Pseudo-division algorithm for matrix multivariable polynomial and its application

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｒｏｂｌｅｍｏｆｇｉｖｅｎｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎａｎｄｓｅｅｋｉｎｇｉｔｓｅｑｕｉｖａｌｅｎｔＨａｍｉｌｔｏｎｉａｎｉａｎｓｙｓｔｅｍｂｅｌｏｎｇｓｔｏｉｎｖｅｒｓｅｐｒｏｂｌｅｍｏｆａｎａｌｙｔｉｃａｌｍｅｃｈａｎｉｃｓ,ａｎｄｔｈｅｌａｔｔｅｒｉｓｏｎｅｏｆｐｒｉｍｅｐｒｏｂｌｅｍｓｉｎｃｌａｓｓｉｃａｌｍｅｃｈａｎｉｃｓ.ＴｈｅｆａｍｏｕｓＰｏｉｎｃａｒｅ＿ＣａｒｔｏｎｐｒｏｂｌｅｍｉｓｔｏｇｅｔｔｈｅＨａｍｉｌｔｏｎｉａｎｉａｎｒｅｇｕｌａ…     

A generalization of recursion operators of differential equations

Ｒｅｃｕｒｓｉｏｎｏｐｅｒａｔｏｒｓａｒｅｑｕｉｔｅｕｓｅｆｕｌｆｏｒｃｏｎｓｔｒｕｃｔｉｎｇｔｈｅｓｙｍｍｅｔｒｉｅｓｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓａｎｄｆｏｒｓｏｍｅｒｅｌａｔｅｄｑｕｅｓｔｉｏｎｓ．ＴｈｅｉｒｇｅｎｅｒａｌｆｏｒｍｓｗｅｒｅｆｉｒｓｔｐｒｅｓｅｎｔｅｄｂｙＯｌｖｅｒ（１９７７）,ｔｈｏｕｇｈｔｈｅｒｅｃｕｒｓｉｏｎｐｒｏｐｅｒｔｉｅｓ,ｅｓｐｅｃｉａｌｌｙｏｎｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ,ｈａｄｂｅｅｎｉｎｖｅｓｔｉｇａｔｅｄｗｅｌｌｂｙｍａｎｙｏｔ…     

变系数偏微分方程组一般解的构造

求解偏微分方程不仅有理论意义,而且有实用价值.本文利用共轭算子的性质给出构造偏微分方程组一般解的方法.     

多套函数有限元逼近与拟协调板元

继[1]、[2]的工作,本文根据多套函数有限元逼近的思想,建立了唐立民等人[3]、[4]提出的拟协调板元的数学基础.     

非线性学习方法与非线性科学

本文立足于中国传统文化提出非线性学习方法, 使数学物理中国化、简单化、个性化. 从八卦的角度统一看待非线性科学中的孤子、混沌、分形和机器学习, 以诗联网引领学科之间的对话, 实现学科之间的乾坤大挪移.     

Lie Reduction and Conditional Symmetries of Some Variable Coefficient Nonlinear Wave Equations＊

Lie symmetry reduction of some truly ＂variable coefficient＂ wave equations which are singled out from a class of （1 ＋ 1）-dimensional variable coefficient nonlinear wave equations with respect to one and two-dimensional algebras is carried out. Some classes of exact solutions of the investigated equations are found by means of both the reductions and some modern techniques such as additional equivalent transformations and hidden symmetries and so on. Conditional symmetries are also discussed.     

Hyperelliptic Function Solutions of Three Genus for KP Equation Using Direct Method

In this paper, we will use a simple and direct method to obtain some particular solutions of （2＋1）- dimensional and （3＋ 1）-dimensional KP equation expressed in terms of the Kleinian hyperelliptic functions for a given curve y^2 = f（x） whose genus is three. We observe that this method generalizes the auxiliary method, and can obtain the hyperelliptic functions solutions.     

源于Fermi-Pasta-Ulam问题的非线性发展方程的相似约化

利用Clarkson和Kruskal引入的直接法和Lou的改进的直接法,给出了源于Fermi-Pasta-Ulam问题的非线性发展方程的4种类型的相似约化.     

An extension of the direct method and similarity reductions of a generalized Burgers equation with an arbitrary derivative function

In this paper,we extend the well-known direct method proposed by Clarkson and Kruskal for finding similarity reductions of partial differential equations.It follows that some new similarity reductions of the generalized Burgers equation,such as travelling wave reduction,logarithmic reduction,power reduction,rational fractional reduction,etc,are derived,in which some of these cannot be obtained solely by using the direct method.The similarity reductions obtained are interpreted by the nonclassical symmetry Lie group.     

The Hamiltonian equations in some mathematics and physics problems

IntroductionHamiltoniansystemsareveryimportantinmathematics,physicsandmechanics.SpeciallyafterFengKang[1]firstproposedthesymplecticnumericalmethodforfinitedimensionalHamiltoniansystemin 1 980s,thestudyofHamiltoniansystemhasbeenfurtherimprovedinboththeoryandapplication .ItisthereforenaturaltoconsidertheproblemaboutwhichkindofpartialdifferentialequationscanberewrittenorrepresentedbyHamiltoniansystem .SuchkindofproblemhasnowbeenwarmlystudiedbymanymathematiciansandphysicianssuchasinRefs.[2 -5] …