Applications of elliptic functions to ion-acoustic plasma waves

New several classes of exact solutions are obtained in terms of the Weierstrass elliptic function for some nonlinear partial differential equations modeling ion-acoustic waves as well as dusty plasmas in laboratory and space sciences. The Weierstrass elliptic function solutions of the Schamel equation, a fifth order dispersive wave equation and the Kawahara equation are constructed. Moreover, Jacobi elliptic function solutions and solitary wave solutions of the Schamel equation are also given. The stability of some periodic wave solutions is computationally studied.     

Wigner function for the Dirac oscillator in spinor space

The Wigner function for the Dirac oscillator in spinor space is studied in this paper.Firstly,since the Dirac equation is described as a matrix equation in phase space,it is necessary to define the Wigner function as a matrix function in spinor space.Secondly,the matrix form of the Wigner function is proven to support the Dirac equation.Thirdly,by solving the Dirac equation,energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.     

Wigner function for the Dirac oscillator in spinor space

The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space. Secondly, the matrix form of the Wigner function is proven to support the Dirac equation. Thirdly, by solving the Dirac equation, energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.     

新环状非球谐振子势的Dirac方程束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下, 给出了Dirac 方程的束缚态解.通过分离变量得到Dirac方程相应的角向方程和径向方程,得出了用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数;获得了精确的束缚态能谱方程并对结果作适当讨论与结论。     

新环状非球谐振子势的Dirac方程束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下, 给出了Dirac 方程的束缚态解.通过分离变量得到Dirac方程相应的角向方程和径向方程,得出了用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数;获得了精确的束缚态能谱方程并对结果作适当讨论与结论.     

On a Liouville-master equation formulation of open quantum systems

The dynamics of open quantum systems is formulated in terms of a probability distribution on the underlying Hilbert space. Defining the time-evolution of this probability distribution by means of a Liouvillemaster equation the time-dependent wave function of the system becomes a stochastic Markov process in the sense of classical probability theory. It is shown that the equation of motion for the two-point correlation function of the random wave function yields the quantum master equation for the statistical operator. Stochastic simulations of the Liouville-master equation are performed for a simple example from quantum optics and are shown to be in perfect agreement with the analytical solution of the corresponding equation for the statistical operator.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.     

Two-dimensional isotropic scattering in a semi-infinite cylindrical medium

Beginning with the integral equation for the source function, the solutions for the source function, flux and intensity at the boundary of a two-dimensional, isotropically scattering cylindrical medium are found. The incident radiation is collimated and normal to the surface of the medium and depends only on the radial coordinate. For a Bessel function boundary condition, separation of variables is used to reduce the source function integral equation to a one-dimensional equation. The resulting integral equation is shown to be the same as that for the two-dimensional planar case. Solutions for other boundary conditions are then shown to be superpositions of the Bessel function solution. Numerical results are presented for a Gaussian distribution of incident radiation which closely models a laser beam. These multiple scattering results are compared to the single scattering approximation. Also, the solution for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.     

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.     

Effect of temperature on $$\mathbf{In}_{{\varvec{x}}} \mathbf{Ga}_{1-{{\varvec{x}}}} \mathbf{As}/\mathbf{GaAs}$$ quantum dots

Analytical solution of the Dirac equation for the modified Pöschl–Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By using asymptotic iteration method, the relativistic energy equation and orbital quantum number (l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number n _r causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativistic energy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.     

铝表面的电子关联函数

根据金属表面的多体波函数,建立了非均匀电子关联函数的积分方程,对此方程消除了因长程关联而出现的发散。在第一次迭代下求得了铝表面的电子关联函数的数值结果。 关键词：     

Relaxation of initial spatially unhomogeneous states of phonon gases scattered by point mass defects embedded in isotropic media

The Cauchy problem for the Boltzmann-Peierls equation describing the rarefied gas of acoustic phonons scattered by point mass defects embedded in an isotropic medium is studied. The equation for the Fourier-Laplace transform of the distribution function obtained from the Boltzmann-Peierls equation is solved. The singularities of the Fourier-Laplace transform of the distribution function are investigated. The explicit time dependence of the Fourier transform is established. The crossover from kinetic to diffusive behaviour and the long-time asymptotics of the distribution function is studied. The spectral decomposition of the collision operator is obtained and several lowest order terms of the Chapman-Enskog series are derived. For comparison the suitable results for two dimensional isotropic media are listed.     

Asymptotics of the binary amplitude for a model Faddeev equation

A model equation obtained from the s-wave Faddeev equation in configuration space for three identical bosons by replacing the inhomogeneous integral term with a known function is studied. This function simulates the asymptotic decrease of the inhomogeneous term in the original Faddeev equation when y → ∞ as ～y^–3/2. The asymptotes of the amplitude functions that approach the scattering amplitudes when y → ∞ are obtained analytically for the model equation using the Green function. It is shown that the asymptotics of the binary channel amplitude function oscillates. The similar oscillating behavior of the binary amplitude function is numerically demonstrated for the original s-wave Faddeev equation describing the neutron–deuteron scattering process.     

Electron transport in semiconductors in the presence of impact ionization

The development of impact ionization in semiconductors has been considered. A transport equation for the electron distribution function in the presence of impact ionization has been derived. It has been found that the collison integral of this equation is nonlinear with respect to the distribution function. The relation between the solutions of this equation and the usual Boltzmann equation have been determined. A Monte Carlo method for numerical calculations in the presence of impact ionization has been developed. The results of numerical calculations carried out using a simplified model of indium antimonide are presented.     

Kinetic theory of hard spheres

Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.     

Method of green's functions in the theory of quantum kinetic equations. II

The kinetic and antikinetic equations are obtained for the single-particle Wigner function in the context of the method of Green's time-temperature functions for an inhomogeneous system of weakly interacting particles situated in a time-dependent electric field. The kinetic equation is derived here from the equation of motion for Green's function, satisfying the causality condition.     

Solution of the Boltzmann equation by expanding the distribution function with several time and coordinate scales in the enskog series in knudsen parameter

A method to solve the Boltzmann equation is analyzed in the case when the distribution function depends on slow and fast time and coordinate scales. Basic relationships for calculating the nonequilibrium multiscale distribution function are shown to differ substantially from those found in the framework of the Chapman-Enskog method: the transfer equations are complemented by the contributions of relaxation processes. The heat and momentum transfer equations derived from the general solution to the Boltzmann equation involve additional terms accounting for relaxation effects. The relaxation effects included in the energy equation result in both a hyperbolic heat conduction equation and a finite rate of heat transfer. In the viscous stress tensor, the Newtonian term of the transfer equation turns out to be supplemented by relaxation terms.     

非线性发展方程的Riemann theta 函数等几种新解

为了构造非线性发展方程的复合型无穷序列精确解, 获得了第二种椭圆方程的Riemann theta 函数等几种新解.在此基础上,利用第二种椭圆方程与Riccati方程的Bäcklund变换和解的非线性叠加公式, 借助符号计算系统 Mathematica, 以mKdV方程为应用实例, 构造了该方程的复合型无穷序列新精确解.这里包括Riemann theta 函数、Jacobi椭圆函数、双曲函数、 三角函数和有理函数,通过几种形式构成的复合型无穷序列新精确解. 关键词： 第二种椭圆方程 Riccati方程 非线性发展方程 Riemann theta 函数无穷序列解