Optically S-polarized surface waves in symmetrical nonlinear sensors: Thermal effects

A surface wave in a planar nonlinear waveguide is a current problem that has several applications in modern electronics and optics such as optical sensors design. The effect of thermal stress on the optical performance of a nonlinear symmetrical sensor is studied. The mathematical forms of the dispersion equation and thermal-stress sensitivity are analytically derived and plotted numerically. It is found that the thermal sensitivity of the sensor can be controlled by tuning the core size, by changing the loading materials, and by carefully selecting the materials.     

Influence of imperfectly bonded micropolar elastic half-space with non-homogeneous viscoelastic layer on propagation behavior of shear wave

A mathematical model is developed in which the effect of imperfect bonding between the constituents of layer and half-space on the phase velocity and damped velocity of SH-wave is discussed. The model consists of a micropolar elastic half-space bonded imperfectly with a heterogeneous viscoelastic layer. The dispersion equation and damping equation of SH-wave propagation in the said model is obtained in the closed form analytically. The effects of imperfect bonding, internal friction, heterogeneity, micropolarity, and complex interface stiffness parameters highlighted through numerical computation and graphical demonstrations. Standard Love-wave equation and dispersion equation as well as damping equation for perfectly bonded micropolar half-space with heterogeneous viscoelastic layer is obtained as a special case of the problem. Through comparative study of homogeneity with heterogeneity in the layer; imperfect bonding of layer and half-space with their welded (perfect) contact; and presence of micropolarity in half-space with its absence in half-space are compared meticulously.     

Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations

This Letter deals with a generalized Camassa-Holm equation and a nonlinear dispersive equation by making use of a mathematical technique based on using integral factors for solving differential equations. The peakons, solitary patterns and periodic solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are highlighted.     

Analytical solution of the generalized smoluchowski problem using a new model kinetic equation

A new kinetic equation whose solution yields a correct Prandtl number is suggested. A technique for analytically solving half-space boundary-value problems is demonstrated with the classical Smoluchowski problem of finding temperature and concentration steps. Numerical results indicating the succession of the equation.     

Transformation of a monochromatic plane wane by the pulse-periodic time modulation of an infinite medium

An exact solution to the problem of the transformation of a monochromatic plane wave by a finite train of equally spaced rectangular pulses of permittivity and conductivity of an infinite medium is considered. The permittivity pulse train is shifted relative to the conductivity pulse train by an arbitrary time. The problem is studied analytically in terms of the second-order Volterra integral equation describing the electromagnetic wave transformation in a medium with time-dependent parameters. The equation is solved using the resolvent technique. Expressions for the amplitude of the transformed electric field component for any time instant at any spatial point are derived and analyzed.     

Bohr-Sommerfeld quantization condition for the Gross-Pitaevskii equation

The discrete spectrum of the nonlinear eigenvalue problem associated to the one-dimensional Gross-Pitaevskii equation with a smooth potential is studied in the quasiclassical limit. We particularly focus on the corrections to the Bohr-Sommerfeld quantization rule for the excited energy levels due to the nonlinearity. Explicit predictions are obtained analytically for these corrections and are supported by numerical computations.     

Analysis of the evolution of perturbations in a magnetized collisionless plasma with an integral-equation technique

Summary A technique recently proposed to study the classical problem of the evolution of small perturbations in a collisionless unmagnetized plasma is extended to a magnetized plasma. A time-convolutive integral equation for the plasma density is obtained from the Vlasov equation for a homogeneous plasma in a uniform, stationary magnetic field. The equation can be solved by means of simple numerical algorithms and, in some cases, analytical solutions can be obtained. The procedure proves to be analytically simpler than the classical one and is more convenient from a numerical point of view. Techniques of solution are presented and analytical and numerical results for electrostatic perturbations are discussed.     

Discontinuities in the effective-mass equation

The validity of the Heisenberg uncertainty principle for the effective-mass equation is analytically verified for the first time. The procedure also provides an unambiguous demonstration of the need for a distributional derivative in the solution of the one-dimensional rectangular quantum-well problem for Hamiltonians with a piecewise-constant mass function .     

Analytical solution to the transient beam loading effects of a superconducting cavity

Transient beam loading is one of the key issues in any high beam current intensity superconducting accelerators, and needs to be carefully investigated. The core problem in the analysis is to obtain the time evolution of effective cavity voltage under transient beam loading. To simplify the problem, the second order ordinary differential equation describing the behavior of the effective cavity voltage is intuitively simplified to a first order one, with the aid of two critical approximations which lack proof of their validity. In this paper, the validity is examined mathematically in some specific cases, resulting in a criterion for the simplification. It is popular to solve the approximate equation for the effective cavity voltage numerically, while this paper shows that it can also be solved analytically under the step function approximation for the driven term. With the analytical solution to the effective cavity voltage, the transient reflected power from the cavity and the energy gain of the central particle in the bunch can also be calculated analytically. The validity of the step function approximation for the driven term is examined by direct evaluations.After that, the analytical results are compared with the numerical ones.     

Eigenvalues for the extremely underdamped Brownian motion in an inclined periodic potential

The eigenvalues for the Brownian motion in a periodic potential with an additive constant force are investigated in the low friction limit. First the Fokker-Planck equation for the distribution function in velocity and position space is transformed to energy and position coordinates. By a proper averaging process over the position coordinate a differential equation for the distribution function depending on the energy only is obtained. Next the eigenvalues and eigenfunctions are calculated from this equation by a Runge-Kutta method. Finally the problem is formulated in terms of an integral equation from which the lowest non-zero eigenvalue is obtained analytically in the bistability region in the zero temperature limit.     

A solution to the problem of the skin effect with a bias current in the Maxwell plasma by the method of expansion in eigenfunctions

A problem of the skin effect in the Maxwell plasma is solved analytically by the method of expansion in eigenfunctions based on the Vlasov–Maxwell kinetic equation with a self-consistent electric field. Specular electron reflection from the boundary is used as a boundary condition. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 33–38, February, 2009.     

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.     

Synthesis of currents on a disk using a directional diagram

The problem of synthesis of currents using a realizable or unrealizable directional diagram is solved. In the former case, the problem is solved analytically: the current is sought as a series in basis, with each of the basis functions satisfying the Meixner condition on an edge. If the diagram is unrealizable, the current is found from a solution to an integral equation with a small parameter. The solution to the integral equation also satisfies the Meixner condition on an edge.     

Radiative transfer through a compton-scattering atmosphere with continuous energy dependence

The method of elementary solutions is applied to an equation of transfer for a Compton-scattering medium. The half-space albedo problem is solved analytically, retaining continuous energy, angle, and position dependence, and numerical results for the emergent distribution are compared to results obtained by an approximate technique.     

留数定理在拓扑相变中的应用

留数定理是高校物理专业必修课程数学物理方法中的一个重要定理.传统教学中关于该定理的讲授着重于数学公式的推导和数学思想的传达,而对于其在具体物理问题上的应用鲜有涉及.本文以一维Su-Schrieffer-Heeger模型的拓扑相变问题为例,阐明了如何利用留数定理解析得到二阶位移量的表达式并用该物理量表征拓扑相变.在讲授留数定理的教学过程中引入具体物理问题的分析实例,可以使学生更深刻地理解数学定理中的物理内涵.     

Analytic Results in the Position-Dependent Mass Schrödinger Problem

We investigate the Schrödinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass V(x)=0 case whose solutions are hypergeometric functions in tanh²x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form V(x)=V₀ sinh²x.     

Theoretical study of electric conductance of atomic contact with the Friedel sum rule

We analyze the electric conductance through nanostructure bridges in terms of phase-shifts, which satisfy the Friedel sum rule. The phase-shifts are given by solving the eigenvalue equation obtained by extending the method applied to a single impurity problem in a metal. We introduce the charge neutrality condition through the Friedel sum rule. It is analytically shown that the electric conductance is quantized and can increase as the two electrodes separate under the condition in which the phase-shifts satisfy the Friedel sum rule.     

Density matrix of two interacting particles with kinetic coupling derived in bipartite entangled state representation

A density matrix is usually obtained by solving the Bloch equation, however only a few Hamiltonians' density matrices can be analytically derived. The density matrix for two interacting particles with kinetic coupling is hard to derive by the usual method due to this coupling; this paper solves this problem by using the bipartite entangled state representation.     

Two half-space problems based on a synthetic-kernel model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve two basic half-space problems based on a new synthetic-kernel model of the linearized Boltzmann equation. In particular, Kramers’ problem and the half-space problem of thermal creep, both basic to the general area of rarefied-gas dynamics, are defined by model equations that are solved (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented to yield numerical results for the slip coefficients and the velocity and heat-flow profiles that compare well with solutions derived from much more computationally intensive techniques.