Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.

Lie symmetries and conserved quantities for generalized Birkhoff system

To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the Lie symmetry of the system is derived. A con- served quantity of Hojman＇ s type and a Noether＇ s conserved quantity are deduced by the Lie symme- try under some conditions. One example is given to illustrate the application of the result.

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.

On Form Invariance,Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.

Symmetry and Exact Solutions of (2+1)-Dimensional Generalized Sasa-Satsuma Equation via a Modified Direct Method

In this paper, first, we employ classic Lie symmetry groups approach to obtain the Lie symmetry groupsof the well-known (2+1)-dimensional Generalized Sasa-Satsuma (GSS) equation. Second, based on a modified directmethod proposed by Lou [J. Phys. A: Math. Gen. 38 (2005) L129], more general symmetry groups are obtained andthe relationship between the new solution and known solution is set up. At the same time, the Lie symmetry groupsobtained are only special cases of the more general symmetry groups. At last, some exact solutions of GSS equationsare constructed by the relationship obtained in the paper between the new solution and known solution.

矩阵方程AXB=C的(R,S)反对称解

利用矩阵的广义奇异值分解,得到了矩阵方程AXB=C有(R,S)反对称解的充要条件,并给出了一般解的表达式,在此基础上给出了最佳逼近解的表达式。

孤子方程的广义对称与S—G型方程的遗传强对称

给出了孤子方程的广义对称概念，求出了几个孤子方程的广义对称族。同时将S－G方程的遗传强对称推广到一般的S—G型方程。

基于广义分数阶算子Birkhoff系统Noether定理

本文就广义分数阶导数算子,提出两个新的“变换公式”,将其应用于分数阶Birkhoff系统,并导出分数阶Birkhoff系统的Noether定理.这个定理提供了一个计算分数阶Birkhoff系统运动常数的方法,克服了先前文献研究分数阶动力学系统守恒量的一些缺陷,并在文末给出两个推论.

偏微分方程广义分量解的对称方法

本文对已知的微分算子,构造一个在该微分算子下不变的有限维线性空间,利用此空间的基得到了具有三个或三个以上自变量的偏微分方程的广义分离变量解.

回到真实的物质世界——皮克林与拉图尔的比较研究

皮克林深受拉图尔的影响,特别表现在拉图尔的广义对称性原则上,这种影响使皮克林像拉图尔一样,抛弃了因果性模型并规避了经典认识论。但拉图尔的符号化趋向使他没有完全摆脱抽象的表征世界。皮克林用局部对称性取代广义对称性、用历时性分析取代共时性分析并在本体论上实现了政治上的彻底性,结果使皮克林超越了拉图尔,使科学实践真正地回到了真实的物质世界。