cauchy problem for ordinary differential equations. Numerical, and in some cases analytical, integration of the Cauchy problem makes it possible to construct explicit solutions of the problem on the level lines (isochrons) of the implicit solution. The method proposed is significantly different from the grid method, finite element method, finite volume method, and, in fact, is more precise. The error of the solution can arise only at the last stage in the numerical integration of the Cauchy problem for ordinary differential equations. Moreover, the method allows us to obtain multivalued solutions, in particular, to study the process of wave breaking in hyperbolic systems. Particular cases of the equations considered here describe diffusion-free approximation in a wide range of mass transport processes in multicomponent mixtures, such as electrophoresis, chromatography, centrifugation. As a simple example, the solution of the electrophoresis problem (separation multicomponent mixture to individual component) is presented.KeywordsGeneralized hodograph methodCommuting flowsDivided differenceMathematical Subject Classification (2000)35L4535L4035L65

Rational general solutions of systems of first-order algebraic partial differential equations

A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In these cases, the envelopes are always fronts. We investigate singular points of envelopes for first-order ordinary differential equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type, respectively.

We theoretically investigate the properties of vector and scalar phonon–polariton cnoidal waves and spatial solitons propagating in a boundless dielectric medium. We obtain analytically the expressions for the envelopes of linearly and circularly polarized nonlinear polariton waves in self-focusing and self-defocusing media. The expressions of spatial solitons and cnoidal waves describe one and several flat flows of polaritons for the linearly polarized wave, respectively. The equation for a right or left circularly polarized polariton scalar wave has a soliton solution and a cnoidal wave solution. The polariton vector wave with right and left polariton spiralities has an analytical solution too. Also we examine the linearly and circularly polarized polariton wave instability in the nonlinear dielectric medium.

Particle nature of light waves in dielectric media

Electromagnetic Environment Associated with Lightning Strikes to Tall Strike Objects