On the effect of a magnetic field on fast charged particles passing through a substance

A transport equation in the small-angle approximation is obtained for a curvilinear beam of fast charged particles passing through a substance in a nonuniform magnetic field. Green functions for this equation are found for an annular beam in a weak-focusing field and for a helical beam in the nonuniform magnetic field.     

An electron in a quantized plane wave and in a constant magnetic field

The Dirac equation is considered for an electron in a constant homogeneous magnetic field and in a quantized electromagnetic plane wave propagating along the direction of the former. This problem is shown to be reduced to the Schrödinger equation for a system of many interacting oscillators. The corresponding Hamiltonian is diagonalized. An equation is established relating the total energy with the moment of a system along the magnetic field. Approximate solutions of this equation are given. A behaviour of a system close to cyclotron resonance is discussed.     

Convection in a slowly diffusing,weakly stratified salt field

We examine thermohaline convection in the small Lewis number limit for a background salt field with a weak gradient. As a result of small diffusivity, the problem cannot be reduced to a simple amplitude equation in the usual manner, but the salt field is described by an advection-diffusion equation coupled to an equation for the amplitude of the convective pattern.     

Localized optical structures in the scheme of a nonlinear layer with a feedback mirror

An analysis is made of the dynamics of the transverse structure of the field in the scheme of a layer of a medium with nonlinear amplitude-phase transmittance and a feedback mirror separated from it by a linear spacing. For the case of small field changes in a single passage, an approximate equation is derived, which is close in form to the equation used in the average-field approximation for nonlinear interferometers excited with external emission. For the nonlinearity of threshold type, an analytical form is presented for the field distribution corresponding to localized dissipative structures (dissitons).     

Density fluctuations and the structure of a nonuniform hard sphere fluid

We derive an exact equation for density changes induced by a general external field that corrects the hydrostatic approximation where the local value of the field is adsorbed into a modified chemical potential. Using linear response theory to relate density changes self-consistently in different regions of space, we arrive at an integral equation for a hard sphere fluid that is exact in the limit of a slowly varying field or at low density and reduces to the accurate Percus-Yevick equation for a hard core field. This and related equations give accurate results for a wide variety of fields.     

Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.     

Pulse propagation in a nonlinear dielectric slab waveguide

The evolution of an optical pulse in a single-mode, step index dielectric slab waveguide which is characterized by an intensity dependent dielectric function in the core and cladding regions is treated by means of differential equation techniques. A cubic order non-linearity is considered. The electromagnetic field distribution in the slab waveguide region satisfies a non-linear wave equation. This field can be represented in terms of even TE guided modes with a slowly varying envelope amplitude function.Then using the well known approximation, based on the slowly varying character of the amplitude function, a non linear partial differential equation is obtained for the amplitude function. As the coefficients of this equation depend on the distance across the transverse direction X, an averaging technique over x is applied to reduce the nonlinear partial differential equation into a form that is easily transformed to the so-called non-linear Scroedinger differential equation.This equation is then attacked by means of the well known Inverse Scattering method in the case of reflection less potentials. The single and double soliton solutions are obtained explicitly for a single-mode slab waveguide. Finally numerical results are presented in the time domain.     

Dynamics of a Quantum Damped Oscillator with Relaxation

The quantum problem of a nonlinear oscillator interacting with the field of harmonic oscillators with continuously distributed frequencies is examined. The Heisenberg representation is used. It is demonstrated that after elimination of field variables from the equation, the oscillator dynamics is described by an integro-differential operator equation. It is proved that in the general nonlinear case, an exact solution for the quantum oscillator demonstrates dissipative behavior under certain limitations on the relaxation kernel.     

Application of a reflector with a nonuniform field in a mass-reflectron

The conditions under which the nonuniform compensating field of the reflector of the mass-reflectron can be generated with an acceptable accuracy at the symmetry axis of the reflector and extrapolated to the radial neighborhood of the axial line are determined. The plots that illustrate the distribution of the calculated nonuniform field of the reflector, the possibilities for implementation, and errors of focusing with respect to time of flight in the radial neighborhood are presented. Analytical expressions for the calculation of the time of flight of ions in the reflector in which the field distribution is described using a power series and analytical expressions for the calculation of the field distribution in the reflector in which the time of flight is determined using a power series are derived. A method for the analytical calculation of the compensating nonuniform field of the reflector based on the given dependence of the time of flight in the absence of such a field is proposed using a solution to the Abel integral equation. The solution to this equation yields analytical expressions for the calculation of the compensating field of the reflector in mass-reflectrons that contain the zero-field drift space and regions of acceleration (deceleration) of ions with a uniform field.     

Amplification of ultimately-short pulses in graphene in the presence of a high-frequency field

The Maxwell equations for an electromagnetic field propagating in graphene are considered taking into account strong Coulomb repulsion between electrons of the same site possessing opposite spin projections. The derived effective equation has the form of a classical 2D sine-Gordon equation. Electrons are treated in terms of quantum formalism with allowance for the dispersion law in the presence of Coulomb interaction. The effective equation is analyzed numerically and the effect of Coulomb repulsion is revealed. It is shown that the system in an external homogeneous electromagnetic field, with its period much shorter than the characteristic pulse length, may show the amplification of an ultimately short pulse.     

Classical spin in a potential field

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrödinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.     

Amplification of electromagnetic pulses in graphene with Hubbard interaction by a uniform high-frequency alternating field

The Maxwell equations for an electromagnetic field propagating in graphene in which electrons with different spin projections experience a strong Coulomb repulsion at a single node are considered. An effective equation analogous to the classical (1+1)-dimensional sine-Gordon equation is derived. Electrons are treated within the framework of the quantum formalism with allowance for changes in the dispersion law in the presence of Coulomb interaction. The effective equation is solved numerically, and the effect of Coulomb repulsion is identified. The amplification of an ultrashort pulse is observed when an external uniform electromagnetic field with a period shorter than the pulse duration is applied.     

Pressure pulse induced by a pulsed electric current in a cylindrical liquid conductor

An analogue of z pinch is considered for a finite-conductivity liquid in a cylindrical nonconducting tube through which an exponentially decaying current pulse is passed. The distributions of magnetic induction, current density, and volume electromagnetic force pressure in the conducting liquid cylinder are found by solving an equation for a quasi-stationary electromagnetic field and the equation of magnetohydrostatics with the operational method.     

On the influence of a weak magnetic field on the drift motion of charged particles

The results obtained in [1] are generalized to the case when an arbitrarily oriented magnetic field is present by allowance for the influence of a magnetic field with cyclotron frequency less than the collision frequency on the dependence of the drift velocity on the velocity of random motion (Langevin equation). A study is made of the influence of a magnetic field on the equation of drift motion derived in [1] and this field is allowed for in the previously obtained results of the solution of that equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 69–72, May, 1972.     

Self-consistent mechanism for sustaining ionization waves in a low-pressure discharge

The possibility of constructing a self-consistent model for the sustaining of ionization waves is demonstrated for a low-pressure discharge in an inert gas. The model is based on the combined solution of the kinetic equation of the electrons and the equation of motion of the ions in a spatially periodic field. The distribution function is constructed in an experimentally measured field and then used to calculate the spatial distributions of the plasma density and the ionization rate. The solution of the equation of motion of the ions makes it possible to reconstruct a field similar to the original one. One specific feature of the mechanism considered is the pronounced nonlocal character of the formation of the electron distribution function by the entire nonuniform potential profile of the ionization wave. Zh. Tekh. Fiz. 67, 24–30 (February 1997)     

Inverted satellites of a solitary oscillating pearl vortex in a thin magnetic superconductor film

An electrodynamic equation is derived for the magnetic field of an isolated Pearl vortex moving along an arbitrary trajectory in an ultrathin film of a magnetic superconductor. This equation is valid for any type of magnetic order in the magnetic subsystem. The magnetic structure of an isolated oscillating Pearl vortex is investigated in a thin magnetic superconductor film. Oscillations of the vortex and the presence of the magnetic subsystem are shown to lead to a significant renormalization of the vortex field in comparison with the Pearl solution. New phenomena of inverted satellites are predicted in which an inverted precursor appears in front of the vortex and an inverted wake is formed behind the latter at a distance of the order of 10λ_eff from the vortex center. These phenomena can be observed in magnetooptical experiments.     

Stochastic diffusion in a periodic potential

This paper deals with two equations for classical stochastic diffusion in a potential. First, the full Fokker-Planck equation in phase-space for a Brownian particle in a periodic potential and linearly coupled to an external field is considered. The solution discussed previously by the author and co-worker is improved upon. An alternative approximation is introduced. Then, the simpler Smoluchowski equation, which is derivable from the Fokker-Planck equation under suitable conditions, is solved using Hill's determinant method. Finally a WKB-type method is proposed to solve the Smoluchowski equation for a general class of potentials.     

Series Solution for Localization and Entanglement of an Exciton in a Quantum Dot Molecule by an ac Electric Field

The Schrödinger equation involving the phenomenon of the localization and entanglement for an exciton in a quantum dot molecule by an ac electric field is analytically investigated. New exact series solutions for the Schrödinger equation have been obtained for the first time. The analytical expressions can further describe the dynamical behaviors of an interacting electron-hole pair in a double coupled quantum dot molecule under an ac electric field accurately.     

Curle's equation and acoustic scattering by a sphere

Recent papers have initiated interesting comparisons between aeroacoustic theory and the results of acoustic scattering problems. In this paper, we consider some aspects of these comparisons for acoustic scattering by a sphere. We give a derivation of Curle's equation for a specific class of linear acoustic scattering problems, and, in response to previous claims to the contrary, give an explicit confirmation of Curle's equation for plane wave scattering by a stationary rigid sphere of arbitrary size in an inviscid fluid. We construct the complete solution for scattering by a rigid sphere in a viscous fluid, and show that the neglect of viscous terms in Curie's equation yields an incomplete prediction of the far field dipole pressure. We also consider the null field solution of the sphere scattering problem, and give its extension to the vorticity modes associated with viscosity. Finally, we construct a solution for an elastic sphere in a viscous fluid, and show that the rigid sphere/null field solution is recovered from the limit of infinite longitudinal and shear wave speeds in the elastic solid.     

A general method for evaluating the current system and its magnetic field of a plane current sheet, uniform except for a certain area of different uniform conductivity, with results for a square area

Summary We derive a linear Fredholm integral equation of the second kind on an arbitrary closed plane contour which divides an infinite plane current sheet into two regions of different uniform integrated conductivities. This integral equation is satisfield along the above-described contour by a certain combination of the limiting values of the electric potential at both sides of the boundary. This electric potential is due to the currents created in the sheet when a uniform electric field is applied to it. The derived integral equation admits exact solutions in closed form for the cases of circular and elliptical insertions. These solutions are identical with those previously obtained, by other methods, for the same cases. A general method is given for the numerical solution of the integral equation. As an illustration, this method is applied to the case of a square insertion where we used the results of Ashour to obtain numerical estimation for the results of the additional magnetic field on and around the square insertion. To speed up publication, the proofs were not sent to the authors and were supervised by the Scientific Commettee.