Multisymplectic Geometry for the Seismic Wave Equation

The multisymplectic geometry for the seismic wave equation is presented in this paper. The local energy conservation law, the local momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Based on the covariant Legendre transform, the multisymplectic Hamiltonian formulation is developed. Multisymplectic discretization and numerical experiments are also explored.     

Multisymplectic Geometry,Local Conservation Laws and a Multisymplectic Integrator for the Zakharov–Kuznetsov Equation

The multisymplectic geometry for the Zakharov–Kuznetsov equation is presented in this Letter. The multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle. Based on the multisymplectic Hamiltonian formulation, we derive a 36-point multisymplectic integrator.     

Multisymplectic Geometry and Its Appiications for the Schrodinger Equation in Quantum Mechanics

Multisymplectic geometry for the Schrodinger equation in quantum mechanics is presented. This formalism of multisymplectic geometry provides a concise and complete introduction to the Schrodinger equation. The Schrodinger equation, its associated energy and momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Some applications are also explored.     

Multisymplectic Structures and Discretizations for One-way Wave Equations

Multisymplectic structures for one-way wave equations are presented in this letter. Based on the multisymplectic formulation, we obtain the corresponding multisymplectic discretizations. The structure-preserving property of a finite difference scheme for the first-order one-way wave equation is proved. Implications and applications of this result are explored.      

Variational Formulation for the Multisymplectic Hamiltonian Systems

A variational formulation for the multisymplectic Hamiltonian systems is presented in this Letter. Using this variational formulation, we obtain multisymplectic integrators from a variational perspective. Numerical experiments are also reported.Mathematical Subject Classifications (2000). 70G50, 58Z05.     

Variational methods, multisymplectic geometry and continuum mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.     

SRLW方程的多辛Fourier谱格式及其守恒律

通过引进正则动量，将对称正则长波方程（简称SRLW方程）转化成多辛形式的方程组，它具有多辛守恒律；介绍了空间方向满足周期边界条件的函数的Fourier谱方法；对SRLW方程的多辛方程组在空间方向利用Fourer谱方法，时间方向上应用Euler中点格式离散，得到其多辛Fourier拟谱格式；证明此格式的一些离散守恒律．用此格式模拟了SRLW方程的单个孤立波，还模拟了多个孤立波的追赶、碰撞和分离过程．     

Total Variation and Multisymplectic Structure for CNLS System

The relation between the toal variation of classical field theory and the multisymplectic structure is shown. Then the multisymplectic structure and the corresponding multisymplectic conservation of the coupled nonlinear Schroedinger system are obtained directly from the variational principle.     

Total Variation and Multisymplectic Structure for CNLS System

The relation between the total variation of classical field theory and the multisymplectic structure is shown. Then the multisymplectic structure and the corresponding multisymplectic conservation of the coupled nonlinear Schrodinger system are obtained directly from the variational principle.     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.     

一维Gross-Pitaevskii方程的高阶紧致分裂步多辛格式

首先把一维Gross-Pitaevskli方程改写成多辛Hamiltonian系统的形式,把形式通过分裂变成2个子哈密尔顿系统.然后,对这些子系统用辛或者多辛算法进行离散.通过对子系统数值算法的不同组合方式,得到不同精度的具有多辛算法特征数值格式.这些格式不仅具有多辛格式、分裂步方法和高阶紧致格式的特征,而且是质量守恒的.数值实验验证了新格式的数值行为.     

计算双电子原子基态能量的坐标张弛变分法

给出了一种计算双电子原子基态能量和波函数的坐标张弛的变分方法.同时,利用Matlab语言开发了一个软件程序,对He原子和类He离子的基态能量进行了变分计算.     

Difference Discrete Variational Principles, Euler-Lagrange Cohomology and Symplectic, Multisyrnplectic Structures Ⅲ: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and H, we have studied the difference discrete variational principle and the EulerLagrange cohomology in the framework of multi-parameter differential approach. W5 have gotten the difference discreteEulcr-Lagrangc equations and canonical ones for the difference discrete versions of classical mechanics and tield theoryas well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessaryand sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangianand Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler-Lagrangecohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonianschemes or Lagrangian ones in both the symplectic and multisymplectic algorithms arc variational integrators and theirdifference discrete symplectic structure-preserving properties can always be established not only in the solution spacebut also in the function space if and only if the related closed Euler Lagrange cohomological conditions are satisfied.     

On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively.We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire grometric object and the noncommutative differential calculus on regular lattice.In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations,the Euler-Lagrange cohomological concepts and content in the configuration space are employed.     

Total Variation in Discrete Multisymplectic Field Theory and Multisymplectic-Energy-Momentum Integrators

A total variation calculus in discrete multisymplectic field theory is developed in this Letter. Using this discrete total variation calculus, we obtain multisymplectic-energy-momentum integrators. The multisymplectic discretization for the nonlinear Schrödinger equation is also presented.     

轴对称谐振势阱中玻色凝聚气体基态和单涡旋态解

对囚禁在轴对称谐振势阱中的玻色凝聚气体，提出一种新的试探波函数，运用Gross-Pitaevskii(G-P)平均场能量泛函和变分的方法，得到玻色凝聚气体基态和单涡旋态波函数的解析表达式，并计算出凝聚原子的平均能量、原子云轴向和径向尺度比，以及产生单涡旋态的临界角速度等重要物理量与凝聚原子数N之间的关系.其结果与Dalfovo等人直接数值求解G-P方程所得到的结果相一致. 关键词： 玻色凝聚气体 G-P泛函 波函数 谐振势阱     

Symplectic And Multisymplectic Fourier Pseudospectral Discretizations for the Klein–Gordon Equation

We present symplectic and multisymplectic formulations of the Klein-Gordon equation in this paper. Based on these two formulations, we investigate the corresponding symplectic and multisymplectic Fourier pseudospectral discretizations. The relationship between these two kinds of Fourier pseudospectral discretizations is discussed. Time discretizations are also presented.     

Multisymplectic Pseudospectral Discretizations for (3+1)-Dimensional Klein-Gordon Equation

We explore the multisymplectic Fourier pseudospectral discretizations for the (3+1)-dimensional Klein-Gordon equation in this paper. The corresponding multisymplectic conservation laws are derived. Two kinds of explicit symplectic integrators in time are also presented.     

Invariance analysis and conservation laws of the wave equation on Vaidya manifolds

In this paper we discuss symmetries of classes of wave equations that arise as a consequence of some Vaidya metrics. We show how the wave equation is altered by the underlying geometry. In particular, a range of consequences on the form of the wave equation, the symmetries and number of conservation laws, inter alia, are altered by the manifold on which the model wave rests. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations on a flat geometry. Finally, we pursue the existence of higher-order variational symmetries of equations on nonflat manifolds.     

Mobility of two-dimensional electron gas in JFETS limited by polar-optic and impurity scattering

The theory of mobility of a two-dimentional electron gas in JFET structures limited by polar-optic phonon and impurity scattering is developed in this work. The energy level and the wave function of the lowest subband are obtained by a variational procedure. The mobility limited by polar-optic phonon scattering is obtained by solving the Boltzmann equation iteratively. The expression for the impurity scattering limited mobility is obtained by using the variational wave function. For numerical calculation, however, the electron gas is assumed to be strictly two-dimensional. It is found that for experimental range of impurity concentration in GaAs JFETs, impurity scattering is the dominant process even at 300K.