微分多项式系统的对合特征集

给出了一类将微分多项式系统关于抽象对合延拓方向约化成对合特征集的算法. 该算法统一了已有的基于Riquier方法和Thomas方法及Pommaret方法延拓方程的算法. 采用我们及其他研究者最近发现的新延拓方向, 给出了计算对合特征集的新方法. 实践表明, 这些新方法可以用于简化Wu-Ritt特征集方法的计算程序.     

判定线性偏微分方程组解的完备性的一个符号计算方法

从微分代数的角度出发,借助于吴微分特征集理论,对于线性偏微分方程组,给出了判定它的解的完备性的一个符号计算方法.这个算法是一个机械化的算法,借助于符号计算软件Maple,可以在计算机上实现.     

一个有限维完全可积系统和MDWW方程解的对合表示

本文讨论高阶ＭＤＷＷ方程的Ｌａｘ对，在位势与特征函数之间的约束条件下，Ｌａｘ系统被非线性化成为有限维Ｌｉｏｕｖｉｌｌｅ完全可积系统．并且获得了高阶ＭＤＷＷ方程解的对合表示．     

一个经典的可积Neumann系统和Boussinesq族Lax对的非线性化

本文在位势与特征函数之间的Neumann约束条件下,经典Boussinesq族的Lax对被非线性化成为自然相容的Lax系统;而且,其为Liouville完全可积的Hamiltonian系统,同时获得了Boussinesq方程解的对合表示。     

线性齐次偏微分方程组的既约化基

本文参照代数Grobner基的思想,提出线性齐次偏微分方程组的既约化基的概念, 并给出了线性齐次偏微分方程组的既约化基的唯一性定理.     

一类拟线性偏微分方程组的Laplace空间解的形式相似性

本作研究了一类拟线性偏微分方程组在不同的外边界条件(无穷大外边界，封闭外边界，定值外边界)和随机时间变化的内边界条件下的初值问题在Laplace空间的解的形式相似性，它能很好地帮助我们认识模型遵从的内在规律及设计相应的应用软件．     

P阶线性偏微分方程组柯西问题的解析解的一般表达式

本文是在[1]—[10]工作的基础上,在复域中讨论P阶线性偏微分方程组柯西问题 (方程中m=∞时,级数在原点附近一致收敛)解析解的结构,获得了方程组的系数矩阵B_4(0)(i=1,2,…,p-1)及微分算子D之间的相互可易,部分可易和互不可易的诸情况下的一般解析解的级数形式显式表达式。     

一类三阶线性微分方程的可积条件

本文给出了三阶交系数线性微分方程具有特解　的充要条件，并由此得到了一类三阶交系数线性微分方程的可积条件。     

高阶非线性常微分方程组的可积类型

线性微分方程在物理学,力学及控制论中有着广泛应用,近年来在国内外一些重要刊物上发表多篇有关求微分方程精确解的论文。本文在这些论文的基础上,借助于Lcibniz求导公式及变换组法,进而提出了高阶非线性常微分方程组的求解法,于是论证了它的可积性,所得结果是相应文献结果的推广。     

Involutive characteristic sets of algebraic partial differential equation systems

This paper presents an algorithm to reduce a nonlinear algebraic partial differential equation system into the involutive characteristic set with respect to an involutive prolongation direction, which covers the existing algorithms based on Riquier method, Thomas method, and Pommaret method. It also provides new algorithms for computing involutive characteristic sets due to the existence of new involutive directions. Experiments show that these new algorithms may be used to significantly reduce the computational steps in Wu-Ritt's characteristic set method for algebraic partial differential equations.     

一个特征值问题的非线性化及其生成的经典可积系统

Under the Bargmann constrained condition, the spatial part of a new Lax pairof the higher order MkdV equation is nonlinearized to be a completely integrable system(R^2N,dp∧dq, H0=1/2F0)(F0=〈Aq,p〉 〈Ap, p〉 〈p,q〉^2). While the nonlinearization of the time part leads to its N-involutive system (Fm).     

一个Bargmann系统及一类演化方程解的对合表示

本文通过一个特征值问题的非线性化,得到了一个Bargmann系统并证明了它是Liouville意义下的完全可积系统,并给出了与这个特征值问题相联系的演化方程解的对合表示。     

The Level Set Method for Systems of PDEs

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996 Evans , L. C. ( 1996 ). A geometric interpretation of the heat equation with multivalued initial data . SIAM J. Math. Anal. 27 ( 4 ): 932 – 958 .[Crossref], [Web of Science ®] , [Google Scholar]) for the heat equation and by Giga and Sato (2001 Giga , Y. , Sato , M.-H. ( 2001 ). A level set approach to semicontinuous viscosity solution for Cauchy problems . Comm. Partial Differential Equations 26 ( 5–6 ): 813 – 839 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for Hamilton–Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each nonzero sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction–diffusion equations.     

Integrable Discretisation of the Lotka-Volterra System

In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka-Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume-preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.     

An Example of (3+1)-Dimensional Integrable System

In this paper an example of the (3+1)-dimensional integrable system is considered and the infinite series of divergent forms are described. The classical symmetries for this system, the factor-equation for these symmetries, and exact solutions of this system are found.     

微分特征列法在拟微分算子和Lax表示中的应用

微分特征列法用于拟微分算子和非线性发展方程Lax表示的计算.首先,利用微分特征列法和微分带余除法计算拟微分算子的逆和方根,由于不必求解常微分方程组,并将解代入,因此,使得计算得以简化.其次,利用微分特征列法,约化从广义Lax方程和Zakharov-Shabat推出的非线性偏微分方程,并得到相应的非线性发展方程.在Mathematica计算机代数系统上,编写了相关程序,从而可以利用计算机辅助完成一些非线性发展方程Lax表示的计算.     

Existence of Conservation Laws and characterization of recursion operators for completely integrable systems

Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we give conditions for (local) existence of conservation laws for analytical quasi-linear systems of two independent variables. This result is applied to characterize the recursion operator (in the sense of Magri) of completely integrable systems.