Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the a...     

Curvature Estimates for the Level Sets of Solutions to the Monge-Amp`ere Equation detD2u = 1

For the Monge-Ampere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.     

Large Solutions to Complex Monge-Amp`ere Equations:Existence, Uniqueness and Asymptotics

The authors consider the complex Monge-Ampère equation det( u_i[`(j)] )\left( {u_{i\bar j} } \right) = ψ(z, u, ∇ u) in bounded strictly pseudoconvex domains Ω, subject to the singular boundary condition u = ∞ on ∂Ω. Under suitable conditions on ψ, the existence, uniqueness and the exact asymptotic behavior of solutions to boundary blow-up problems for the complex Monge-Ampère equations are established.     

Bretherton方程的对称约化和精确解

运用李群对称方法解决Bretherton方程问题,得到方程的对称约化和群不变解,比如幂级数解,最后得出该问题的守恒率.     

Kadomtsev-Petviashvili方程的精确解

齐次平衡法是把非线性偏微分方程转换成带约束条件的线性偏微分方程的一种很好的方法 .本文在齐次平衡法的基础上具体讨论了KP方程的精确解 ,包括孤波解 ,一般的行波解 ,有理函数解和一种新类型的解 .     

Riemann流形上的抛物型Monge-Ampère方程

This paper deals with a class of parabolic Monge-Ampère equation on Riemannian manifolds. The existence and uniqueness of the solution to the first initial-boundary value problem for the equation are established.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

RLW—Burgers方程的精确解

借助未知函数的变换，ＲＬＷ－Ｂｕｒｇｅｒｓ方程和ＫｄＶ－Ｂｕｒｇｅｒｓ方程化为易于求解的齐次形式的方程，从而得到ＲＬＷ－Ｂｕｒｇｅｒｓ方程和ＫｄＶ－Ｂｕｒｇｅｒｓ方程的精确解。     

广义变系数Kdv方程的对称及其群不变解

利用经典李对称的方法对广义变系数Kdv方程进行研究,利用这种方法得到了该方程的一个新的精确解,这种方法的基本思路是通过对称约化将原来较难求解的偏微分方程转化为较易求解的常微分方程进行求解.实例证明这种方法具有一般性,适合于求一大类变系数的非线性演化方程.     

Burgers-BBM方程新的精确解

借助两个推广形式的Riccati方程组和Mathematica软件，求出了Burgers-BBM方程，BBM方程，KDV—Burgers方程的大量新的精确解，包括各种形式的孤立波解和三角函数周期解．     

Exact Vacuum Solutions to the Einstein Equation

In this paper,the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation.This procedure of resolution is based on a canonical form of the metric.According to this procedure,the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.     

广义KdV方程的若干新的精确解

本文利用守恒律方程等价的概念和求精确解的方法,对广义KdV方程求出了若干新解。     

二类变式Boussinesq方程的对称性约化和精确解

将Ｃｌａｒｋｓｏｎ等最近发展的直接法推广应用于变式Ｂｏｕｓｓｉｎｅｓｑ方程组,给出四种类型对称性约化方程和三组显式精确解．结果表明：在适当变换下变式Ｂｏｕｓｓｉｎｅｓｑ方程组可约化为具有椭圆函数解的Ｄｕｆｆｉｎｇ型方程和Ｐａｉｎｌｅｖ　Ⅱ方程,并且约化结果包含有关于时间ｔ的二种类型奇点;极点和代数支点．     

一类抛物方程初边值问题的精确解及其应用

本文建立了多层合采油藏压力分布新模型,并在无限大、有界定压和有界封闭三种边界条件下,求得了均质弱可压缩液体朝向具有井筒储集和表皮效应井渗滤时,各层内压力分布的精确表达式;给出了该压力解在油藏试井分析中的应用.     

Exact solutions of semilinear radial wave equations in n dimensions

Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system of group-invariant variables given by group foliations of the wave equation, using the one-dimensional admitted point symmetry groups. (These groups comprise scalings and time translations, admitted for any nonlinearity power, in addition to space-time inversions admitted for a particular conformal nonlinearity power.) This is shown to yield not only group-invariant solutions as derived by standard symmetry reduction, but also other exact solutions of a more general form. In particular, solutions with interesting analytical behavior connected with blow-ups as well as static monopoles are obtained.     

Exact Solutions of the Wick-typ e KdV-Burgers Equation

In this paper, we consider the wick-type KdV-Burgers equation with variable coe?cients. By using Tanh method with the aid of Hermite transformation, we deduce the exact solutions which include hyperbolic-exponential, trigonometric-exponential and expo-nential function solutions for the considered equation.     

高阶非线性薛定谔方程的精确解

通过修正的映射方法和推广的映射方法，我们得到了高阶非线性薛定谔方程新的精确解，它们是两个不同的雅可比椭圆函数的线性组合．并研究了在极限情况下高阶非线性薛定谔方程的解．     

二维RLW方程和二维SRLW方程的显式精确解

本文讨论了二维RLW方程和二维SRLW方程孤立波解的性态，通过直接积分的方法求出了这两个方程的显式精确孤立波解，并通过选取初始条件的方法求出了二维RLW方程和二维SRLW方程的另一类精确行波解．     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.