非齐次Toda晶格的对称,精确解和可积性

研究非齐次Toda晶格，即一类非齐次非线性微分差分方程的对称与可积性。给出了这一类方程的Lie点对称，条件对称和精确解。给出这类方程与Toda晶格之间的可逆点变换，从而表明这一类方程是可积的。     

一类具有高阶奇性解的Hilbert核奇异积分方程

本文利用指数变换研究了一类解具高阶奇性的周期黎曼边值问题,通过转化法研究了一类解具有高阶奇性的Hilber核奇异积分方程,获得了相应的解和可解条件表达式.推广了Hilber核奇异积分方程的结果.     

非线性非单调二元算子方程组的解及其应用

利用非线性泛函分析中的锥理论和单调迭代的方法,研究了一类非线性非单调二元算子方程组的解的存在性,并给出了收敛于解的迭代序列,然后作为应用,得到了B anach空间中的一类非线性V olterra型积分方程组的解,改进了最近的许多结果.     

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Itô integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.     

滞后型泛函微分方程的有界变差解

本文研究了一类滞后型泛函微分方程的有界变差解.利用Henstock-Kurzweil积分与Schauder不动点定理,在Henstock-Kurzweil积分下,得到了这类滞后型泛函微分方程有界变差解的存在性定理,推广了一些相关的结果.     

Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.     

Gevrey Hypoellipticity for a Class of Kinetic Equations

In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.     

一类系数为间隙级数的高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of a class of higher order linear differential equations with coefficients being gap series. In this case, we remove the condition that the order of coefficients in equations is less than 1/2, and obtain some results which improve the previous results.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching

In this paper, a class of stochastic age-dependent population equations with Markovian switching is considered. The main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age-dependent population equations with Markovian switching. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions. An example is given for illustration.     

On a class of ill-posed non-linear integral equations arising in the interpretation of indirect measurements

In this paper we consider a class of specific Urysohn integral equations for which the solutions are only determined with the exception of rearrangements of function values and associated arguments. As an alternative to Tikhonov's regularization method approximating minimum-norm solutions for this ill-posed class of inverse problems, a constrained least-squares approach is presented. This approach is aimed at finding decreasing rearrangements serving as appropriate solution representatives. It is shown that the inverses of these decresing solutions solve a Fredholm linear integral equation of the first kind.     

Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process satisfying some moment condition and an independent Brownian motion is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given.     

Long-time behaviour of solutions of a class of nonlinear parabolic equations

In this paper, the long-time behaviour of solutions of a class of nonlinear parabolic equations is studied. It is shown that the solutions of initial-boundary value problem to the equations converge to a travelling wave solution of the equation or a self-similar solution of a Hamilton–Jacobi equation under certain conditions on initial and boundary values of the solutions.     

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.     

关于一类拟线性拟抛物方程解的Blow─up问题

本文利用Ｆｏｕｒｉｅｒ空间的比较原理研究一类拟线性拟抛物方程解的Ｂｌｏｗ－ｕｐ问题，并给出了其解在有限时刻Ｂｌｏｗ－ｕｐ的条件。     

爱因斯坦场方程的一类新解

作者构造了真空爱因斯坦场方程的一类新解,并给出了两个具体的例子,其中一个例子是时间周期解,而时间周期解和循环宇宙有着密切的关系.作者证明了这一类新解不是Minkowski的,并且与其它已有的解有本质上的不同.作者期望这种解可以在现代宇宙学和广义相对论中有所应用.     

Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations

In this paper, we consider nonnegative weak solutions for a class of degenerate parabolic equations. Using Moser’s method, we get the local boundedness of solutions to equations of this class. Then we prove that the solutions are locally Hölder continuous.     

Oscillatory Behavior of Solutions for a Class of Second Order Nonlinear Differential Equation with Perturbation

In this paper, a class of second order nonlinear differential equations with perturbation is studied. By using the generalized Riccati transformation, the integral averaging technique and the method of classification, new oscillation criteria are obtained for all solutions of the equations, which generalize and improve some known results.     

系数A_0起支配作用的高阶线性微分方程解的增长性

该文在方程系数A_0的增长性起支配作用的条件下,研究了一类高阶线性齐次微分方程解的增长性,得到两个主要结果,一个是全平面的,一个是单位圆的,改进和推广了原有的一些结果.     

解一类抽象积—微分方程的积分半群方法

在这篇文章内，研究一类含参数线性一算子族Ａ（ｔ）和线性算子Ｂ的积－微分方程，应用积分半群方法证明了该方程存在指数有界解，和含Ａ（ｔ），Ｂ的二阶微分方程存在指数有界和分解。这里Ａ（ｔ），Ｂ不必满足Ｈｉｌｌｅ－Ｙｏｓｉｄａ条件。