Dust acoustic rogue waves of fractional-order model in dusty plasma

In this paper, the fractional-order model is used to study dust acoustic rogue waves in dusty plasma. Firstly, based on control equations, the multi-scale analysis and reduced perturbation method are used to derive the (3+1)-dimensional modified Kadomtsev-Petviashvili (MKP) equation. Secondly, using the semi-inverse method and the fractional variation principle, the (3+1)-dimensional time-fractional modified Kadomtsev-Petviashvili (TF-MKP) equation is derived. Then, the Riemann-Liouville fractional derivative is used to study the symmetric property and conservation laws of the (3+1)-dimensional TF-MKP equation. Finally, the exact solution of the (3+1)-dimensional TF-MKP equation is obtained by using fractional order transformations and the definition and properties of Bell polynomials. Based on the obtained solution, we analyze and discuss dust acoustic rogue waves in dusty plasma.     

A new complex line soliton for the two-dimensional KdV–Burgers equation

Making use of a extended tanh method with symbolic computation, we find a new complex line soliton for the two-dimensional (2D) KdV–Burgers equation. Its real part is the sum of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D KdV (KP) equation, and its imaginary part is the product of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D MKdV (MKP) equation.     

Instantaneous Bethe-Salpeter Equation and Its Exact Solution

We present an approach to solve Bethe-Salpeter (BS) equations exactly without any approximation if the kernel of the BS equations exactly is instantaneous, and take positronium as an example to illustrate the general features of the exact solutions. The key step for the approach is from the BS equations to derive a set of coupled and well-determined integration equations in linear eigenvalue for the components of the BS wave functions equivalently, which may be solvable numerically under a controlled accuracy, even though there is no analytic solution. For positronium, the exact solutions precisely present corrections to those of the corresponding Schrödinger equation in order v¹ (v is the relative velocity) for eigenfunctions, in order v² for eigenvalues, and the mixing between S and D components in J^PC=1^-- states etc., quantitatively. Moreover, we also point out that there is a questionable step in some existent derivations for the instantaneous BS equations if one is pursuing the exact solutions. Finally, we emphasize that one should take the O(v) corrections emerging in the exact solutions into account accordingly if one is interested in the relativistic corrections for relevant problems to the bound states.     

Lax pair, Bäcklund transformation, solitary wave solution and finite conservation laws of the general KP equation and MKP equation with variable coefficients

In this paper, we obtain the Lax pair, Bäcklund transformation, solitary wave solution and infinite conservation laws of the general KP equation and MKP equation with variable coefficients.     

Optimal mass transport for higher dimensional adaptive grid generation

In this work, we describe an approach for higher dimensional adaptive grid generation based on solving the L² Monge–Kantorovich problem (MKP) which is a special case of the classical optimal mass transportation problem. Two methods are developed for computing the coordinate transformation used to define the grid adaptation. For the first method, the transformation is determined by solving a parabolic Monge–Ampère equation for a steady state solution. For the second method, the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the L² MKP. Several numerical experiments are presented to demonstrate the performance of the MKP methods and to compare them with some related adaptive grid methods. The experimental results demonstrate that the MKP methods show promise as effective and reliable methods for higher dimensional adaptive grid generation.     

双稳激光器的不稳定性本质研究

本文从非线性方程稳定性理论出发,给出了双稳激光器速率方程的稳定性三条件,指出了其中两个条件分别是产生双稳和自脉动的不稳定性条件,证明了双稳与光功率曲线出现负斜率的区域相对应,这时小信号分析方程有单调上升解,自脉动则与无稳定解的区域相对应,这时小信号分析方程有振荡上升解。另外两者都属于“三次”非线性现象,但是要在g达到一定的非线性时才可能产生自脉动。 关键词：     

Reduction of Einstein's field equations for twisting typeN-vacuum fields with anH 2

The problem of vacuum typeN solutions of Einstein's field equations with Killing vector and homothetic group can be reduced to a single ordinary differential equation of the third order for a single real function. Hauser's solution [4] is an exceptional case. If the homothetic parameterN takes the valueN=2 the third order differential equation becomes surprisingly simple. This caseN=2 is therefore the most promising one for the search of exact solutions.     

The kink-soliton and antikink-soliton in quasi-one-dimensional nonlinear monoatomic lattice

1 Introduction During recent decades great progress has been made in the soliton formation and its application in one-dimensional (1D) system. The interest in the soliton modes has been intensified due to their observation in the energy propagating along bio- logical molecule chain. The new achievement in studying the topological soliton in polyacetylene molecule chain and the phenomena of transformation with the action of electric field are continuously reported[1]. The electric field domain…     

Strict Positivity of a Solution to a One-Dimensional Kac Equation Without Cutoff

We consider the solution of a one-dimensional Kac equation without cutoff built by Graham and Méléard. Recalling that this solution is the density of a Poisson driven nonlinear stochastic differential equation, we develop Bismut's approach of the Malliavin calculus for Poisson functionals in order to prove that this solution is strictly positive on ]0,[x.     

KdVB方程行波解的渐近分析

本文利用奇异摄动的理论和方法,研究v²?4μ时KdVB方程的行波解,得到行波解的三阶渐近展开式的显式,同时得到行波解的一般渐近展开式的表达式:u≈u⁽⁰⁾+εu⁽¹⁾+ε¹u⁽²⁾+…+εⁿu⁽ⁿ⁾+…;并且证明u^(j)(j=1,2,…,n,…)都是有界函数。 关键词：     

Exact and approximate solutions for the electron nonsinglet structure function in QED

Two exact, valid up to infinite perturbative order, numerical solutions of the Lipatov equation for the nonsinglet electron structure function in the QED are presented. One of them is of the Monte Carlo type and another is based on the numerical inversion of the Mellin transform. They agree numerically to a very high precision (better than 0.05%). Within the leading logarithmic approximation, the exact solution is compared with the perturbative second and third order exponentiated solutions. It is shown that the perturbative second order solution inspired by the Yennie-Frautschi-Suura (exclusive) exponentiation is much closer to the exact solution than the other ones. New compact analytical formula for the third order exponentiated solution is given. It is shown to be in perfect numerical agreement with the infinite order solution of the Monte Carlo and Mellin type.     

Darboux Transformation and N-soliton Solution for Extended Form of Modified Kadomtsev-Petviashvili Equation with Variable-Coefficient

The extended form of modified Kadomtsev-Petviashvili equation with variable-coefficient is investigated in the framework of Painlevé analysis. The Lax pairs are obtained by analysing two Painlevé branches of this equation. Starting with the Lax pair, the N-times Darboux transformation is constructed and the N-soliton solution formula is given, which contains 2n free parameters and two arbitrary functions. Furthermore, with different combinations of the parameters, several types of soliton solutions are calculated from the first order to the third order. The regularity conditions are discussed in order to avoid the singularity of the solutions. Moreover, we construct the generalized Darboux transformation matrix by considering a special limiting process and find a rational-type solution for this equation.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

偏离朗伯比尔定律NaCl水溶液的红外光谱合成

氯化钠水溶液中,水分子之间的氢键相互作用随氯化钠浓度变化而发生改变,导致不同浓度的水溶液光谱形状也不同,因此NaCl溶液的光谱不满足朗伯比尔定律。如果浓度c与吸光率A不满足朗伯比尔定律,则利用光谱差减技术扣除溶剂水的吸收峰问题就会遇到瓶颈,差减效果不能令人满意。利用二元线性回归分析法和双边夹原理合成杂化光谱,期望在偏离朗伯比尔定律体系,可以理论上获得各种浓度的NaCl水溶液的杂化光谱,并且保证杂化光谱与其代表的真实光谱高度相似。主要结论如下：(1)浓度c₁和c₂的NaCl水溶液的红外光谱分别为A_c1和A_c2,则浓度介于c₁和c₂之间的溶液光谱,均可用A_hc(c₁, c₂)=âA_c1 +A_c2来表示。A_hc(c₁, c₂)与浓度为c(c₁＜c＜c₂)的溶液的真实光谱A_c高度类似。(2)c₂和c₁间隔越小,杂化光谱A_hc(c₁, c₂)与真实光谱A_c的相似度越高,因此可根据精度要求选择合适的A_c1和A_c2来拟合A_hc 。(3)杂化光谱A_hc(2%, 30%)=âA_2%+A_30%,适用浓度区间2%～30%,涵盖浓度范围宽,效果令人满意。(4)回归系数â和与浓度c关系为â=-3.592c+1.058 9,=3.565c-0.0551 5。在2%~30%浓度范围内,选取浓度c数值,即可获得相应回归系数â和,再将â和代入方程A_hc(2%, 30%)=âA_2%+A_30%,即可获得NaCl水溶液的杂化光谱。杂化光谱与相对应的真实光谱高度近似,A_hc(2%, 30%)≈A_c,完全可替代真实光谱。     

Gravity on a Parallelizable Manifold. Exact Solutions

The wave type field equation^a =^a, where ^ais a coframe field on a space-time, was recentlyproposed to describe the gravity field. This equationhas a unique static, spherically symmetric, asymptotically flat solution, whichleads to the viable Yilmaz–Rosen metric. We showthat the wave type field equation is satisfied by thepseudo-conformal frame if the conformal factor isdetermined by a scalar 3D-harmonic function. This functioncan be related to the Newtonian potential of classicalgravity. So we obtain a direct relation between thenon-relativistic gravity and the relativistic model: every classical exact solution leads to asolution of the field equation. With this result weobtain a wide class of exact, static metrics. We showthat the theory of Yilmaz relates to thepseudo-conformal sector of our construction. We derive also aunique cosmological (time dependent) solution of thedescribed type.     

直流和交流Josephson效应的研究

基于Feynman模型得到的Josephson方程,在求解过程中采取了ρ1=ρ2的假设,但这个假设在数学处理上并不严格。为了得到严格的结果,未进行上述假设,通过解微分方程得到了严格的Josephson方程,结果显示在一定条件下才能得到通常所说的Josephson方程。     

关于干涉反演光谱技术能否扩大F-P干涉仪光谱范围的探讨

本文从物理学的角度提出了Fredholm第一类积分方程数值解的可靠性概念,证明了在被称为Fabry-Perot干涉反演光谱技术中,当△σ=2/x,△σ=2/σ时,若取样点数为一个适当的奇数,那么积分方程的数值解是稳定的.但是进一步的计算机模拟实验表明,该数值解不是原积分方程的可靠解,因此,干涉反演光谱技术不能扩大Fabry-Perot干涉仪的光谱范围.理论分析表明,在前述条件下,积分方程数值解不可靠的根本原因在于该积分方程本身没有唯一解.     

Traffic Flow Models Considering an Internal Degree of Freedom

We solve numerically the integrodifferential equation for the equilibrium case of Paveri–Fontana's Boltzmann-like traffic equation. Beside space and actual velocity, Paveri–Fontana used an additional phase space variable, the desired velocity, to distinguish between the various driver characters. We refine his kinetic equation by introducing a modified cross section in order to incorporate finite vehicle length. We then calculate from the equilibrium solution the mean-velocity–density relation and investigate its dependence on the imposed desired velocity distribution. A further modification is made by modeling the interaction as an imperfect showing-down process. We find that the velocity cumulants of the stationary homogeneous solution essentially only depend on the first two cumulants, but not on the exact shape of the imposed desired velocity distribution. The equilibrium solution can therefore be approximated by a bivariate Gaussian distribution which is in agreement with empirical velocity distributions. From the improved kinetic equation we then derive a macroscopic model by neglecting third and higher order cumulants. The equilibrium solution of the macroscopic model is compared with the cumulants of the kinetic equilibrium solution and shows good agreement, thus justifying the closure assumption.     

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     

Smooth Positons of the Second-Type Derivative Nonlinear Schrödinger Equation

We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schr¨odinger(DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the positon solution, and display its approximate orbits and variable "phase shift". The second and third order breather-positon solutions are also constructed.