完整力学系统的高阶运动微分方程

从质点系的牛顿动力学方程出发,引入系统的高阶速度能量,导出完整力学系统的高阶Lagrange方程、高阶Nielsen方程以及高阶Appell方程,并证明了完整系统三种形式的高阶运动微分方程是等价的.结果表明,完整系统高阶运动微分方程揭示了系统运动状态的改变与力的各阶变化率之间的联系,这是牛顿动力学方程以及传统分析力学方程不能直接反映的.因此,完整系统高阶运动微分方程是对牛顿动力学方程及传统Lagrange方程、Nielsen方程、Appell方程等二阶运动微分方程的进一步补充. 关键词： 高阶速度能量 高阶Lagrange方程 高阶 Nielsen方程 高阶Appell方程     

均匀磁场中二维各向同性带电谐振子的守恒量与对称性研究

由牛顿第二定律得到二维各向同性带电谐振子在均匀磁场中运动的运动微分方程,通过对运动微分方程的直接积分得到系统的两个积分(守恒量).利用Legendre变换建立守恒量与Lagrange函数间的关系,从而求得系统的Lagrange函数,并讨论与守恒量相应的无限小变换的Noether对称性与Lie对称性,最后求得系统的运动学方程.     

变质量完整力学系统的高阶运动微分方程

从Мещерский方程出发,建立变质量力学系统的高阶D'Alembert-Lagrange原理,导出变质量完整力学系统的各类高阶运动微分方程.结果表明,它们扩充和优化了完整力学的相关理论. 关键词： 变质量完整力学系统 高阶力变率 高阶D'Alembert-Lagrange原理 高阶运动微分方程     

相对论性非球谐振子势场中的赝自旋对称性

求解了非球谐振子势场中1/2自旋粒子满足的Dirac方程,Dirac哈密顿量包含有标量非球谐振子势S(r)和矢量非球谐振子势V(r).在Σ(r)=S(r)+V(r)=0和Δ(r)=V(r)-S(r)=0的条件下,解析地得到了Dirac旋量波函数的束缚态解和能谱方程,结果表明非球谐振子势 关键词： 非球谐振子势 Dirac方程 赝自旋对称性 束缚态     

类Quesne环状球谐振子势场中的赝自旋对称性

提出了一种新的类Quesne环状球谐振子势,应用二分量方法求解1/2-自旋粒子满足的Dirac方程, Dirac哈密顿量由标量和矢量类Quesne环状球谐振子势构成.在Σ=S(r)+V(r)=0的条件下,得到了Dirac旋量波函数下分量的束缚态解和能谱方程, 显示出类Quesne环状球谐振子势场中的赝自旋对称性.讨论了束缚态波函数和能谱方程的有关性质. 关键词： 类Quesne环状球谐振子势 Dirac方程 赝自旋对称性 束缚态     

外加直流电场作用下高阶弱非线性复合介质的电势分布

利用同伦分析方法, 研究了一类由柱形杂质随机嵌入基质所形成的、电场和电流密度满足J = σ E + χ |E|²E + η|E|⁴E 形式本构关系的高阶弱非线性复合介质在外加直流电场作用下的电势分布问题. 首先利用模函数展开法, 将本构方程及边界条件化成了一系列非线性常微分方程的边值问题; 再利用同伦分析方法进行求解, 给出了电势在基质和杂质区域的渐近解析解. 关键词： 高阶弱非线性复合介质 模函数展开法 同伦分析方法 电势分布     

非理想约束曲线运动的质点动力学

本文从牛顿第二运动定律出发，导出了受非理想约束的质点作平面曲线运动的动力学方程；给出了质点处于平衡状态的临界条件；求解了几个对应的平面曲线运动问题。     

氮分压对氮化铜薄膜结构及光学带隙的影响

在不同的氮分压r(r=N₂/［N₂+Ar］)和射频功率P下，使用反应射频磁控溅射法，在玻璃基片上制备了氮化铜薄膜样品.用台阶仪测得了薄膜的厚度，用原子力显微镜、X射线衍射仪、紫外-可见光谱仪对薄膜的表面形貌、结构及光学性质进行了表征分析.结果表明，薄膜的沉积速率随P和r的增加而增大.薄膜表面致密均匀，晶粒尺寸为30nm左右.随着r的增加，薄膜颗粒增大，且薄膜由(111)晶面转向(100)晶面择优生长.薄膜的光学带隙E_g在1.47—1.82eV之间，随r的增加而增大. 关键词： 氮化铜薄膜 反应射频磁控溅射 晶体结构 光学带隙     

基于太极形介质柱六角光子晶体禁带特性研究

提出了一种新型的非对称性散射体的二维六角晶格光子晶体结构–-太极形介质柱光子晶体. 利用平面波展开法从理论研究这种光子晶体结构的能带特性以及结构参数对完全禁带的影响. 研究表明:散射体对称性的打破, TE模和TM模能带宽度和数目都会有所增加, 有益于获得更宽的完全禁带以及更多条完全禁带.通过参数优化, 发现在ε = 17, R=0.38 μm, r=0.36R, θ = 0° 时, 获得最大完全带隙宽度0.0541(ωa/2πc); 在ε = 16, R=0.44, r=0.2R, θ = 0°时, 光子晶体完全带隙数目最多达到8条. 关键词： 光子晶体 禁带 平面波展开     

运动阻力对牛顿第二定律验证实验的影响及修正

在考虑滑块和导轨之间空气粘滞阻力的情况下，本对牛顿第二定律验证实验进行了深入的研究，提出了包括运动阻力在内的牛顿第二定律验证的实验方法。结果表明，在实验误差允许的范围内牛顿第二定律均能得到较好地验证。     

General N-soliton solution of the AKNS class on arbitrary background

An explicit determinant formula for the N-fols Bäcklund transform of any pair of AKNS fields q(x, t), r(x, t) is derived. In particular this formula includes all N-soliton solutions of nonlinear evolution equations which belong to the AKNS class.     

Analytical time-like geodesics in Schwarzschild space-time

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r = r(λ), t = t(χ), and τ= τ (χ), where λ is the true anomaly and χ is a parameter along the orbit. A very simple relation between λ and χ is also shown. These solutions are very useful for modelling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.     

Integrable Equations of the Form q t =L 1(x,t,q,q x,q xx )q xxx +L 2(x,t,q,q x,q xx )

Integrable equations of the form q _t =L ₁(x,t,q,q _x ,q _xx )q _xxx +L ₂(x,t,q,q _x ,q _xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given.     

Integrable Nonlinear Evolution Equations on a Finite Interval

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ⁿ_xq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q_t+q_xxx and q_t−q_xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q_x(0,t) and q_x(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).     

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system are investigated. Appell equations and differential equations of motion for a variable mass holonomic system are established. A new expression of the total first derivative of the function with respect of time t along the systematic motional track curve, and the definition and the criterion of Mei symmetry for Appell equations under the infinitesimal transformations of groups are given. The expressions of the structural equation and Mei conserved quantity for Mei symmetry in Appell are obtained. An example is given to illustrate the application of the results.     

Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q ₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g ₀(t) and the unknown boundary value q _x (0,t)=g ₁(t). Furthermore, it has been shown that given q ₀ and g ₀, the function g ₁ can be characterized through the solution of a certain 'global relation' coupling q ₀, g ₀, g ₁, and (t,k), where satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of , the global relation can be explicitly solved for g ₁.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Cosmological applications in Kaluzaben Klein theory

The field equations of Kaluza-Klein (KK) theory have been applied in the domain of cosmology. These equations are solved for a flat universe by taking the gravitational and the cosmological constants as a function of time t. We use Taylor's expansion of cosmological function, Λ(t), up to the first order of the time t. The cosmological parameters are calculated and some cosmological problems are discussed.     

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.