Algebraic structure and Poisson integrals of a rotational relativistic Birkhoff system

We have studied the algebraic structure of the dynamical equations of a rotational relativistic Birkhoff system. It is proven that autonomous and semi-autonomous rotational relativistic Birkhoff equations possess consistent algebraic structure and Lie algebraic structure. In general, non-autonomous rotational relativistic Birkhoff equations possess no algebraic structure, but a type of special non-autonomous rotational relativistic Birkhoff equation possesses consistent algebraic structure and consistent Lie algebraic structure. Then, we obtain the Poisson integrals of the dynamical equations of the rotational relativistic Birkhoff system. Finally, we give an example to illustrate the application of the results.     

二维三温热传导方程求解中的非线性迭代初值选取

由于二维三温热传导方程具有很强的非线性特性,因此采用全隐格式对该方程离散后,所得非线性代数方程组的求解将变得非常困难.针对二维三温热传导方程离散所得非线性代数方程组的迭代求解,提出了一种有效的选取初值的方法.对两种不同性质的介质进行数值实验,结果表明,所设计的初值选取方法不仅大大提高了计算效率,而且能够降低非线性解法器对时间步长的影响.     

Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations

By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.     

New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov Equations Using General Projective Riccati Equation Method

Applying the generalized method, which is a direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraic system, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, we can not only successfully recover the previously known travelling wave solutions found by existing various tanh methods and other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons, and periodic solutions.     

非线性长波方程组和Benjamin方程的新精确孤波解

在齐次平衡法、双曲正切函数法和辅助方程法的基础上引入一个新的辅助方程,并借助符号计算系统Mathematica来构造了非线性长波方程组和Benjamin方程的新精确孤波解, 这种方法也可用于寻找其他非线性发展方程的新的孤波解. 关键词： 新的辅助方程 非线性长波方程组 Benjamin方程 孤波解     

A new method based on the harmonic balance method for nonlinear oscillators

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.     

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for （2＋1）-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton： 1） the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2） the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.     

基于稳态辐射传输方程的小动物尺寸组织体扩散光学层析法

在Newton-Raphson逆模型框架下,发展了基于稳态辐射传输方程光子输运正模型和代数重建技术的扩散光学层析图像重建算法,并针对小尺寸组织体内部吸收和散射系数同时重建情况下的标准代数重建技术算法进行了改进,数值结果表明改进后的代数重建技术算法重建图像效果优于标准代数重建技术算法.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Sinc Collocation Solutions for the Integral Algebraic Equation of Index-1

In this article, Sinc collocation method is considered to obtain the numerical solution of integral algebraic equation of index-1 by reducing it to an explicit system of algebraic equation. It is shown that Sinc collocation solution can produce an error of order $\mathcal{O}(√Ne^{−k√N})$. Moreover, Sinc method is applied to several examples to illustrate the accuracy and implementation of the method.     

物理方程的迭代法计算

对一些需要作近似计算了物理方程（非线性方程，超越方程等）用解析方法确定根所在区间，采用Ｘ＝φ（Ｘ）的形式进行迭代计算。     

核类似铜酸盐超导模型的sus(1,1)⊕sud(1,1)代数结构

由Iachello提出的核类似铜酸盐超导模型具有su(3)代数结构,其平均场近似下的Hamilton量可以写成su(3)生成元的线性组合.通过代数生成元的实现,该模型约化的Hamilton量具有su_s(1,1)⊕su_d(1,1)代数结构,利用相干算子U(θ,φ)的幺正变换,可得到系统约化Hamilton量的能隙方程及本征值,发现应用不同代数结构求解会得到不同侧重点的分析结果.     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.``     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.     

Reduced dynamics and the master equation of open quantum systems

An exact reduced density operator of a quantum system interacting with a bosonic thermal reservoir is derived by means of the simple algebraic method. The necessary and sufficient condition is found that the time-convolutionless master equation becomes exact up to the second order with respect to the system-reservoir interaction. The result is examined by means of the boson-detector model. The reduced dynamics of a quantum system interacting with a classical reservoir is also discussed.     

New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation

Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e.,the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.     

核类似铜酸盐超导模型的su_s(1,1)su_d(1,1)代数结构

由Iachello提出的核类似铜酸盐超导模型具有su(3)代数结构,其平均场近似下的Hamilton量可以写成su(3)生成元的线性组合.通过代数生成元的实现,该模型约化的Hamilton量具有Su_s(1,1)SU_d(1,1)代数结构,利用相干算子U(θ,φ)的幺正变换,可得到系统约化Hamilton量的能隙方程及本征值,发现应用不同代数结构求解会得到不同侧重点的分析结果.     

Colored Quantum Algebra and Its Bethe State

We investigate the colored Yang-Baxter equation. Based on a trigonometric solution of colored Yang-Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.     

一些直接代数方法的几何解释及应用

基于微分方程多边形的概念,就一些直接代数方法给出了几何解释,同时将这种解释推广到方程组的情形,最后以变形Boussinesq方程组为例说明了该几何解释的合理性。 关键词： 微分方程 多边形 几何解释 精确解     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f(ξ), is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.