入射平面电磁波两类球面波函数展开式的等价性证明

入射平面电磁波的球面波函数展开是求解不同圆球结构的平面波散射问题的重要工具,相关文献分别利用场的坐标分解和矢量势法得到了入射平面波的球面波函数的两种不同形式的展开式.利用偏微分方程边值问题解的存在唯一性定理,给出了这两种展开式的等价性的一个简洁的解析证明,并进行了数值验证.     

N-dimensional Schrödinger equation at finite temperature using the Nikiforov–Uvarov method

The N-radial Schrödinger equation is analytically solved. The Cornell potential is extended to finite temperature. The energy eigenvalues and the wave functions are calculated in the N-dimensional form using the Nikiforov–Uvarov (NV) method. At zero temperature, the energy eigenvalues and the wave functions are obtained in good agreement with other works. The present results are applied on the charmonium and bottomonium masses at finite temperature. The effect of dimensionality number is investigated on the quarkonium masses. A comparison is discussed with other works, which use the QCD sum rules and lattice QCD. The present approach successfully generalizes the energy eigenvalues and corresponding wave functions at finite temperature in the N-dimensional representation. In addition, the present approach can successfully be applied to the quarkonium systems at finite temperature.     

一类非线性波动方程位势井深度函数的连续性

本文研究一类非线性波动方程位势井深度函数的连续性.通过引入位势井深度函数并给出其性质,给出了位势井深度函数连续性的证明.而位势井深度函数连续性保证了在其基础上得到的位势井族有意义.     

Smooth finitely computable functions are affine,or why quantum systems cause waves

Abstract—It is shown that the class of all C ²-smooth real functions that can be computed (in a new, but natural sense precisely defined below) on Mealy machines (letter-to-letter transducers or, briefly, transducers) consists of affine functions only. Moreover, it turns out that all these functions can be naturally associated with wave functions of particles. Accordingly, this work can be regarded as a mathematical reasoning that the wave properties of quantum systems are caused by the discreteness of matter and causality, since transducers are a mathematical formalization of the law of causality.     

SKYRMIONS IN GROSS-PITAEVSKII FUNCTIONALS

In Bose-Einstein condensates （BECs）, skyrmions can be characterized by pairs of linking vortex rings coming from two-component wave functions. Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence, and describe the configurations of skyrmions in such BECs.     

A theoretical description for solitons in polyacetylene

The bond-alternation domain walls or the solitons in the dimerized polyacetylene are analyzed theoretically. The width of the soliton is many times the period of the chain, so that the soliton can be reasonably well described by a continuum model. Because of the existence of the bond-alternation domain walls, the electron density is different definitely. Thus the electron density can be used to describe the formation of the domain walls, and a self-trapped potential is discussed and introduced in the Hamiltonian. It is shown that the envelope of the wave functions of the chain is governed by the nonlinear Schr?dinger equation which has soliton solutions. Then the shape of the soliton is determined analytically which is in accordance with the numerical calculations by Su, Schrieffer and Heeger. This implies that the bond-alternation domain wall or the soliton is observed as the envelope of the wave function.     

Harmonic Analysis and Expansion Formulas for Two-Variable Hypergeometric Functions

The authors show how their Lie-theoretic characterization of two-variable hypergeometric functions can be employed to derive expansion theorems involving these functions. In particular, the functions arise as solutions of the Laplace, wave, heat, Helmholtz, and Schrooinger equations, and new bases can be constructed from the functions with which to expand general solutions of these physically important equations.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

Exactly solvable two-dimensional complex model with a real spectrum

Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh ⁻² term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator. Dedicated to the 80th birthday of Yuri Victorovich Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 102–111, July, 2006.     

应用F展开法求KdV方程的周期波解

提出了求非线性数学物理演化方程周期波解的F展开法,该方法可看作最近提出的扩展的Jacobi椭圆函数展开方法的浓缩.直接利用F展开法而不计算Jacobi椭圆函数,我们可同时得到著名的KdV方程的多个用Jacobi椭圆函数表示的周期波解.当模数m→1 时,可得到双曲函数解(包括孤立波解).     

Wave Kernels with Bi-Inverse Square Potentials on Euclidean Plane

The Cauchy problemfor the wave equation with bi-inverse square potential on Euclidean plane is solved in terms of the two variables Appell F₂ hypergeometric functions. Our principal tools are the Hankel transforms and the special functions of mathematical physics.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

The plane wave expansion, infinite integrals and identities involving spherical Bessel functions

This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identities for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Spectral properties of boundary-value problem with a shift for wave equation

We consider a differential operator determined by wave equation with potential in characteristic triangle, and boundary-value conditions with shift on the characteristics, and with oblique derivative on non-characteristic part of a boundary. We obtain condition for validity of the Volterra property, and show completeness of the root functions in the rest cases. We study basis property for the system of root functions under assumption that the potential depends on a single variable.     

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the new Hamiltonian amplitude equation introduced by Wadati et al. When the modulus m approaches to 1 and 0, then the hyperbolic function solutions (including the solitary wave solutions) and trigonometric function solutions are also given respectively. As the parameter ε goes to zero, the new Hamiltonian amplitude equation becomes the well-known nonlinear Schrödinger equation (NLS), and at least there are 37 kinds of solutions of NLS can be derived from the solutions of the new Hamiltonian amplitude equation.     

Dodd-Bullough-Mikhailov方程的精确解

利用(G'/G)法求解了Dodd-Bullough-Mikhailov的精确解,得到了Dodd-Bullough-Mikhailov方程的用双曲函数,三角函数和有理函数表示的三类精确行波解.由于方法中的G为某个二阶常系数线性ODE的通解,故方法具有直接、简洁的优点;更重要的是,方法可用于求得其它许多非线性演化方程的行波解.如果对其中双曲函数表示的行波解中的参数取特殊值,那么可得已有的孤波解.     

Potential Renormalization Method for a Model of the Hartree–Fock–Slater Type

We propose a new potential renormalization method for a semiclassical model of the Hartree–Fock–Slater type. The method allows easily constructing the wave functions and, in contrast to most of the similar methods, does not require an explicit form of the potential.     

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.