Torque equations for spin and orbital angular momenta of radiation fields and Faraday effect

The spin angular momentum S of light has never been linked to the Faraday rotation of light traveling in an optically active medium possessing a rotational invariance of a crystal, because there was no helicity term associated with the phase shift in the previous torque equation for S. In order to relate the change in S with time to the Faraday rotation, therefore, we derived an exact torque equation for S. As a result, a magnetic helicity term appeared in a new torque equation for S, so that one-half of the phase shift derived from the helicity term was equivalent to the Faraday rotation angle. However, the orbital angular momentum L had no relation to the Faraday rotation. It was thus clarified that the change in S with time is related to the Faraday rotation angle of light traveling in an optically active medium, owing to the appearance of the helicity term without a rotational invariance around the optical axis. It was also demonstrated theoretically that the Faraday rotation is accompanied by a torque acting on the crystal so that the total angular momentum of light and matter is conserved.     

INVARIANCE OF THE KAUP–KUPERSHMIDT EQUATION AND TRIANGULAR AUTO-BÄCKLUND TRANSFORMATIONS

We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup–Kupershmidt equation derived by E. G. Reyes in his paper titled "Nonlocal symmetries and the Kaup–Kupershmidt equation" [J. Math. Phys. 46 (2005) 073507, 19 pp.]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup–Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialization of the Kupershmidt equation.     

Development of the Evans Wave Equation in the Weak-Field Limit: The Electrogravitic Equation

The Evans wave equation [1-3] is developed in the weak-field limit to give the Poisson equation and an electrogravitic equation expressing the electric field strength E in terms of the acceleration g due to gravity and a fundamental scalar potential ⁽⁰⁾ with the units of volts (joules per coulomb). The electrogravitic equation shows that an electric field strength can be obtained from the acceleration due to gravity, which in general relativity is non-Euclidean spacetime. Therefore an electric field strength can be obtained, in theory, from scalar curvature R. This inference is supported by recent experimental data from the patented motionless electromagnetic generator [5].     

On the Cauchy Problem for a Nonlinearly Dispersive Wave Equation

Abstract

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.     

Quantum condensates classified in superconductivity with topology in the Minkowski space-time

A substantial problem in the macroscopic theory of pure superconductivity has been left forgotten for a long time since London and London in 1935. An impression survived that the Meissner effect is more substantial than the zero-resistivity. But, the London equation [I], the Newtonian equation of motion, was abandoned, whereas the London equation [II], derived from the Maxwell equations, was postulated. The London equation [II] included the logical gap [ α ] in real time, whereas the London equation [I] has been ignored without even noting the logical gap [ β ] in space. Microscopically, after the publication of F. London's book and the discovery of the isotope effect in 1950, the success of the Bardeen--Cooper--Schrieffer (BCS) theory in 1957 was likely to have finally given the definitive explanation on superconductivity by proving only the London equation [II] that claimed the coherent condensation of Cooper pairs in the momentum space. Since then, these arguments have been regarded to be a standard among various preceding theories. Meanwhile, the London equation [I] has faded away and has been long-forgotten. But we must not abandon the London equation [I], and, rather, retrieve it. We later recognized also that the DC-component of a persistent current can never be determined by using the Fourier transform analysis, because of its singularity at ω?=?0 and q ?=?0 with huge differences of space-time domain. Quite recently, in 2003, we first recognized a proper and harmonious view to simultaneously account for (i) the zero-resistivity in an open system with (i-c) the resultant persistent current in a closed system, and (ii) the perfect diamagnetism at T???0?K in the space-time aspects in terms of the gauge field theory. Here, we further clarify where and how we have lost and found a properly perspective view of the superconductivity. Here, we eliminate two logical gaps [ α ] and [ β ] by using the gauge field theory for further clarifying a position of the previous and present works. We especially classify superconductors with topology which eventually leads us such as (ii-2D) magnetic flux quantization in a ring. By projecting the 3-dimensional BCS-theory with the concept of ‘coherence’ among an enormous number of Bosons like Cooper pairs onto the (1?+?3)-dimensional Minkowski space-time [β?=?(v/c)?=?0], we clarify responses of the ground state Ψ _macro at T???0?K in a set of the basic equations, for (i) the zero-resistivity, [E _K ???qφ( R )]?=?0 at ω?=?0 and (ii) the perfect diamagnetism [?K ???qA ( R )]?=?0 at q ?=?0 as an inevitable consequence at the gauge fields in the proper theory of superconductivity.     

Non-Lie and Discrete Symmetries of the Dirac Equation

Abstract

New algebras of symmetries of the Dirac equation are presented, which are formed by linear and antilinear first–order differential operators. These symmetries are applied to decouple the Dirac equation for a charged particle interacting with an external field.     

Quantum kinetic Boltzmann equation taking into account the resonant exchange of excitations

A derivation of the quantum Boltzmann equation is given for identical particles with internal degrees of freedom. It is shown that the off-diagonal (with respect to the internal degrees of freedom) term of the equation contains an energy pole term, which is not present in the most commonly used kinetic equation, known as the Waldmann-Snider equation. The physical conditions underlying the occurrence of the pole term in the quantum kinetic equation are analyzed. Zh. éksp. Teor. Fiz. 111, 831–837 (March 1997)     

Is it possible to infer the equation of state of a mixture of hard discs from that of the one-component system?

Based on exact asymptotic properties of the composition-independent virial coefficients of a binary mixture of hard discs in the limits α = σ₂/σ₁ → 0, α → 1 and α → ∞, R. J. Wheatley (1998, Molec. Phys., 93, 965) has recently proposed an approximate interpolation equation for these coefficients. In this note, the equation of state equivalent to this interpolation is obtained, expressing the compressibility factor of the mixture in terms of that of the pure system. An extension to an arbitrary number of components is also given. The equation of state derived here is compared with another one recently proposed by following a different route (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) and with Monte Carlo simulation results. It is shown that the latter equation is more accurate than the former one, at least for not too disparate mixtures (0.7 < α < 1).     

The design of metamaterial cloaks embedded in anisotropic medium

By using coordinate transformation method, this paper obtains an useful equation of designing meta-material cloaks embedded in anisotropic medium. This equation is the generalization of what was introduced early by Pendry et al (2006 \textit{Science} {312 1780) and can be more widely used. As an example of its applications, this paper deduces the material parameter equation for cylinder cloaks embedded in anisotropic medium, and then offers the numerical simulation. The results show that such a cylinder cloak has perfect cloaking performance and therefore verifies the method proposed in this paper.     

Temperature of a thermally insulated surface in a gas flow

An equation is derived for determining the temperature of a thermally insulated surface in a gas flow. The equation does not contain any empirical coefficients. The derivation is based on allowance for the work done by a jet arrested at an obstructing surface on the surrounding flow layers. The application of the equation to subsonic and supersonic flows is discussed. Zh. Tekh. Fiz. 68, 134–135 (April 1998)     

Renormalization of the Lorentz-Abraham-Dirac equation for radiation reaction force in classical electrodynamics

Dirac’s analysis of radiation reaction force in classical electrodynamics suggested that a 4-momentum not collinear with 4-velocity could be introduced for a radiating electron. This would be equivalent to renormalization of the electron mass as an operator relating these 4-vectors. Dirac also pointed to an arbitrary choice made in deriving the Lorentz-Abraham-Dirac (LAD) equation. It was shown that renormalization substantially modifies the LAD equation under the additional requirement that the standard relativistic relation ℰ² = p ² c ² + m ² c ⁴ holds for the renormalized energy and momentum. The renormalized LAD equation is more rigorous than the LAD equation, because the drawbacks of the latter are eliminated, and is simpler than a well-known approximation of the LAD equation. The renormalized LAD equation appears to be better suited for numerical simulations of processes in ultrahigh-intensity laser-pulse fields.     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

The Scattering Approach for the Camassa–Holm equation

Abstract

We present an approach proving the integrability of the Camassa–Holm equation for initial data of small amplitude.     

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

Abstract

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.     

Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

Abstract

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We show that the Lie method is only suitable for an equation of maximal symmetry. We indicate the applicability of the method to cases in which the parametric function depends also upon the time.     

Energy loss of relativistic, multiply charged ions in an electron plasma

A simple equation is derived for calculating the energy loss of a relativistic, multiply charged ion moving in an electron plasma in the region where the Born approximation fails. The contribution of the energy losses from collisions with solitary electrons is calculated using the exact Dirac equation for relativistic Coulomb scattering. Zh. Tekh. Fiz. 68, 9–12 (February 1998)     

Statistical Approach of Modulational Instability in the Class of Derivative Nonlinear Schrödinger Equations

The modulational instability (MI) in the class of NLS equations is discussed using a statistical approach (SAMI). A kinetic equation for the two-point correlation function is studied in a linear approximation, and an integral stability equation is found. The modulational instability is associated with a positive imaginary part of the frequency. The integral equation is solved for different types of initial distributions (δ-function, Lorentzian) and the results are compared with those obtained using a deterministic approach (DAMI). The differences between MI of the normal NLS equation and derivative NLS equations is emphasized. PACS: 05.45.     

Numerical simulation methods for rough surface scattering

Abstract

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form

ABSTRACT

The Klein–Gordon equation plays an important role in mathematical physics. In this paper, a direct method which is very effective, simple, and convenient, is presented for solving the conformable fractional Klein–Gordon equation. Using this analytic method, the exact solutions of this equation are found in terms of the Jacobi elliptic functions. This method is applied to both time and space fractional equations. Some solutions are also illustrated by the graphics.     

A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

Abstract

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.