对求解非线性方程方法的探索

BaiCL在2001年提出了双曲函数法，并把解设成具体的双曲函数，即sinhw和coshw的线性组合．本文把解设成是由2个函数f(ξ)和g(ξ)组成的线性组合，它们可以有多种取法，从而使方程更直接、更有效．借助数学软件Maple，用改进后的方法和吴文俊消元法，求解Bai C L文中的一个例子，获得了包含Bai文结果在内的更为丰富、精确的行波解．     

双相干点光源干涉的一般分析

讨论了双相干点光源的三维空间干涉场的一般数学描述,分析了3个不同截面处的干涉强度的空间分布情况,并给出了相干相长点的计算机模拟图样.该结果也适用于其他类型波的相干叠加场分布.     

Density computations for real quadratic units

In order to study the density of the set of positive integers for which the negative Pell equation is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in as the factorization of is a natural part of the input for the values of we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm .

     

Extended Hyperbola Function Method and Its Application to Nonlinear Equations

An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.     

On Properties of Rectangular Hyperbolas

This paper improves an old theorem about a rectangular hyperbola : its centre lies on the pedal circle of any point on with respect to any triangle inscribed in . We also prove that an analogous result holds for Cevian circles. These results are used to obtain new characterisations of the Feuerbach, Jarabek, and Kiepert hyperbolas of a triangle.     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.     

与对角格空时码相关的一类mathbb{Z}[zeta_{m}]上不可约多项式的判别式

为实现信号在空间的分集, 关于格的空时分组码的设计近年来备受关注.通过研究与对角的格空时码相关的$\mathbb{Z}[\zeta_{m}]$上的一类二次不可约多项式的判别式$|\Delta|$,确定了$\mathbb{Z}[\zeta_{m}]$上的格空时编码的正规分集乘积的大小.进而, 利用Pell方程的解的性质, 构造性地证明了$m=5, 8, 10, 12$时,$|\Delta|$的值可以任意小. 最后,提出几个关于$\mathbb{Z}[\zeta_{m}]$上的二次不可约和三次不可约多项式的判别式大小的猜想.     

组合KdV方程的显式精确解

借助计算机代数系统Mathematica，利用双曲函数法找到了组合KdV方程(Combined KdV Equation)的精确孤立波解，包括钟型孤立波解和扭结型孤立波解.在此基础上又对双曲函数法的思想进行了推广，从而获得了其更多的显式精确解，包括间断型激波解和指数函数型解.这种方法也适用于求解其他非线性发展方程(组). 关键词： 组合KdV方程 双曲函数法 孤立波解 精确解     

Variable-Coefficient Hyperbola Function Method and Its Application to (2+1)-Dimensional Variable-Coefficient Broer-Kaup System

Based on a new intermediate transformation, a variable-coefficient hyperbola function method is proposed.Being concise and straightforward, it is applied to the (2 1)-dimensional variable-coefficient Broer Kaup system. As a result, several new families of exact soliton-like solutions are obtained, besides the travelling wave. When imposing some conditions on them, the new exact solitary wave solutions of the (2 1)-dimensional Broer-Kaup system are given. The method can be applied to other variable-coefficient nonlinear evolution equations in mathematical physics.     

联立Pell方程组x~2-ay~2=1和y~2-bz~2=1的解数

设a,b是正整数.我们研究了联立Pell方程组x~2-ay~2=1,y~2-bz~2=1的正整数解(x,y,z)的个数.本文利用Bennett关于联立Padé逼近的一个结果,证明了该方程组至多只有两组正整数解(x,y,z),从而改进了Bennett(1998),袁平之(2004)等人的结论.