A brief history and a mathematical description of the dynamic projection operators technique is presented. An example of the general Cauchy problem for evolution equations in 1 + 1 dimensions is studied in detail. A boundary regime propagation is formulated in terms of operators and illustrated by the simplest one-dimensional difusion equation. The problem of temperature waves is discussed.
We consider a boundary regime problem for 1D wave propagation in a metamaterial medium with simultaneously negative dielectric permittivity and magnetic permeability. We apply a projecting operator method to the Maxwell system in the time domain that allows the space of the linear propagation problem to be split into subspaces of directed waves for the relations of a given material with general dispersion. After projection, the equations for directed waves have a maximally simplified form, which is most convenient for numerical and analytical integration. Matrix elements of the projectors act as integral operators. For a given nonlinearity and dispersion we derive a general system of interacting rightłeft waves with combined (hybrid) amplitudes. The result is specified for the Drude metamaterial model for both permittivity and permeability coeffiients and the Kerr nonlinearity. We also discuss and investigate singular solitary wave solutions of the system as a limit stationary elliptic system related to some boundary regimes.
A numerical model of propagation of internal gravity waves in a stratified medium is applied to the problem of tsunami wave run-up onto a shore. In the model, the ocean and the atmosphere are considered as a united continuum in which the density varies with height with a saltus at the water-air interface. The problem solution is sought as a generalized (weak) solution; such a mathematical approach automatically ensures correct conditions of matching of the solutions used on a water-air interlayer. The density stratification in the ocean and in the atmosphere is supposed to be described with an exponential function, but in the ocean a scale of the density stratification takes a large value and the density changes slightly. The initial wave running to a shore is taken in the form of a long solitary wave. The wave evolution is simulated with consideration of the time-varying vertical wave structure. Near the shore, the wave breaks down, and intensive turbulent mixing develops in the water thickness. The wave breakdown effect depends on the bottom shape. In the case when the bottom slope is small and the inshore depth grows slowly with the distance from the shore, mixing happens only in the upper stratum of the fluid due to the formation of a quiet region near the bottom. When the bottom slope takes a suffiiently large value, the depth where fluid mixing takes place goes down up to 50 meters. The developed model shows that the depth of the mixing effects strongly depends on the bottom shape, and the model may be useful for investigation of the impact strong gales and hurricanes on the coastline and beaches.
The propagation of X-ray waves through an optical system consisting of many X-ray refractive lenses is considered. Two differential equations are contemplated for solving the problem for electromagnetic wave propagation: first – an equation for the electric field, second – an equation derived for a complex phase of an electric field. Both equations are solved by the use of a finite-difference method. The simulation error is estimated mathematically and investigated. The presented results for equations show that in order to establish a high accuracy computation a much smaller number of points is needed to solve the problem of X-ray waves propagation through a multi-lens system when the method for the second equation is used. The reason for such a result is that the electric field of a wave after passing through many lenses is a quickly oscillating function of coordinates, while the electric field phase is a quickly increasing, but not oscillating function. Therefore, a very detailed difference grid, which is necessary to approximate the considered electric field can be replaced by not such a detailed grid, when computations are made for the complex wave of the electric field. The simulation error of both suggested methods is estimated. It is shown that the derived equation for a phase function allows effiient simulation of propagation of X-rays for the multi-lens optical system.
General properties of ladder operators applied to inhomogeneous problems are studied in the context of their usefulness for solving practical problems with stress put on the possibility of embedding the interwine relation onto a wider class of operators. From those general remarks an algorithm using the Darboux transform for construction of the Green function for linear partial differential equations is formed and a sample implementation thereof is shown along with some examples of solutions.
This paper presents calculation of the electron-impurity scattering coeffiient of Bloch waves for one dimensional Dirac comb potential. The impurity is also modeled as delta function pseudopotential that allows explicit solution of the Schrodinger equation and scattering problem for Bloch waves.
A general problem of monochrome plane electromagnetic wave reflection and refraction at the interface between the conducting medium and the dielectric is formulated and solved by symbolic computation for given incident wave polarization. The conductivity account via the Ohm law directly in the Maxwell equation leads to a complex wavenumber and hence complex amplitudes of the reflected and refracted waves. Atomic absorption is taken into account via the imaginary part of permittivity. The general formula for the time-averaged Pointing vector in the conducting media as a function of the medium parameters and the incident angle is derived and used for the refraction angle definition. The result is compared with textbooks and recent publications. The dependence of intensity as a function of the angle to the interface is determined also via the Pointing vector as a function of the incident wave and medium parameters.
A problem of wave identification is formulated. We propose a diagnostic analysis of medium disturbances based on distinguishing of components of a wave vector that is specific for each kind of the wave mode. Mathematically it is realized by projection operator technique. An example is considered in conditions of a one-dimensional Cauchy problem for a conventional wave equation in the matrix form and its version with weakly x-dependent coeffiients as a demonstration of the method application for the simplest adiabatic theory of one-dimensional acoustics. The case of acoustics in a gas with a dissipation account is also discussed from the point of view of the wave and entropy mode diagnostics.
Non-stationary thermal self-action of a periodic or impulse acoustic beam containing shock fronts in a thermoconducting Newtonian fluid is studied. Self-focusing of a saw-tooth periodic and impulse sound is considered, as well as that of a solitary shock wave which propagates with the linear sound speed. The governing equations of the beam radius are derived. Numerical simulations reveal that the thermal conductivity weakens the thermal self-action of the acoustic beam.
Modeling of the electromagnetic interaction with different homogeneous or inhomogeneous objects is a fundamental and important problem. It is relatively easy to solve Maxwell equations analytically when the scattering object is spherical or cylindrical, for example. However, when it loses these properties all that is left for us is to use approximation models, to acquire the solution we need. Modeling of complex, non-spherical, asymmetric particles is used to study cosmic, cometary dust, aerosols, atmospheric pollution etc. Few analytical, surface-based and volume-based methods of light scattering modeling, most commonly used by scientists, are reviewed here.
In this article we look at a conundrum that the Boussinesq-type equations pose for mathematicians allowing a Miura-type transformation while at the same time exhibiting no trace of a supersymmetric structure. We demonstrate that this riddle should be unraveled by dropping the standard supersymmetric approach in favor of its generalization: the “parasupersymmetry”.
The free convection heat transfer from an isothermal vertical plate in open space is investigated theoretically. In contrast to conventional approaches we use neither boundary layer nor self-similarity concepts. We base on expansion of the fields of velocity and temperature in a Taylor Series in x coordinate with coeffiients being functions of the vertical coordinate (y). In the minimal version of the theory we restrict ourselves by cubic approximation for both functions. The Navier-Stokes and Fourier-Kirchhoffequations that describe the phenomenon give links between coeffiient functions of y that after exclusion leads to the ordinary differential equation of forth order (of the Mittag-Leﬄeur type). Such construction implies four boundary conditions for a solution of this equation while the links between the coeffiients need two extra conditions. All the conditions are chosen on the basis of the experience usual for free convection. The choice allows us to express all the theory parameters as functions of the Rayleigh number and the temperature difference. To support the conformity of the theory we derive the NusseltRayleigh numbers relation that has the traditional form. The solution in the form of velocity and temperature profiles is evaluated and illustrated for air by examples of plots of data.
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