Stability boundaries of a Mathieu equation having PT symmetry

I have applied multiple-scale perturbation theory to a generalized complex PT-symmetric Mathieu equation in order to find the stability boundaries between bounded and unbounded solutions. The analysis suggests that the non-Hermitian parameter present in the equation can be used to control the shape and curvature of these boundaries. Although this was suggested earlier by several authors, analytic formulas for the boundary curves were not given. This paper is a first attempt to fill this gap in the theory.     

光格中玻色-爱因斯坦凝聚的自旋磁子和自旋波

研究了一维光格中旋量玻色—爱因斯坦凝聚体的自旋磁子和自旋波。求出了自旋磁子和自发磁化强度。用经典方法求得了均匀体系的自旋波和能级以及激光调制下的自旋波和能级。特别是在连续极限近似下,求得了马丢函数波及其能级。     

光格中玻色-爱因斯坦凝聚的自旋磁子和自旋波

研究了一维光格中旋量玻色—爱因斯坦凝聚体的自旋磁子和自旋波。求出了自旋磁子和自发磁化强度。用经典方法求得了均匀体系的自旋波和能级以及激光调制下的自旋波和能级。特别是在连续极限近似下,求得了马丢函数波及其能级。     

Stability regions for Mathieu equation with imperfect periodicity

We consider a mean square stability for Mathieu equation with a random phase modulation in parametric excitation. An efficient numerical scheme is proposed for obtaining the stability charts for this equation. The influence of the random phase modulation on the shape of parametric resonance regions is studied. It is found that this influence can lead to stabilization under some conditions. A comparison with a case of Gaussian parametric excitation is presented.     

Mathieu Equation and Elliptic Curve

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q<<1 and q>>1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We make a few conjectures about the general structure of these differential operators.     

无衍射 Mathieu光束自重建特性的理论和实验研究

首次对无衍射Mathieu光束的自重建特性进行理论和实验研究, 利用Mathieu-Hankel波理论分析了Mathieu光束的自重建机理. 基于菲涅尔衍射积分理论, 推导出了高斯吸收型圆形障碍物部分遮挡后的Mathieu光场重建后的解析表达式, 并数值模拟了无衍射Mathieu光束的经圆形障碍物部分遮挡后光场的自重建过程. 采用柱透镜-轴棱锥组合光学系统产生近似零阶无衍射Mathieu光束, 实验分别验证了轴上和轴外障碍物遮挡时近似零阶无衍射Mathieu光束的自重建特性. 理论模拟与实验结果相符.     

Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution

Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope.     

Scattering of Mathieu beams by a rigid sphere

Based on the recent results on the scattering of Bessel beams by a sphere and using the Whittaker integral, the scattering by a rigid sphere centred on a Mathieu beam is derived. The scattering field is expressed as a partial wave series involving the scattering angles relative to the beam axis and Mathieu beam parameters. Some numerical calculations are performed and it is shown that the illumination of a rigid sphere by a Mathieu beam produces asymmetrical scattering as a function of scattering angles θ and ?. The geometrical properties of the scattering Mathieu beam are noted.     

Quasi-Periodic Waves and Asymptotic Property for Boiti-Leon-Manna-Pempinelli Equation

In this paper, multi-periodic （quasi-periodic） wave solutions are constructed for the Boiti-Leon-Manna- Pempinelli （BLMP） equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Quasi-Periodic Waves and Asymptotic Property for Boiti-Leon-Manna-Pempinelli Equation

In this paper, multi-periodic (quasi-periodic) wave solutions are constructed for the Boiti-Leon-Manna-Pempinelli (BLMP) equation by using Hirotabilinear method and Riemann theta function. At the same time, weanalyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass-spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker-Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.     

Transmission Modes of Closed Elliptic-Groove Waveguides for Millimeter Waves

In this paper, a new type of horizontal-flat and vertical-flat closed elliptic-groove guides is firstly presented. The field components and characteristic equations for some main transmission modes of the closed elliptic-groove guides are derived by with the help of the mode-matching method and Mathieu function. The variations of the cut-off wavelengths along with eccentricities, widths and lengths of the parallel plates for a number of lower order transmission modes of the closed elliptic-groove guides are also analysed in detail. The calculated results in good agreement with ones in the relevant references are of very important values in theoretical researches and actual applications of closed elliptic-groove guides for millimeter and submillimeter waves.     

Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the first-order solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.     

The Complex Bateman Equation

The general solution to the complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.     

Dynamical Localization II with an Application to the Almost Mathieu Operator

Several recent works have established dynamical localization for Schrödinger operators, starting from control on the localization length of their eigenfunctions, in terms of their centers of localization. We provide an alternative way to obtain dynamical localization, without resorting to such a strong condition on the exponential decay of the eigenfunctions. Furthermore, we illustrate our purpose with the almost Mathieu operator, H _{, ,} =–+ cos(2(+x)), 15 and with good Diophantine properties. More precisely, for almost all , for all q>0, and for all functions ²( ) of compact support, we show that The proof applies equally well to discrete and continuous random Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes, supplied in the random case by the multi-scale analysis.     

Bäcklund Transformations,Solitary Waves,Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

Some simple special Bäcklund transformation theorems are proposed and utilized to obtain exact solutions for the (2+1)-dimensional Euler equation. It is found that the (2+1)-dimensional Euler equation possesses abundant soliton or solitary wave structures, conoid periodic wave structures and the quasi-periodic Bessel wave structures on account of the arbitrary functions in its solutions. Moreover, all solutions of the arbitrary two dimensional nonlinear Poisson equation can be used to construct exact solutions of the (2+1)-dimensional Euler equation.     

The Multi-field Complex Bateman Equation

The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered.