非线性扩散方程广义解的有限传播速度性质

本文考虑具强退缩性的非线性扩散方程的第一初边值问题。在一定的结构条件下证明了广义解具有有限传播速度。该条件在某种意义下不可改进。     

Rangwala-Rao方程和几个非线性导数Schrdinger型方程的一类显式精确孤波解

首先求出了Lienard方程的显式精确解,进而求出了Rangwala-Rao方程,Ablowitz方程,Chen-Lee-Lin方程,以及Gerdjikov-Ivanov方程的型如的显式精确孤波解。     

一致分数阶导数意义下NLS方程和CNLS方程的精确解

在一致分数阶导数意义下,利用新的复变换研究了二阶NLS方程和CNLS方程,得到了两类带约束条件的薛定谔方程复值函数形式的新精确解.通过图像模拟,给出了两类方程的精确解在特定参数条件下随时间和空间变化的模值函数三维坐标图,实部三维坐标图和虚部三维坐标图.     

导数Manakov方程的Darboux变换及其精确解

本文给出了导数Manakov方程新的Darboux变换.利用此Darboux变换得到了导数Manakov方程的精确解.最后,通过选择适当的参数,作出了孤子解的图形.     

混合Boussinesq 方程的有限亏格解

借助于Lenard 递推方程和定态零曲率方程, 我们给出与3 × 3 矩阵谱问题相联系的一族混合Boussinesq 方程. 利用Lax 矩阵的特征多项式, 引入一条三角曲线K_m-1, 由此构造出相应的Baker-Akhiezer 函数、亚纯函数和Dubrovin- 型方程. 混合Boussinesq 流在Abel 映射下被拉直. 基于三角曲线和三类Abel 微分的理论, 我们得到了Baker-Akhiezer 函数、亚纯函数的Riemann θ 函数表示, 特别地, 给出了混合Boussinesq 方程的有限亏格解.     

非稳定非线性薛定谔方程的马尔钦科方程

该文基于对非稳定非线性薛定愕方程作反散射变换得到的Ｚａｋｈａｒｏｖ－Ｓｈａｂａｔ方程,直接对积分核作变换,导出马尔钦科方程．得到的马尔钦科方程在形式上与一般非线性薛定谔方程得到的一样简单明了,且不存在逆变换的自洽困难．     

一类具广义边界条件参数方程的解

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参数方程所确定的函数的高阶导数的一种逐次求导法

求由参数方程所确定的函数的高阶导数,提出了一种较为直观、简便的逐次求导方法.     

具有局部时间分数阶导数的非线性Zoomeron方程的精确解

In this paper, we are concerned with the nonlinear Zoomeron equation with local conformable time-fractional derivative. The concept of local conformable fractional derivative was newly proposed by R. Khalil et al. The bifurcation and phase portrait analysis of traveling wave solutions of the nonlinear Zoomeron equation are investigated. Moreover, by utilizing the exp(-?(ε))-expansion method and the first integral method, we obtained various exact analytical traveling wave solutions to the Zoomeron equation such as solitary wave, breaking wave and periodic wave.     

Existence Results for Fractional Differential Equations with the Riesz-Caputo Derivative

In this paper, we apply some fixed point theorems to attain the existence of solutions for fractional differential equations with the space-time Riesz-Caputo derivative. We study the boundary value problems that the nonlinearity term $f$ is relevant to fractional integral and fractional derivative. In addition, the boundary conditions involve integral. Two examples are given to show the effectiveness of theoretical results.     

双曲型方程具有奇性斜导数的混合问题<英>

本文讨论了2m阶双曲型方程具有奇性斜导数的边值问题。在边界奇点(即不满足Lopatinsky边界条件的点)子流形的一定假设下,证明了所论问题在Sobolev空间H~(s,s)(Q)中解的存在性和唯一性,从而将二阶双曲方程的相应问题的已有的结果(例如[1]、[4—6])推广到了高维的情形。     

Decay Estimates of Solutions to Wave Equations in Conical Sets

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.     

一类Monge-Amp\`{e}re 方程解的二阶导数估计

In this paper, we consider a class of Monge-Ampere equations in relative differential geometry. Given these equations with zero boundary values in a smooth strictly convex bounded domain, we obtain second order derivative estimates of the convex solutions.     

关于有限域上一类方程解的个数

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N(a1x12 + ···+anxn2 = bx1 ···xs) denote the number of solutions(x1,...,xn) of the equation a1x12 +···+ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F*q,b ∈ F*q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.     

Counting Solutions to Equations in Many Variables over Finite Fields

We present a polynomial-time algorithm for computing the zeta function of a smooth projective hypersurface of degree d over a finite field of characteristic p, under the assumption that p is a suitably small odd prime and does not divide d. This improves significantly upon an earlier algorithm of the author and Wan which is only polynomial-time when the dimension is fixed.     

某些对角方程在有限域上的解数

主要运用Gauss和以及Jacobi和的相关性质给出两类对角方程在有限域上的解数公式,分别是形如s∑(i=1) a_ix_i~(m_i)=c的对角方程,其中a_i,c∈F_q~2~*,(m_i,m_j)=1,m_i|(q+1),m_i为奇数或(q+1)/(m_i)为偶数,i=1,2,…,s,以及形如s∑(i=1) x_i~m=c的对角方程,其中c∈F_q~*,m|(q+1),m为奇数或(q+1)/m为偶数.     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．