Identical motion in relativistic quantum and classical mechanics

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.     

Application of the complex point source method to the Schrödinger equation

The paraxial wave equation is a reduced form of the Helmholtz equation. Its solutions can be directly obtained from the solutions of the Helmholtz equation by using the method of complex point source. We applied the same logic to quantum mechanics, because the Schrödinger equation is parabolic in nature as the paraxial wave equation. We defined a differential equation, which is analogous to the Helmholtz equation for quantum mechanics and derived the solutions of the Schrödinger equation by taking into account the solutions of this equation with the method of complex point source. The method is applied to the problem of diffraction of matter waves by a shutter.     

Wave propagation in a random medium: The solution of the m-nth moment equation with different wavenumbers

The moment equation with different wavenumbers and different transverse coordinates for wave propagation in a random medium is a linear differential equation. It often appears in the study of problems related to wave propagation in a random medium. The differential equation can be converted into an integral equation by using Green's functions and the integral equation can be solved by iteration. The moment equation is solved by the method of successive scatters, too. The solution of the moment equation is a Dyson expansion. The physical implication of the successive solution of the moment equation with different wavenumbers is explained.     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发，推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示，三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到，表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径. 关键词： Lagrange方程 加速度能量 广义坐标     

输油管道油流静电带电数值计算

对输油管道内油品流动带电问题的数值计算进行了研究.紊流条件下的电荷输运方程是一个对流占优的对流扩散反应方程,采用算子分裂法,将该方程分解为纯对流方程、纯扩散方程和纯反应方程,分别采用特征线法和差分法求解.算例证明,该方法能准确描述管道内电荷分布,因而提供了一种获取冲流电流的可靠方法.     

The electromagnetic coupling in Kemmer-Duffin-Petiau theory

We analyze the electromagnetic coupling in the Kemmer-Duffin-Petiau (KDP) equation. Since the KDP equation which describes spin-0 and spin-1 bosons is of Dirac type, we examine some analogies with and differences from the Dirac equation. The main difference with the Dirac equation is that the KDP equation contains redundant components. We will show that as a result certain interaction terms in the Hamilton form of the KDP equation do not have a physical meaning and will not affect the calculation of physical observables. We point out that a second order KDP equation derived by Kemmer as an analogy to the second order Dirac equation is of limited physical applicability as (i) it belongs to a class of second order equations which can be derived from the original KDP equation and (ii) it lacks a back-transformation which would allow one to obtain solutions of the KDP equation out of solutions of the second order equation.     

天线理论中的第二类积分方程

本文提出一个决定柱状天线电流分布的新的积分方程,这是一个一维的Fredholm第二类积分方程。它不同于天线理论中惯用的Hallen积分方程。本文着重分析了在天线为无穷长时,由两个方程所解得的电流和磁场,说明第二类积分方程比Hallen方程更适于描述天线的实际情况。还初步进行了有限长度天线的数值计算,结果表明用第二类积分方程进行天线的数值计算是可行的。     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

利用连续介质力学方法研究超光速现象

从可压缩连续介质角度出发,给出一种可以在空间二级准确度上兼容的相对论然而又允许超光速介质运动存在的数学描述.先给出无粘不可压缩流动的Euler方程可以改写成和电磁场方程相同的表达形式：在空间二阶准确度的意义上来说,不可压缩流+相对论近似等于可压缩流,它们都是和现在的实验结果相融的;给出可压缩流动的广义相对论.这也说明协变不变原理不过是可压缩流动的一种近似处理方法.新模型可以解释索么菲（A.Sommerfeld）提出的粒子在超过光速后减小能量反而加速的现象.     

基于孔洞体胞模型铸造镁合金ML308的本构方程

铸造镁合金不可避免地包含许多微孔洞,这些微孔洞在材料的后续加工及服役过程中将发生演化,并对材料的力学行为产生重要影响.基于球形孔洞体胞模型,提出微孔洞长大及形核方程,它们构成微孔洞的演化方程.根据孔洞演化将造成材料性质弱化的物理机制,将微孔洞演化以弱化函数的形式引入到非经典弹塑性本构方程,得到考虑孔洞演化的铸造镁合金弹塑性本构方程.发展与本构方程相应的有限元数值分析程序,用其模拟了铸造镁合金ML308的微孔洞演化及力学行为,计算结果与实验结果符合较好. 关键词： 铸造镁合金 孔洞体胞模型 孔洞演化方程 本构方程     

On the modified discrete KP equation with self-consistent sources

The modified discrete KP equation is the Bäcklund transformation for the Hirota’s discrete KP equation or the Hirota-Miwa equation. We construct the modified discrete KP equation with self-consistent sources via source generation procedure and clarify the algebraic structure of the resulting coupled modified discrete KP system by presenting its discrete Gram-type determinant solutions. It is also shown that the commutativity between the source generation procedure and Bäcklund transformation is valid for the discrete KP equation. Finally, we demonstrate that the modified discrete KP equation with self-consistent sources yields the modified differential-difference KP equation with self-consistent sources through a continuum limit. The continuum limit of an explicit solution to the modified discrete KP equation with self-consistent sources also gives the explicit solution for the modified differential-difference KP equation with self-consistent sources.     

THE IMPROVED MURNAGHAN EQUATION

The improved Murnaghan equation is derived by integrating B(P)=-v(∂p/∂v)_T=B₀⁽¹⁾(P)·P and expanding B⁽¹⁾(P) and simllar function in series of (1-K) , where K is the compression ratio (K≡(v/v₀)^1/3. The new equation obtained is compared with Murnagnan equation, Keane equation and Birch equation.It is found that, the improved Murnaghan equation has better convergency in comparison with the Birch equation. The characteristics of the new equation and the reasons of its better convergency are discussed.     

概率波和非概率波

对于把克莱因-戈尔登方程当作是玻色子的方程的看法提出异议,认为它是所有微观粒子均要满足的方程,但它却不能成为任何一类粒子的波动方程.提出了克-戈方程中包含着概率和非概率两类波的概念,认为概率波还要遵从一个对时和空都是一阶导数的方程,这才是粒子的波动方程.不同种类粒子性质的不同则体现在他们概率波类型的不同上.     

On the dynamics of a quantum-statistical condensate

The dynamical equation for a condensate at finite temperatures is derived. The derivation of the equation is based on the model due to Wu. The equation derived is the nonlinear Schrödinger equation of diffusion type.     

The superposition method in seeking the solitary wave solutions to the KdV-Burgers equation

In this paper, starting from the careful analysis on the characteristics of the Burgers equation and the KdV equation as well as the KdV-Burgers equation, the superposition method is put forward for constructing the solitary wave solutions of the KdV-Burgers equation from those of the Burgers equation and the KdV equation. The solitary wave solutions for the KdV-Burgers equation are presented successfully by means of this method.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

Fractional Fokker-Planck equation for fractal media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.     

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.