Discrete phase space I. Finite difference operators and lattice Schrödinger wave equation

A new representation of quantum mechanics involving finite difference operators is presented. The time-dependent Schrödinger wave equation is furnished as a partialdifference-differential equation. This wave equation is solved exactly for a three-dimensional oscillator.It is emphasized that this representation is exact and not a numerical approximation.     

Study of diffusion of wave packets in a square lattice under external fields along the discrete nonlinear Schrödinger equation

The object of the present work is to analyze the effect of nonlinearity on wave packet propagation in a square lattice subject to a magnetic and an electric field in the Hall configuration, by using the Discrete Nonlinear Schrödinger Equation (DNLSE). In previous works we have shown that without the nonlinear term, the presence of the magnetic field induces the formation of vortices that remain stationary, while a wave packet is introduced in the system. As for the effect of an applied electric field, it was shown that the vortices propagate in a direction perpendicular to the electric field, similar behavior as presented in the classical treatment, we provide a quantum mechanics explanation for that. We have performed the calculations considering first the action of the magnetic field as well as the nonlinearity. The results indicate that for low values of the nonlinear parameter U the vortices remain stationary while preserving the form. For greater values of the parameter the picture gets distorted, the more so, the greater the nonlinearity. As for the inclusion of the electric field, we note that for small U, the wave packet propagates perpendicular to the applied field, until for greater values of U the wave gets partially localized in a definite region of the lattice. That is, for strong nonlinearity the wave packet gets partially trapped, while the tail of it can propagate through the lattice. Note that this tail propagation is responsible for the over-diffusion for long times of the wave packet under the action of an electric field. We have produced short films that show clearly the time evolution of the wave packet, which can add to the understanding of the dynamics.     

An FDTD scheme on a face-centered-cubic (FCC) grid for the solution of the wave equation

A method is proposed to improve the numerical dispersion characteristics for simulations of the scalar wave equation in 3D using the FDTD method. The improvements are realized by choosing a face-centered-cubic (FCC) grid instead of the typical Cartesian (Yee) grid, which exhibits non-physical distortions of the wavefront due to the FD stencil. FCC grids are the logical extension of hexagonal grids in 2D, and have been shown previously to provide optimal sampling of space based on close packing of spheres (highest density). The difference equations are developed for the wave equation on this alternative grid, and the dispersion relationship and stability for grids of equal and non-equal aspect ratios are derived. A comparison is made between FCC and Cartesian formulations, based upon having an equal volume density of gridpoints in each method (i.e. the computational storage requirements of each method would be the same for the same simulated space). The comparison shows that the FCC grid exhibits a much more isotropic dispersion relation than the Cartesian grid of equivalent density. Furthermore, for an equivalent density, the FCC method has a more relaxed stability criterion by a factor of approximately 1.35, resulting in a further reduction in computational resources.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

变系数Burgers方程的分类及相似解(英文)

应用经典李群理论考虑了描述非平面冲击波形成和衰减现象的(1 1)维变系数Burgers方程,得到该方程的群分类及相应的对称.对于某些具体形式的色散项系数a(t)和非线性项系数b(t),给出了对应方程的对称约化及相似解.本文在已有基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Eigen spectra and wave functions of the massless Dirac fermions under the nonuniform magnetic fields in graphene

In this research, firstly, by using the new form of Dirac-Weyl equation and the series method with submitting more suitable details, the energy spectrum and wave functions of the massless Dirac fermions are calculated under the inhomogeneous and q-deformed spatially magnetic fields. Although, we discussed about the results of the energy levels, further, we obtained the wave function as the Hessenberg determinant with calculating the elements of it as exact. On the other hand, by using the Mellin-Barnes integral representation and Hurwitz zeta function, we have achieved the thermodynamic physical quantities of the Dirac-Weyl fermions in the absence of a magnetic field for inside of the graphene quantum dot. Finally, our numerical results for the wave functions and probability densities are presented too.     

变系数(2+1)维Broer-Kaup方程的精确解

利用齐次平衡原则,导出了变系数(2+1)维Broer-Kaup方程的B?cklund变换(BT),并由该BT,求出了(2+1)维Broer-Kaup方程的各种形式的精确解.     

变系数（2＋1）维Broe-Kaup方程的新精确解

通过一个简单的变换，变系数(2 1)维Broer-Kaup方程被简化为人们熟知的变系数Burgers方程。利用近年来广泛使用的齐次平衡法和tanh-函数法，获得了变系数(2 1)维Broer-Kaup方程的一些新的精确解。     

Solitons in a generalized space- and time-variable coefficients nonlinear Schrödinger equation with higher-order terms

We introduce an approach that combines a similarity method with several transformations to find analytical solitary wave solutions for a generalized space- and time-variable coefficients of nonlinear Schrödinger equation with higher-order terms with consideration of varying dispersion, higher nonlinearities, gain/loss and external potential. One of these transformations is constructed in such a way that allows study of the width of localized solutions. Solitary-like wave solutions for front, bright and dark are given. The precise expressions of the soliton?s width, peak, and the trajectory of its mass center and the external potential which are symbol of dynamic behavior of these solutions, are investigated analytically. In addition, the dynamical behavior of moving, periodic, quasi-periodic of breathing, and resonant are discussed. Stability of the obtained solutions is analyzed both analytically and numerically.     

变系数(2+1)维Broer-Kaup方程的新精确解P

通过一个简单的变换,变系数(2+1)维Broer-Kaup方程被简化为人们熟知的变系数Burgers方程.利用近年来广泛使用的齐次平衡法和tanh-函数法,获得了变系数(2+1)维Broer-Kaup方程的一些新的精确解.     

A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev——Petviashvili equation

Using the solution of general Korteweg--de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev--Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.     

变系数(2+1)维Broer-Kaup方程新的类孤子解

基于齐次平衡原则和分离变量法的思想,通过两个推广的Riccati方程组和Mathematica软件,求出了变系数(2+1)维Broer-kaup方程的一些精确解,包括各种类孤立波解、类周期解,其中许多解是新的.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.     

Evolution of the exact spatiotemporal periodic wave and soliton solutions of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients

In this paper the spatiotemporal evolution of the periodic wave is investigated analytically when the laser passes through the inhomogeneous nonlinear medium. Firstly, the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients is solved analytically by an improved homogeneous balance principle and F-expansion technique. A number of exact periodic traveling wave and spatiotemporal soliton solutions are obtained. Then, their propagation characteristics are analyzed in detail. It is found that the evolutions of propagation of spatiotemporal soliton and periodic wave solutions are regular when the diffraction and dispersion coefficients are the identical distributed coefficients, but the evolutions of propagation of these solutions are irregular with other coefficients.     

(2+1)维色散长波方程的新精确解及其复合波激发

利用Riccati方程展开法和变量分离法,得到了(2+1)维色散长波方程的变量分离解.根据得到的孤波解,构造出该方程新颖的复合波局域结构,研究了复合波随时间的演化.     

N维氢原子四类升降算子的归一化系数

计算了N(d≥2)维氢原子四类升降算子的矩阵元,得到了N维氢原子归一化系数的普适表示式.当d=2或d=3时,得到二维与三维氢原子升降算子的归一化系数.     

New integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations with constant and time-dependent coefficients

In this work, we mainly address two new integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations, which naturally appear in surface theory and fluid dynamics. The first equation includes constant coefficients, while the other is characterized with time-dependent coefficients. It is of further value to investigate the integrability of each model. This study puts forward a Painlevé test to reveal the Painlevé integrability. We show that the first equation passes the Painlevé test to confirm its integrability. However, the compatibility conditions of the second model with time-dependent coefficients provides the relation between these coefficients to ensure its integrability. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where four sets of multiple soliton are derived for these equations.