一类Camassa-Holm型方程的不变子空间及其精确解

研究一类三阶CH型方程的解的问题.利用不变子空间理论,讨论了方程在不同参变量取值情况下所允许的不变子空间,从而得出了此类方程的两个特殊的精确解.所得结果不仅描述了该方程的一些特性,而且丰富了文献中关于此类CH型方程的内容.     

空间分数阶Schr?dinger方程调制不稳定性的数值研究

调制不稳定性在数学和物理等学科中应用十分广泛.本文主要通过分裂谱方法对空间分数阶薛定谔方程进行数值计算,并根据Benjamin-Feir-Lighthill准则推导了非线性薛定谔方程的调制不稳定条件.文中分别研究了空间分数阶薛定谔方程在不同初值条件下的不稳定行为,并与整数阶薛定谔方程的不稳定性行为作比较,通过数值比较分析,发现整数阶薛定谔方程的这种不稳定行为对于空间分数阶薛定谔方程同样存在.     

非齐次非线性扩散方程的高维不变子空间和等价变换

利用与不变子空间方法相关的等价变换和变换v=enu给出了非齐次非线性扩散方程的等价方程,并得到了等价方程的高维不变子空间.最后给出一些例子构造了非齐次非线性扩散方程的广义泛函分离变量解.     

带有微结构的连续统中新的能量守恒定律和C-D不等式

对带有微结构的连续统中现有的基本定律、均衡方程和Clausius-Duhem不等式进行了系统的再研究,发现所有的能量均衡方程和相关的C-D不等式都是不完整的.本文对现有的结果进行了评注,并提出新的能量均衡方程和相关的C-D不等式.     

试论统计物理基本方程

提出时间反演不对称的Liouville空间反常Langevin 方程或其等价的广义Linouville方程,作为统计物理的基本方程.此方程反映了统计热力学的运动规律是随机性的而非确定性的.由它出发推导出了非平衡熵、熵增加原理、平衡态系综、BBGKY扩散方程链、流体力学方程,如质量漂移扩散方程、广义Navier-StokeS方程等.所有这一切都是统一的严格的,不需增补任何假设.但是难以普遍证明所有非均匀的远离平衡态的孤立系统内各处的熵产生密度σ≥0.     

一般非齐次非线性扩散方程的等价变换和高维不变子空间

结合压力变换和不变子空间方法中的等价变换,给出了一般非齐次非线性扩散方程的等价方程,并给出了等价方程的高维不变子空间.由此构造了一般非齐次非线性扩散方程的广义分离变量解,并给出了几个例子解释这个过程.     

无限Fuzzy关系方程的解集

对(0,1)格上的无限Fuzzy关系方程,给出其可解的充要条件.利用可达解与不可达解的概念,给出方程存在可达解与不可达解的充要条件.进一步,在解集非空时,刻画了方程的解集的结构及性质.     

Morse理论及其在一类N-Laplacian方程中的应用

研究了一类次临界增长的N-Laplacian方程.利用Trudinger-Moser不等式和Morse理论,证明了在不具有山路引理的几何性质、不满足Ambrosetti-Rabinowitz条件及整体符号条件下,该方程非平凡解的存在性.     

通过解题训练 强化逆向思维

如果方程f(x)=0的根为a,很多学生很容易知道f(a)=0.但是,反过来,由f(a)=0,学生就很不容易想到a是方程f(x)=0的根.究其原因,是由于不善于反向运用方程根的定义,不习惯按逆向展开思维.下面通过一些例子,谈谈如何强化学生的逆向思维.     

随机Navier-Stokes方程数值解的收敛性

Navier-Stokes方程在流体力学中有广泛的应用.通常情况下,大多数Navier-Stokes方程没有精确解,数值方法显得尤为重要.本文根据BDM法,利用It公式,Burkholder-Davis-Gundy不等式,Doob不等式和Gronwall引理对随机Navier-Stokes方程数值解的收敛性进行了讨论,得出数值解均方意义下收敛到解析解.     

Symmetry analysis, conservation laws of a time fractional fifth-order Sawada-Kotera equation

In this paper, we intend to study the symmetry properties and conservation laws of a time fractional fifth-order Sawada-Kotera (S-K) equation with Riemann-Liouville derivative. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation. Based on this, we find that the S-K equation can be reduced to a fractional ordinary differential equation with Erdelyi-Kober derivative by the similarity variable and transformation. Furthermore, we construct some conservation laws for the S-K equation using the idea in the Ibragimov theorem on conservation laws and the fractional generalization of the Noether operators.     

特征值问题的应用简述

从矩阵的特征问题入手,引出常系数线性齐次微分方程求解的特征方程方法;利用分离变量法求解热传导方程,引入拉普拉斯方程的特征问题,给出求解过程,并给出热方程的解的渐近稳定性.     

An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.} \amsclass{49L20, 35G20, 93E20.} Accepted 11 September 2000. Online publication 16 January 2001.     

Existence and asymptotic behavior of traveling wave solution for Korteweg-de Vries-Burgers equation with distributed delay

In this paper, we investigate the existence and asymptotic behavior of traveling wave solution for delayed Korteweg-de Vries-Burgers (KdV-Burgers) equation. Using geometric singular perturbation theory and Fredholm alternative, we establish the existence of traveling wave solution for this equation. Employing the standard asymptotic theory, we obtain asymptotic behavior of traveling wave solution of the equation.     

Long time behavior of a mean-field model of interacting neurons

We study the long time behavior of the solution to some McKean–Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean–Vlasov equation.     

Limit behavior of the global solutions to the Degasperis-Procesi-type equation

In this paper, we study the Degasperis-Procesi equation with a physically perturbation term—a linear dispersion. Based on the global existence result, we show that the solution of the Degasperis-Procesi equation with linear dispersion tends to the solution of the corresponding Degasperis-Procesi equation as the dispersive parameter goes to zero. Moreover, we prove that smooth solutions of the equation have finite propagation speed: they will have compact support if its initial data has this property.     

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.     

Sharp conditions of global existence and scattering for a focusing energy-critical Schrödinger equation

We consider a focusing energy-critical Schrödinger equation with subcritical perturbations and address question related to the sharp criterion of global existence and scattering. By analyzing the variational characteristics of this equation, we established two types of invariant flows. Then approximating this equation by the energy-critical nonlinear Schrödinger equations with the same initial data and combining the properties of the invariant flows, we obtain the sharp conditions of global existence for this equation. Moreover, when the solution is globally defined, we prove the scattering.     

On solutions of a quadratic integral equation of Hammerstein type

We study the solvability of a nonlinear quadratic integral equation of Hammerstein type. Using the technique of measures of noncompactness we prove that this equation has solutions on an unbounded interval. Moreover, we also obtain an asymptotic characterization of these solutions. Several special cases of this integral equation are discussed and applications to real world problems are indicated.     

Stability of Peakons for a Nonlinear Generalization of the Camassa-Holm Equation

In this paper, by using the dynamic system method and the known conservation laws of the gCH equation, and underlying features of the peakons, we study the peakon solutions and the orbital stability of the peakons for a nonlinear generalization of the Camassa-Holm equation (gCH). The gCH equation is first transformed into a planar system. Then, by the first integral and algebraic curves of this system, we obtain one heteroclinic cycle, which corresponds to a peakon solution. Moreover, we give a proof of the orbital stability of the peakons for the gCH equation.