LIE SYMMETRY ANALYSIS AND PAINLEV(E) ANALYSIS OF THE NEW (2+1)-DIMENSIONAL KdV EQUATION

Lie point symmetries associated with the new (2 l)-dimensional KdV equation ut 3uxuy uxxy = 0 are investigated. Some similarity reductions are derived by solving the the soliton solution is obtained directly from the B(a)cklund transformation.     

LIE SYMMETRY ANALYSIS AND PAINLEVE ANALYSIS OF THE NEW(2+l)-DIMENSIONAL KdV EQUATION

Lie point symmetries associated with the new (2 1)-dimensional KdV equation ut 3uxuy uxxy= 0 are investigated. Some similarity reductions are derived by solving the corresponding characteristic equations. Painleve analysis for this equation is also presented and the soliton solution is obtained directly from the Backlund transformation.     

NEW EXPLICIT SOLUTIONS TO THE （2＋1）-DIMENSIONAL BROER-KAUP EQUATIONS

Applying the homogeneous balance method, we have found the explicit and soliton solutions and given a successive formula of finding explicit solutions to the(2+1)-dimensional Broer-Kaup equations. Moreover, by using the Lie group method, we have discussed the similarity solutions to the (2+1)-dimensional Broer-Kanp equations.     

高阶广义KDV方程和KP方程的数值解法

采用一种线性隐格式来解3-阶和5-阶的广义Korteweg-De Vries(KDV)方程,对这种方法做一推广,就能应用到它的二维形式广义Kadomtsew-Petviashvili(KP)方程．这种方法是无条件稳定的,且是无损耗的．数值实验描述了一个线性孤波运动的情形以及两个孤波交互的情形,从结果来看,它们满足孤立子解的两个守恒-动量守恒和能量守恒．     

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

（2＋1）维色散长波方程的扩展椭圆函数有理展开解法

在一个新的更一般的假设下,借助于符号计算,提出了一个椭圆函数有理展开法,并用它统一地求出许多非线性发展方程新的双周期精确解.本文选择(2+1)维色散长波方程作为此方法的应用来加以说明.得到了Yan方法所得的所有解,并且得到更多的一般形式的解.在m取它的极限时,可得到许多冲击波解和孤立波解.     

LUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES

Taking a class of linear(4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial differential equations via symbolic computations,and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.     

A GENERALIZED QUASI-NEWTON EQUATION AND COMPUTATIONAL EXPERIENCE

The quasi-Newton equation has played a central role in the quasi-Newton methods forsolving systems of nonlinear equations and/or unconstrained optimization problems.In-stead,Pan suggested a new equation,and showed that it is of the second order while thetraditional of the first order,in certain approximation sense[12].In this paper,we makea generalization of the two equations to include them as special cases.The generalizedequation is analyzed,and new updates are derived from it.A DFP-like new update out-performed the traditional DFP update in computational experiments on a set of standardtest problems.     

EXACT DARK SOLITON AND ITS PERSISTENCE IN THE PERTURBED(2+1)-DIMENSIONAL DAVEY-STEWARTSON SYSTEM

For the Davey-Stewartson system,the exact dark solitary wave solutions,solitary wave solutions,kink wave solution and periodic wave solutions are studied.To guarantee the existence of the above solutions,all parameter conditions are determined.The persistence of dark solitary wave solutions to the perturbed Davey-Stewartson system is proved.     

广义混合非线性Schr(o)dinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义混合非线性Schrödinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

Fisher方程和Burgers-Fisher方程新的精确行波解

本文研究了Fisher方程和Burgers-Fisher方程.运用一种辅助微分方程方法,得到了这两种非线性偏微分方程新的精确行波解.     

广义Lienard方程边值问题

By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equation a(t)x＂+F(x,x′)x′+g(x)=e(t),x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α| y|β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.     

非饱和土壤水流方程的CN广义差分法

首先给出二维非饱和土壤水流方程时间二阶精度的Crank-Nicolson(CN)时间半离散化格式,然后直接从CN时间半离散化格式出发,建立具有时间二阶精度的全离散化CN广义差分格式,并给出误差分析,最后用数值例子验证全离散化CN广义差分格式的优越性.这种方法能提高时间离散的精度,极大地减少时间方向的迭代步,从而减少实际计算中截断误差的积累,提高计算精度和计算效率.而且该方法可以绕开对空间变量的半离散化广义差分格式的讨论,使得理论研究更简便.     

对流扩散方程的一种新型差分格式

对流扩散方程可以描述众多的物理化学现象，因而对其寻求稳定的，实用的数值解法有着重要的现实意义。本文针对形式较一般的一维非定常对流扩散方程，构造了对角元严格占优的Crank-Nicholson差分格式，然后对其分别用分离变量的方法以及能量估计的方法作了稳定性的分析，最后给出了数值试验的结果，数值结果表明本文构造的格式能够较好的处理经典的Crank-Nicholson格式所不能处理的对流项系数较大的对流扩散方程，并具有较好的精度。     

非线性Schr(o)dinger方程的一个新的守恒差分格式

本文对非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程提出了一种新的带参数的守恒差分格式,并证明了该格式的收敛性与稳定性,通过数值计算获得如下结论,本文提出的差分格式在取适当的参数后,精度上比ＺｈａｎｇＦｅｉ等人（１９９５）的格式有较大幅度的提高。     

RLW方程的一个新的显式多辛格式

以多辛Euler-box格式为基础对正则长波(RLW)方程的初边值问题进行了讨论,推导了一个新的显式10点格式.模拟孤立波的数值实验表明,这个新的多辛格式是行之有效的,能很好的反映出RLW方程的非弹性性质.     

A NEW FORM OF DISCRIMINANT FOR HILL'S EQUATION

1IntroductionHill'sequationoftenhastwoforms.OneistheotheriswhereaandAareconstants,p(t)satisfiesp(t)30pECIorjorp(t)dt>0(see[2]).Obviously,thesetwoformscanbeinterChangedtoeachother.Letgi(t)andyZ(t)betwosolutionsofEq.(1.1)(orEq.(1.2))whichsatisfyinitialvalueconditions:yi(0)=1,yi(0)=0andy2(0)=0,yi(0)=l,respectively.WriteA=yi(r) yi(r).(l.3)ThenaiscalledthediscriminantofEq.(l.1)(orEq.(l.2)).ItiswellknownthatEq.(l.l)(orEq.(1.2))isstableifandonlyifIII<2orlal=2andyi(r)=yi(r)=0.Thetraditionalmet…     

对流扩散方程的数值计算

本文研究了对流扩散方程的一种并行格式.利用一组saul'yev型非对称格式进行二次构造,分别得到了一类并行GE格式和GEL、GER格式;进一步推广,得到绝对稳定的交替分组显式AGE格式,并用数值例子检验AGE格式的数值计算效果.