含时Schroedinger方程的高阶辛FDTD算法研究

提出了一种新的算法一高阶辛时域有限差分法（SFDTD（3,4）：symplectic finite—difference time-domain）求解含时薛定谔方程．在时间上采用三阶辛积分格式离散,空间上采用四阶精度的同位差分格式离散,建立了求解含时薛定谔方程的高阶离散辛框架;探讨了高阶辛算法的稳定性及数值色散性．通过理论上的分析及数值算例表明：当空间采用高阶同位差分格式时,辛积分可提高算法的稳定度;SFDTD（3,4）法和FDTD（2,4）法较传统的FDTD（2,2）法数值色散性明显改善．对二维量子阱和谐振子的仿真结果表明：SFDTD（3,4）法较传统的FDTD（2,2）法及高阶FDTD（2,4）法有着更好的计算精度和收敛性,且SFDTD（3,4）法能够保持量子系统的能量守恒,适用于长时间仿真．     

含时Schrdinger方程的高阶辛FDTD算法研究

提出了一种新的算法—高阶辛时域有限差分法(SFDTD(3,4):symplectic finite-difference time-domain)求解含时薛定谔方程.在时间上采用三阶辛积分格式离散,空间上采用四阶精度的同位差分格式离散,建立了求解含时薛定谔方程的高阶离散辛框架;探讨了高阶辛算法的稳定性及数值色散性.通过理论上的分析及数值算例表明:当空间采用高阶同位差分格式时,辛积分可提高算法的稳定度;SFDTD(3,4)法和FDTD(2,4)法较传统的FDTD(2,2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明:SFDTD(3,4)法较传统的FDTD(2,2)法及高阶FDTD(2,4)法有着更好的计算精度和收敛性,且SFDTD(3,4)法能够保持量子系统的能量守恒,适用于长时间仿真.     

含时Schrödinger方程的高阶辛FDTD算法研究

提出了一种新的算法——高阶辛时域有限差分法(SFDTD(3, 4): symplectic finite-difference time-domain)求解含时薛定谔方程.在时间上采用三阶辛积分格式离散, 空间上采用四阶精度的同位差分格式离散, 建立了求解含时薛定谔方程的高阶离散辛框架;探讨了高阶辛算法的稳定性及数值色散性.通过理论上的分析及数值算例表明:当空间采用高阶同位差分格式时, 辛积分可提高算法的稳定度;SFDTD(3, 4)法和FDTD(2, 4)法较传统的FDTD(2, 2)法数值色散性明显改善.对二维量子阱和谐振子的仿真结果表明: SFDTD(3, 4)法较传统的FDTD(2, 2)法及高阶FDTD(2, 4)法有着更好的计算精度和收敛性, 且SFDTD(3, 4)法能够保持量子系统的能量守恒, 适用于长时间仿真.     

高阶辛算法的稳定性与数值色散性分析

利用Maxwell方程的哈密尔顿函数,导出对应的欧拉-哈密尔顿方程.利用辛积分技术与高阶交错差分技术,建立求解三维时域Maxwell方程的高阶辛算法;结合电磁场中的物理概念,借助矩阵分析和张量分析理论,获得高阶时域方法及高阶辛算法的稳定性和数值色散性的统一处理新方法.用数值结果证实方法的正确性,与FDTD算法和其它时域高阶方法相比,高阶辛算法具有较大的计算优势,为电磁计算提供了新的途径.     

粒子自旋问题的辛算法研究

将辛算法应用于求解量子力学中自旋问题的含时薛定谔方程,自编程序在微机上进行了计算。结果表明,辛算法是用于求解含时薛定谔方程等一类偏微分方程的一种好的数值计算法。     

两量子位Grover量子算法NMR脉冲序列参量的研究

从两量子位核磁共振量子计算机物理模型出发,通过解单体含时薛定谔方程和解两体含时薛定谔方程,提出了Grover量子算法核磁共振脉冲序列参量设定的两种规则,给出了具体参量取值,并进行了数值仿真,仿真结果表明:解两体薛定谔方程给出的参量设定规则,能使两量子位量子搜索的目标态是纯基态,目标态的z分量期望值精确度达到在小数点后三位与理论值完全相同,验证了我们提出的参量设定规则的正确性.     

辛算法在限制性三体问题数值研究中的应用

限制性三体问题是太阳系动力学中常采用的一种力学模型,是一哈密顿(Hamilton)系统.由于数学工具的不够,一些重要问题只能进行数值研究,但要了解系统的演化状况,必须进行长期跟踪计算.因此,对算法要求极高,应能保持运动的整体特征,而Hamilton系统的辛算法正符合这一要求,文章将利用算法合成构造旋转坐标系中圆型和椭圆型限制性三体问题(对应不可分Hamilton系统)的显式辛差分格式,并以计算实例表明方法的有效性.     

两量子位Grover量子算法NMR脉冲序列参量的研究

从两量子位核磁共振量子计算机物理模型出发,通过解单体含时薛定谔方程和解两体含时薛定谔方程,提出了Grover量子算法核磁共振脉冲序列参量设定的两种规则,给出了具体参量取值,并进行了数值仿真,仿真结果表明：解两体薛定谔方程给出的参量设定规则,能使两量子位量子搜索的目标态是纯基态,目标态的z分量期望值精确度达到在小数点后三位与理论值完全相同,验证了我们提出的参量设定规则的正确性.     

辛和非辛算法求解薛定谔方程误差分析

在哈密尔顿体系下,提出气体声波传播的一种新的谐振子模型,并引入群论确定气体声波传播过程中的分子振动模式、能级简并.新模型将气动声学声传播问题与分子振动关联起来。由于发展高效的薛定谔方程的数值计算方法,有利于联系分子的性质来解释声的传播.本文从此出发,用二阶有限差分格式和生成函数法构造的二阶辛格式分别计算一维定态谐振子势场和含时谐振子势场的薛定谔方程,分析了数值解的误差以及传播能量误差.结果表明辛算法具有明显的优势.     

求解波动方程的准粒子离散修正辛算法

针对波动方程求解,在Hamilton体系下建立了对空间离散的准粒子体系,该准粒子体系实现简单,物理意义明确;在时间离散方面,构造了一种适合高效声波模拟的修正辛格式,该格式是在常规的二阶Partitioned Runge-Kutta(PRK)基础之上构造而成,其具有三阶时间精度,从理论上分析了修正辛格式的数值稳定性和频散性能.数值结果表明,本文提出的方法在计算时间,计算精度和计算存储量等各方面性能都有相应改善。     

超短脉冲在光学介质中传输的B积分特性

从非线性薛定谔方程出发,利用分步傅里叶算法对其进行求解,数值模拟了在超短脉冲激光系统中几种光学介质中的B积分传输规律,并对其特性进行了简要分析。数值计算结果表明,初始脉冲的输入光强与光学介质的增益系数的增大会使B积分值增加。脉冲形状对B积分值有一定影响,与飞秒高斯脉冲比较,当输入为ps级的啁啾脉冲时,B积分较小。针对所选计算模型,适当的群速色散与高阶色散可以减小B积分值。这对超短脉冲放大系统的设计有参考意义。     

Periodic Structure of Equatorial Envelope Rossby Wave Under Influence of Diabatic Heating

A simple shallow-water model with influence of diabatic heating on aβ-plane is applied to investigate the nonlinear equatorial Rossby waves in a shear flow. By the asymptotic method of multiple scales, the cubic nonlinear Schrodinger (NLS for short) equation with an external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial envelope Rossby waves are obtained with the help of Jacobi elliptic functions and elliptic equation. It is shown that phase-locked diabatic heating plays an important role in periodic structures of rational form.     

高阶效应对光纤系统中孤子和艾里脉冲传输特性影响的数值研究

以变系数耦合高阶非线性薛定谔方程为理论模型,采用分步傅里叶方法,讨论了孤子和艾里脉冲在光纤中的传输特性。研究表明三阶色散、自频移和自陡峭效应导致孤子和艾里脉冲所分离出的俘获孤子的中心位置发生偏移,且三阶色散会影响俘获孤子的强度,并讨论了两脉冲的中心位置及强度对艾里脉冲截断系数的依赖关系。     

Coordinate Transformation and Exact Solutions of Schr(o)dinger Equation with Position-Dependent Effective Mass

Using the coordinate transformation method, we solve the one-dimensional Schr(o)dinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigcnfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term Vm, which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of Vm for several mass functions are discussed.     

横向非周期调制的五次非线性薛定谔方程的精确孤子解

本文首先应用Adomian分解法给出了横向非周期调制的五次非线性薛定谔方程的精确孤子解,并将其同数值结果进行了比较,它们吻合得很好.进而针对不同介质的传播常数k研究了精确解的线性稳定性和非线性稳定性,k＜1.724时孤子解不稳定,1.724≤k＜2.264时其具有非线性稳定性,但是不具有线性稳定性;k≥2.264时孤子解既是非线性稳定的,也是线性稳定的.     

A Study of Infrared Photonic Crystal Devices Using New ADI-FDTD Algorithm

To investigate the infrared photonic crystal devices numerically, the traditional finite-difference time-domain (FDTD) method has been modified by combining with a new alternating direction implicit (ADI) algorithm. An improvement of two-five in speed over previous FDTD methods can be obtained by calculating the envelope rather than the fast-varying field, and the numerical errors are minimized. Consider the isolated localized coupled-cavity modes, the phenomenon of eigenmode splitting has been observed when the coupled-cavity structures in two dimension triangular dielectric photonic crystals are simulated. The results are in good agreement with experiments.     

非线性色散渐减光纤中高峰值脉冲的激发和传输

本文基于变系数的非线性薛定谔方程,数值地讨论高峰值脉冲在色散渐减光纤中的激发和传输。首先,基于变系数非线性薛定谔方程的Peregrine孤子解,解析和数值地讨论精确的Peregrine孤子在色散渐减光纤中的传输特性。其次,通过输入不同的平面波背景上的局域脉冲,研究高峰值脉冲在非线性色散渐减光纤中的激发和传输。结果显示Peregrine孤子在色散渐减光纤中传输时,会产生一个空间和时间都局域化的高峰值单脉冲,并且当啁啾为负时,脉冲的幅值增加,脉宽被压缩。若光纤系统存在增益,脉冲的幅值也会增加。由于非线性光纤中的调制不稳定性过程,不同平面波背景上的小局部扰动都可激发出高峰值脉冲,除了峰值和宽度略有不同外,激发脉冲的形状几乎相同。     

三维各向同性谐振子波函数的递推关系(英文)

本文应用拉普拉斯变换得到了三维各向同性谐振子波函数边界的精确解,同时,利用同种方法还得到了利用产生算符和湮灭算符表达的该波函数的递推关系.     

Applications of Extended Hyperbolic Function Method for Quintic Discrete Nonlinear Schr(o)dinger Equation

By using the extended hyperbolic function method,we have studied a quintic discrete nonlinear Schr(o)dinger equation and obtained new exact localized solutions,including the discrete bright soliton solution,dark soliton solution,bright and dark soliton solution,alternating phase bright soliton solution,alternating phase dark soliton solution,and alternating phase bright and dark soliton solution,if a special relation is bound on the coefficients of the equation.     

Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation.