A Schr?dinger formulation research for light beam propagation

The wave equation of light beam propagation was written in the form of an axial-coordinate-dependent Schr?dinger equation, and the expectation value of a dynamical variable, the trial function of variational approach and the ABCD law were discussed by use of quantum mechanics approach. In view of the evolution equations of expectation values of dynamical variables in the framework of quantum mechanics, the definition of a potential function representing the beam propagation stability and its universal formula with the quality factor, the universal formula of beam width and curvature radius for a paraxial beam and cylindrically symmetric non-paraxial beam, the general formula of second derivative of beam width with respect to the axial coordinate of beam for a paraxial beam, and the general criteria of the conservation of beam quality factor and the existence of a potential well of a potential function for a paraxial beam, were given or derived, respectively. Starting with the same trial function, the comparative research of our formulation with variational approach was done, which gave some further insight into the physical nature of a beam propagation parameters. The ABCD law of nonparaxial beam was discussed in terms of the definition of the non-paraxial expectation value of a dynamical variable for the first time. The applications to the media of constant second derivative of beam width with respect to the axial coordinate of a beam, square law media and the media of constant refractive index in the momentum representation were discussed, respectively.     

A Schrodinger formulation research for light beam propagation through the media of complex refractive index

The Helmhotz equation of light beam propagating through a medium of complex refractive index is reduced to the axial-coordinate-dependent Schr?dinger equation of complex potential. The new bra vector, the new expectation value of a dynamical variable and the extended Heisenberg picture are defined by the inverse of the evolution operator instead of its Hermitian adjoint, and the complex beam propagation parameters defined in terms of the new expectation value, the complex ABCD law and the ABCD formulation of the Huygens' integral are discussed in terms of quantum mechanics. It is shown that the evolution equations of the complex beam propagation parameters are the same as those of the beam propagation parameters of beam propagating through a medium of real refractive index. The research on an optical system of the conservative complex beam quality factor shows that the complex ABCD law holds, the evolution of its coordinate operator and the momentum operator is linear, and the Huygens' integral is of the ABCD formulation.     

Geometrical study of paraxial light beam transmission in free space

By introducing an imaginary space transform curvature ρ_s, a complex space called Riemannian space is constructed, in which the light propagating in free space has the trajectory of straight line while propagating. Moreover, this curvature couples with that of the wave front of the paraxial beam ρ_w, and therefore a complex curvature ρ_c is constructed, which can be employed to investigate the behavior of the light transmission and to generalize the ABCD law. Project supported by the National Hi-Tech Inertial Confinement Fusion Committee, the Guangdong Natural Science Foundation the Postdoctoral Foundation of Guangdong and National Postdoctoral Foundation of China.     

Quasidifferentiability in nonsmooth,nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.     

Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation

Based on the nonlocal nonlinear Schrödinger equation that governs phenomenologically the propagation of laser beams in nonlocal nonlinear media, we theoretically investigate the propagation of sinh-Gaussian beams (ShGBs). Mathematical expressions are derived to describe the beam propagation, the intensity distribution, the beam width, and the beam curvature radius of ShGBs. It is found that the propagation behavior of ShGBs is variable and closely related to the parameter of sinh function (PShF). If the PShF is small, the transverse pattern of ShGBs keeps invariant during propagation for a proper input power, which can be regarded as solitons. If the PShF is large, it varies periodically, which is similar to the evolution of temporal higher-order solitons in nonlinear optical fiber. Numerical simulations are carried out to illustrate the typical propagation characteristics.     

用变分方法求解大变形对称弹性力学问题

文中以经典力学的数学理论和陈氏定理为基础,用变分的方法求解大变形对称弹性力学问题,得出了以瞬时位形为基准的位能广义变分原理和余能广义变分原理,以及两个变分原理的等价性;此外,还给出了以瞬时位形为基准的动力学问题的广义变分原理.     

功能梯度材料Timoshenko梁的热过屈曲分析

研究了功能梯度材料Timoshenko梁在横向非均匀升温下的热过屈曲．在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了功能梯度Timoshenko梁在热-机械载荷作用下的几何非线性控制方程,将问题归结为含有7个基本未知函数的非线性常微分方程边值问题A·D2其中,假设功能梯度梁的材料性质为沿厚度方向按照幂函数连续变化的形式．然后采用打靶法数值求解所得强非线性边值问题,获得了横向非均匀升温场内两端固定Timoshenko梁的静态非线性热屈曲和热过屈曲数值解．绘出了梁的变形随温度载荷及材料梯度参数变化的特性曲线,分析和讨论了温度载荷及材料的梯度性质参数对梁变形的影响．结果表明,由于材料在横向的非均匀性,均匀升温时的梁中存在拉-弯耦合变形．     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

一类指定应力问题的变分原理与应用

针对有限元分析中对应力或内力有指定条件的问题,引入非弹性应变作为实现指定应力条件的附加未知量,在小变形条件下描述了指定应力条件应当满足的弹性力学控制方程;以位移和未知非弹性应变作为独立变量,建立了具有指定应力条件问题的势能变分原理和虚功方程;以位移、弹性应变、未知非弹性应变和应力为独立变量,建立了一个含四类变量的广义变...     

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

A novel perspective to the local fractional Zakharov–Kuznetsov-modified equal width dynamical model on Cantor sets

In this work, the local fractional Zakharov–Kuznetsov-modified equal width dynamical (LFZKMEWD) model is investigated on Cantor sets by using the local fractional derivative (LFD). The fractal variational wave method (FVWM) is employed to obtain the exact traveling wave solutions of the nondifferentiable type for the LFZKMEW model. The numerical example illustrates the FVWM is efficient and straightforward. The properties of exact traveling wave solutions are also elaborated by some figures.     

水波的界带有限元

研究了水波计算的位移法.采用物质坐标,以位移为基本未知量,考虑小变形条件,引入流函数满足不可压缩条件.于是,分析力学的变分原理可以运用了,界带有限元、正则变换、保辛积分等有效手段可使数值求解方便得多.     

Effect of two photon absorption on nonlinear pulse propagation in gain medium

The effects of two photon absorption (TPA) and gain dispersion on soliton propagation in amplified medium are investigated. For finite gain bandwidth, the effect of gain dispersion becomes significant along with TPA and is treated as perturbation in fundamental soliton propagation. Including these perturbing effects an analytical expression of integrated intensity is formulated applying a completely new methodology by adopting Rayleigh’s dissipation function in the framework of variational approach. With classical analogy, the Euler–Lagrange equation in non-conservative system is used to solve the problem analytically. In order to justify the analytical prediction a numerical verification is made by split-step beam propagation method following Ginzburg–Landau equation.     

Representation of the Variational Sequence by Differential Forms

In the paper the representation of the finite order variational sequence on fibered manifolds in field theory is studied. The representation problem is completely solved by a generalization of the integration by parts procedure using the concept of the Lie derivative of forms with respect to vector fields along canonical maps of prolongations of fibered manifolds. A close relationship between the obtained coordinate invariant representation of the variational sequence and some familiar objects of physics, such as Lagrangians, dynamical forms, Euler–Lagrange mapping and Helmholtz–Sonin mapping is pointed out and illustrated by examples.Mathematics Subject Classifications (2000) 58E99, 49F99.Jana Musilová: Research of both authors supported by grants MSM 0021622409 and 201/03/0512.     

具有分数导数本构关系的粘弹性Timoshenko梁的静动力学行为分析

利用粘弹性材料的三维分数导数型本构关系,建立粘弹性Timoshenko梁的静、动力学行为研究的数学模型;分析Timoshenko梁在阶跃载荷作用下的准静态力学行为,得出了问题的解析解,考察了一些材料参数对梁的挠度的影响。基于模态函数讨论了粘弹性Timoshenko梁在横向简谐激励作用下的动力响应,并考察了剪切和转动惯性对梁振动响应的影响。     

Dynamical principle

We suggest a formulation of the dynamical principle for mechanics in which time is not a preferred evolution parameter but plays the role of a new generalized coordinate. The advantage of this approach is the possibility of extending it to dynamical systems in which there is no natural evolution parameter (thermodynamics, equilibrium economics, and the like). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 18–28, October, 2007.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

The Stochastic Quasi-chemical Model for Bacterial Growth: Variational Bayesian Parameter Update

We develop Bayesian methodologies for constructing and estimating a stochastic quasi-chemical model (QCM) for bacterial growth. The deterministic QCM, described as a nonlinear system of ODEs, is treated as a dynamical system with random parameters, and a variational approach is used to approximate their probability distributions and explore the propagation of uncertainty through the model. The approach consists of approximating the parameters’ posterior distribution by a probability measure chosen from a parametric family, through minimization of their Kullback–Leibler divergence.     

Fractionalization of the complex-valued Brownian motion of order n using Riemann–Liouville derivative. Applications to mathematical finance and stochastic mechanics

The (complex-valued) Brownian motion of order n is defined as the limit of a random walk on the complex roots of the unity. Real-valued fractional noises are obtained as fractional derivatives of the Gaussian white noise (or order two). Here one combines these two approaches and one considers the new class of fractional noises obtained as fractional derivative of the complex-valued Brownian motion of order n. The key of the approach is the relation between differential and fractional differential provided by the fractional Taylor’s series of analytic function , where E is the Mittag–Leffler function on the one hand, and the generalized Maruyama’s notation, on the other hand. Some questions are revisited such as the definition of fractional Brownian motion as integral w.r.t. (dt), and the exponential growth equation driven by fractional Brownian motion, to which a new solution is proposed. As a first illustrative example of application, in mathematical finance, one proposes a new approach to the optimal management of a stochastic portfolio of fractional order via the Lagrange variational technique applied to the state moment dynamical equations. In the second example, one deals with non-random Lagrangian mechanics of fractional order. The last example proposes a new approach to fractional stochastic mechanics, and the solution so obtained gives rise to the question as to whether physical systems would not have their own internal random times.