Caudrey–Dodd–Gibbon–Kotera–Sawada方程的对称、精确解和守恒律

应用改进的CK直接方法,得到了（2+1）维Caudrey–Dodd–Gibbon–Kotera–Sawada（CDGKS）方程的对称群定理。利用对称群理论和方程的旧解得到了该方程新的精确解,扩大了解的范围。最后根据对称和共轭方程求出了（2+1）维CDGKS方程的无穷多守恒律。     

Camassa-Holm方程的等价变换

利用改进的直接方法得到了一类Camassa-Holm方程的等价变换和对称群定理，建立了方程新解与旧解之间的关系，最后在已有的一些精确解的基础上利用对称群定理得到了Camassa-Holm方程的许多新的显式精确解。     

（2+1）维非线性发展方程的对称约化和显式解

利用相容方法,得到了(2+1)维非线性发展方程的对称,并根据相应的特征方程组得到了(2+1)维非线性发展方程的相似约化,同时得到了一些新的显式解.     

两个非线性方程的势对称及其不变解

通过计算NTT方程和Burgers方程的势对称扩大了其古典对称,并获得了 Burgers 方程的一系列新的精确解. 首先,基于微分特征列集算法确定了NTT方程和Burgers方程的古典对称和势对称,并确定了 Burgers 方程的两个势对称对应的单参数Lie变换群. 其次,利用推广的简单方程方法构造了 Burgers方程的不变解,这些解分别以含任意两个参数的双曲函数、三角函数和有理函数表示. 最后,将势对称对应的Lie变换群(14)作用于Burgers方程的不变解上获得了新的精确解,重要的是这些解都不能由方程的古典对称得到.     

(2+1)维Gardner方程的对称、约化及其群不变解

利用经典李群方法,得到了一类(2+1)维Gardner方程的显式解,推广了唐和陈勇的某些结果,并且得到了该方程的对称、约化及其群不变解.     

广义四阶色散方程的对称约化和精确解

运用经典对称方法解决广义四阶色散方程问题,得到对称约化和群不变解,包括双曲函数解,三角周期解和孤立子解.最后得出了该问题的守恒律.     

广义KdV-Zakharov-Kuznetsev方程的对称约化、精确解和守恒律

利用直接对称方法得到了广义KdV-Zakharov-Kuznetsev方程(简写为mKdV-ZK)的对称约化、精确解,其中包括椭圆函数解,幂级数解,艾米儿函数解等。利用得到的对称,求出了该方程的守恒律。     

GF(p)上q元旋转对称弹性函数的一个等价刻画

基于旋转对称弹性函数l值支撑矩阵的性质,给出了GF(p)上q变元旋转对称弹性函数的一个等价刻画,证明了GF(p)上q变元旋转对称一阶弹性函数的构造问题等价于一个方程组的求解问题,并且利用方程组的所有解给出这类函数计数结果的一个表示。     

广义变系数(3+1)-维非线性薛定谔方程的有限对称群解

基于推广的对称群方法和符号计算,研究了变系数非线性薛定谔方程的有限对称群解。 我们构造了标准的(3+1)-维非线性薛定谔方程和带色散项、非线性项和增益或损耗项的(3+1)-维非线性薛定谔方程的对称变换。 利用该变换,我们从标准的(3+1)-维非线性薛定谔方程中得到了(3+1)-维变系数非线性薛定谔方程丰富的精确解。     

（3+1）维广义KdV-Zakharov-Kuznetsev方程的对称、约化和精确解

利用直接对称方法得到了广义KdV-Zakharov-Kuznetsev方程（简写为mKdV-ZK）的对称约化、精确解，其中包括椭圆函数解，幂级数解，艾米儿函数解等. 利用得到的对称，求出了该方程的守恒律.     

An open surface integral formulation for electromagnetic scattering by material plates

Using the surface equivalence theorem, four coupled integral equations are developed for electromagnetic scattering by a thin material plate. Using symmetry properties, it is shown that these equations can be written as open surface integral equations. Surface impedance relationships are obtained and used to eliminate two of the four integral equations. The remaining two equations are solved using the method of moments (MM). Numerical results for penetrable and impenetrable material plates are in reasonable agreement with measurements.     

Theorems of bianisotropic media

Theorems concerning electromagnetic fields in linear nonconducting bianisotropic media are investigated. After establishing their symmetry properties, the constitutive relations are examined under time reversal and spatial inversion transformations. Conditions under which the image method and the reciprocity relationships can be applied are discussed. Dyadic Green's functions and duality relations are also derived. With a postulated Lagrangian density, Maxwell's equations are obtained from Hamilton's principle, and energy momentum tensors are obtained from Noether's theorem. Introducing a quantum postulate in addition to the Maxwell's equations, electromagnetic fields in bianisotropic media are quantized.     

Jimbo-Miwa方程的对称约化及不变解

基于李群理论利用直接对称法得到了(3 1)-维Jimbo-Miwa方程的对称性.在此基础上,对相应的李代数进行优化,得到了方程的七种相似约化,通过变量分离以及借助辅助函数的方法,对得到的约化方程进行求解,并且得到了方程的一些新的不变解.     

基于对称约化的偏微分方程相似解研究

由于非线性系统的复杂性,对于其求解问题的研究目前还没有通用的方法,为了丰富非线性系统的求解方法,在此通过偏微分方程的决定方程确定点对称无穷小生成元,结合对称约化中的非经典Lie群法得到热方程新的相似解,并基于符号计算系统Maple给出相应的符号计算方法和实现步骤。结果表明,该算法能够有效求解PDEs的相似解,并且不需要显示地求解对应于不变曲面条件的特征方程,同时也适用于其他的发展方程。     

Semiclassical theory of multimode operation of a distributedfeedback laser

Nonlinear self-consistency equations, including for the first time the effects of mode competition, are derived for index and gain gratings. Additional terms to those contained in self-consistency equations for lasers with a conventional resonator are found. Approximate analytic solutions for single- and two-mode operation of distributed-feedback (DFB) lasers are obtained with the help of the energy theorem. The analysis includes nonvanishing end reflectors of DFB structure     

Unified theory of near-field analysis and measurement: Nonmathematical discussion

A new unified theory of near-field analysis and measurement emphasizes highly accurate, extremely efficient data processing to yield, e.g., radiation, receiving, and scattering patterns, and absolute gain values. The theory includes 17 types of plane rectangular and plane radial scanning, a more accurate and efficient plane polar scanning, spherical scanning, various types of circular cylindrical scanning, many procedures for determining complex dyadic scattering patterns, the extrapolation method for gain and effective area, and application of symmetry analysis to scattering and inverse scattering analysis. High accuracy is obtained by expressing the fields as linear combination of exact solutions of the differential equations involved (Maxwell's in the electromagnetic (EM) cases) and by using exact expressions for their transformations under coordinate changes. High efficiency is obtained with natural orthogonalities of both the solutions and transformation coefficients with respect to integration, especially summation, and implemented with the fast Fourier transform (FFT) as an approximation-free symmetry decomposition. The unified theory is based upon relativistic and gauge invariances, symmetry analysis, and the scattering matrix theory; it yields all the preceding facets and systems, both electromagnetic and scalar, and the single unified notation, general equations, and explicit expressions for the quantities which vary with the physical or scanning system. A nonmathematical discussion of other papers on the theory is provided. Many of the conceptual errors of the literature are corrected. The advantages and limitations of near-field measurements are described, and scanning systems are compared.