Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

Algebraic dynamics algorithm:Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.     

Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation

Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations are applied to the nonlinear advection equation. The results show that the approach is effective for the exact analytical solution and the algorithm has higher precision than other existing algorithms in numerical computation for the nonlinear advection equation. Supported by the National Natural Science Foundation of China (Grant Nos. 90503008 and 10775100), the Doctoral Program Foundation from the Ministry of Education of China, and the Center of Theoretical Nuclear Physics of HIRFL of China     

Symplectic algebraic dynamics algorithm

Based on the algebraic dynamics solution of ordinary differential equations andintegration of  ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.     

两粒子自旋系统量子纠缠的时间演化

研究了两个具有海森伯耦合的自旋为1/2的粒子在随时间变化的磁场中的运动情况.系统的哈密顿量具有SU（2）代数结构,运用代数动力学方法对此系统进行求解,得到了时间演化算子的严格解.基于严格解,求得两粒子体系随时间变化的波函数,从而计算得到两粒子体系的纠缠.对不同初始波函数,研究了系统纠缠随时间的变化情况.讨论了外场影响纠缠的条件. 关键词： 二粒子系统纠缠 代数动力学解法     

含时旋转磁场中的两耦合粒子的纠缠演化(英文)

我们研究了含时旋转磁场中海森堡XXX模型下的双量子比特的动力学演化情况.基于此非自治系统的代数结构,我们用代数动力学方法求得了系统的精确解析解.在此基础上,进一步研究了在不同初态下系统的纠缠测度随时间的演化,发现纠缠测度由系统的初态的系数和耦合强度决定.     

Entanglement Evolution of Two-Coupled Spins in a Time-Dependent Rotating Magnetic Field

We study the dynamics evolution of a two-qubit Heisenberg XXX spin chain under a time-dependent rotating magnetic field. Based on the algebraic structure of the non-autonomous system, the exact solution of the Schrödinger equation is obtained by using the method of algebraic dynamics. Based on the time-dependent analytical solution, we further study the entanglement evolution between the two coupled spins for different initial states, and find that the entanglement is determined by the coefficients of the initial state and the coupling constant $J$ of the system.     

基于代数动力学方法实现一类相干态的构建

最近提出的一个构建相干态的方案中,需要精确求解一个时间相关的常微分方程.基于代数动力学理论,利用该方程具有的SU(1,1)动力学对称性,提出了对此方程在含时系数取任意函数形式时的统一的精确求解方法,并且得到了严格的解析解.运用这个精确解,就可以构造相应物理系统的精确相干态的具体表达式.给出了一个解例,即频率取"快变"函数的情形.利用得到的精确结果,讨论了这个系统的量子涨落(量子噪声)随时间演化的情况.针对动量算符不确定度随时间演化的曲线的性态,指出在制备这个系统压缩态时可以利用的一些性质.最后,讨论了这个量子系统的不确定关系随时间演化的情况.     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

精确Cosserat弹性杆动力学的分析力学方法

以脱氧核糖核酸和工程中的细长结构为背景, 大变形大范围运动的弹性杆动力学受到关注. 将分析力学方法运用到精确Cosserat弹性杆动力学, 旨在为前者拓展新的应用领域, 为后者提供新的研究方法. 基于平面截面假定, 在弯扭基础上再计及拉压和剪切变形形成精确Cosserat弹性杆模型. 用刚体运动的概念描述弹性杆的变形, 导出弹性杆变形和运动的几何关系; 在定义截面虚位移及其变分法则的基础上, 建立用矢量表达的d’Alembert-Lagrange原理, 在线性本构关系下化作分析力学形式, 并导出Lagrange方程和Nielsen方程, 定义正则变量后化作Hamilton正则方程; 对于只在端部受力的弹性杆静力学, 导出了将守恒量预先嵌入的Lagrange方程, 并讨论了其首次积分. 从弹性杆的d’Alembert-Lagrange原理导出积分变分原理, 在线性本构关系下化作Hamilton原理. 形成的分析力学方法使弹性杆的全部动力学方程具有统一的形式, 为弹性杆动力学的对称性和守恒量的研究及其数值计算铺平道路. 关键词： 精确Cosserat弹性杆 分析动力学方法 变分原理 Lagrange方程     

Lie algebraic approach for Fokker-Planck dynamics with space-dependent diffusion and mean-reverting drift

Using the Lie algebraic approach we have derived the exact diffusion propagator of the Fokker-Planck equation with a time-dependent variable diffusion coefficient and a time-dependent mean-reverting force between two absorbing boundaries. The exact diffusion propagator not only enables us to study the time evolution of the corresponding stochastic system, but the knowledge of the propagator can also provide a benchmark for testing approximate numerical or analytical procedures. Furthermore, the Lie algebraic method is very simple and could be easily extended to the more general Fokker-Planck equations with well-defined algebraic structures. Received 18 December 2002 / Received in final form 3 March 2003 Published online 24 April 2003     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f(ξ), is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

量子微腔中运动原子的辐射压力

利用全量子理论研究了量子微腔中运动原子的辐射压力. 从量子微腔与运动原子相互作用模型出发, 利用代数动力学方法对系统的哈密顿量进行规范变换, 推导出系统的时间演化算符和原子内态约化密度算符的表达式, 在此基础上给出辐射压力的解析解, 并讨论了驻波场和行波场中运动二能级原子和三能级原子的辐射压力, 数值结果与实验符合. 关键词： 量子微腔 运动原子 代数动力学 辐射压力     

Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations

We investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations. The proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems. We examine dynamical scaling characteristics in the short-time and the long-time evolution regime providing only a reduced number of degrees of freedom to the evolution process.     

由各向异性海森伯自旋环链组成的量子位及其通用量子逻辑门

研究了各向异性耦合的三粒子海森伯自旋环链团簇在随时间变化的磁场中的运动.该系统的哈密顿量具有SU(2)代数结构.用代数动力学方法对此系统进行求解，得到了严格的解析解.基于严格解, 可以构造一位量子逻辑门.通过调节磁场强度和频率, 就可以控制该量子逻辑门, 实现一位量子逻辑门的任何操作. 关键词： 代数动力学 自旋环链团簇 一位量子逻辑门