理想磁流体气球模本征方程的数值解法

一、基本方程 Tokamak等离子体高n气球模的不稳定性研究,归结为求解下述常微分方程的最大本征值Γ。     

Strong "quantum" chaos in the global ballooning mode spectrum of three-dimensional plasmas

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a k-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.     

Comments on curve veering in eigenvalue problems

The dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci. When two loci approach each other, they often cross or abruptly diverge. The latter case, called “curve veering”, has been observed in approximate solutions associated with discretized models. The influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions. The existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem. Veering is then examined in a general eigenvalue problem. Criteria are established to distinguish veerings from crossings in both continuous and discretized models. The application of the criteria is illustrated by examples.     

Eigenvalue degeneracy of the eight-vertex transfer matrix

We show for finite or infinite volume that the degeneracy of the maximum eigenvalue of the eight-vertex transfer matrix is the same above and below the critical temperature. A degeneracy of the maximum eigenvalue and a symmetry change in the eigenvectors of the transfer matrix provide the “mathematical mechanism” for the phase transition.     

Model for Risk Calculation and Reliability Comparison of Level Crossings

There is not a single country in the world that is so rich that it can remove all level crossings or provide their denivelation in order to absolutely avoid the possibility of accidents at the intersections of railways and road traffic. In the Republic of Serbia alone, the largest number of accidents occur at passive crossings, which make up three-quarters of the total number of crossings. Therefore, it is necessary to constantly find solutions to the problem of priorities when choosing level crossings where it is necessary to raise the level of security, primarily by analyzing the risk and reliability at all level crossings. This paper presents a model that enables this. The calculation of the maximal risk of a level crossing is achieved under the conditions of generating the maximum entropy in the virtual operating mode. The basis of the model is a heterogeneous queuing system. Maximum entropy is based on the mandatory application of an exponential distribution. The system is Markovian and is solved by a standard analytical concept. The basic input parameters for the calculation of the maximal risk are the geometric characteristics of the level crossing and the intensities and structure of the flows of road and railway vehicles. The real risk is based on statistical records of accidents and flow intensities. The exact reliability of the level crossing is calculated from the ratio of real and maximal risk, which enables their further comparison in order to raise the level of safety, and that is the basic idea of this paper.     

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Mode theory of plane Fabry-Pérot resonator with inhomogeneous active medium

A new method of the eigenvalue problem solution for a resonator with inhomogeneous active medium is given. The approach is based on Maxwell's wave equation and impedance boundary conditions of resonance-type at open resonator ends. A resonator equipped with mirrors in the form of infinite long strips is studied as an example. A rigorous solutions for the cases of stepped and bounded parabolic active medium profiles are obtained. Transcendental eigenvalue equations are investigated, distributions of field amplitude of active resonator modes are found. Asymptotic behavior of rigorous solutions is investigated. A multilayer approximation method is proposed for the eigenvalue problem solution for a resonator with an arbitrary gradient profile of active medium. The testing of this method was carried out with the rigorous solutions for the bounded parabolic profile.     

Stability of tokamak plasmas with internal transport barriers against high n ideal magnetohydrodynamic ballooning mode

A ballooning mode equation for tokamak plasma, with the toroidicity and the Shafranov shift effects included, is derived for a shift circular flux tokamak configuration. Using this equation, the stability of the plasma configuration with an internal transport barrier （ITB） against the high n （the toroidal mode number） ideal magnetohydrodynamic （MHD） ballooning mode is analysed. It is shown that both the toroidicity and the Shaftanov shift effects are stabilizing. In the ITB region, these effects give rise to a low shear stable channel between the first and the second stability regions. Out of the ITB region towards the plasma edge, the stabilizing effect of the Shaftanov shift causes the unstable zone to be significantly narrowed.     

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(εz + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω_i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.     

The Taylor-Couette eigenvalue problem with independently rotating cylinders

The eigenvalue problem for disturbances of the laminar Couette flow with co- or counterrotating cylinders of infinite height is considered. We solve the eigenvalue equation numerically and also give some analytical approximations for the eigenvalues and eigenmodes. Flow field patterns and spectra are offered.     

托卡马克理想磁流体不稳定性的统一描述 (I)

在与平衡磁面相联系的坐标系下，用剪切阿尔芬波近似给出了统一描述托卡马克等离子体理想磁流体线性运动的本征模方程。利用此方程，可以进一步给出大尺度扰动（扭曲模、低模数气球模、阿尔芬模）和小尺度扰动（高模数气球模、Mercier模）的本征模方程。本文详细讨论了小尺度扰动的本征模方程。     

The positivity and other properties of the matrix of capacitance: Physical and mathematical implications

We prove that the matrix of capacitance in electrostatics is a positive-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for such a matrix. Many properties are easily visualized by constructing a “potential space” isomorphic to the euclidean space. The problem of minimizing the internal energy of a system of conductors under constraints is considered, and an equivalent capacitance for an arbitrary number of conductors is obtained. Moreover, some properties of systems of conductors in successive embedding are examined. Finally, we discuss some issues concerning the gauge invariance of the formulation.     

Role of flow shear in the ballooning stability of tokamak transport barriers

A tokamak's confinement time is greatly increased by a transport barrier (TB), a region having a high pressure gradient and usually also a strongly sheared plasma flow. The pressure gradient in a TB can be limited by ideal magnetohydrodynamic instabilities with a high toroidal mode number n ("ballooning modes"). Previous studies in the limit n--> infinity showed that arbitrarily small (but nonzero) flow shears have a large stabilizing influence. In contrast, the more realistic finite n ballooning modes studied here are found to be insensitive to sub-Alfvénic flow shears, provided the magnetic shear s approximately 1 (typical for TBs near the plasma's edge). However, for the lower magnetic shears that are associated with internal transport barriers, significantly lower flow shears will influence ballooning mode stability, and flow shear should be retained in the analysis of their stability.     

Time series analysis and Monte Carlo methods for eigenvalue separation in neutron multiplication problems

A time series approach has been applied to the nuclear fission source distribution generated by Monte Carlo (MC) particle transport in order to calculate the non-fundamental mode eigenvalues of the system. The novel aspect is the combination of the general technical principle of projection pursuit for multivariate data with the neutron multiplication eigenvalue problem in the nuclear engineering discipline. Proof is thoroughly provided that the stationary MC process is linear to first order approximation and that it transforms into one-dimensional autoregressive processes of order one (AR(1)) via the automated choice of projection vectors. The autocorrelation coefficient of the resulting AR(1) process corresponds to the ratio of the desired mode eigenvalue to the fundamental mode eigenvalue. All modern MC codes for nuclear criticality calculate the fundamental mode eigenvalue, so the desired mode eigenvalue can be easily determined.     

Unified Solutions of the Hard-Core Fermi-and Bose-Hubbard Models

A unified algebraic approach to both the hard-core Fermi- and Bose-Hubbard models is extended to boththe finite- and infinite-site with periodic condition cases. Excitation energies and the corresponding wavefunctions ofboth the models with nearest neighbor hopping are exactly derived by using a new and simple algebraic method. It isfound that spectra of both the models are determined simply by eigenvalue problem of N × N hopping matrix, where Nis the number of sites for finite system or the period of sites for infinite system.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

Spectrum of a Problem of Elasticity Theory in the Union of Several Infinite Layers

The essential spectrum of the Dirichlet problem for the system of Lamé equations in a three-dimensional domain formed by three mutually perpendicular elastic layers occupies the ray [Λ_?,+∞). The lower bound Λ_? > 0 is the least eigenvalue (its existence is established) of the problem of elasticity theory in an infinite two-dimensional cross-shaped waveguide. It is proved that the discrete spectrum of the spatial problem is nonempty. Other configurations of layers and the scalar problem of the junction of quantum waveguides are also considered.     

Diagonalisation of finite-size corner transfer matrices and related spin chains

We study Baxter's corner transfer matrices for anisotropic Ising models of finite size. They are related to spin one-half chains with coefficients which increase linearly along the chain. The operators are diagonalised with the help of special polynomials and the eigenvalue spectrum is discussed. The relation to the infinite lattice limit is outlined.     

Linewidth of a detuned single mode laser near threshold

The linewidth of a detuned single mode laser is calculated. Using the Fokker-Planck equation method, the problem is reduced to a nonhermitian eigenvalue problem. Numerical results near threshold and an approximate linewidth formula are given.     

Impact parameter calculation of electron impact excitation of hydrogen

The convenience of a diabatic or an adiabatic interpolation of frequencies in variational transition state calculations involving interpolation methodologies is discussed. The gas phase proton transfer between butanone and a hydroxide anion has been used to illustrate the theoretical discussion. For example, it has been shown that if vibrational normal mode crossings exist the simple adiabatic interpolation can produce incorrect entropy contributions and, as a consequence, a spurious displacement of the generalized transition state.