一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

二阶Camassa-Holm方程行波解

对二阶Camassa-Holm方程行波解的情况进行了讨论.利用解的唯一性,得到了如下结论:二阶CH方程的行波解唯一存在,但不具有u(x,t)=kem(x-ct)形式.还为二阶CH方程行波解的研究提供了一种新途径和方法.     

一类反应扩散方程的行波解

1　引　　言由于反应扩散方程涉及的大量问题来自物理学、化学、生物学和人口动力学中众多的数学模型,因而有广阔的实际背景.其行波解引起了人们的兴趣,行波解是某个常微分方程的解,对某些传播速度,利用几何方法可以建立其解的存在性(见[1][2][3]).在文[4]中J.Canosa讨论了Fisher方程ut=2u2x+u(1-u)(1)行波解的存在性、逼近解和误差估计.所谓方程(1)的行波解是指形为u(x,t)=u(x-ct)=u(z)的解.众所周知,行波解u(x,t)=u(x-ct)=u(z)是方程(1)的行波解的充要条件是d2udz2+cdudz+u(1-u)=0(2)若u(z)是单调有界且不恒为常数,则u(z)叫做(1)的波前…     

用Hirota双线性导数变换求MNLS方程的Rogue波解

修正的非线性薛定谔方程(MNLS方程)与导数非线性薛定谔方程(DNLS方程)是两个紧密相关且完全可积的非线性偏微分方程.该文通过Hirota双线性导数变换方法,首先求得MNLS方程在平面简谐波背景下的空间周期解,即Akhmediev型呼吸子解,再通过长波极限得其Rogue波解.根据简单的参数归零法使之自然地约化为DNLS方程的Rogue波解,并借助于一个积分变换将其变换为Chen-Lee-Liu方程的Rogue波解.文章还简要讨论了MNLS方程和DNLS方程在非局域情形整体解的存在性问题.     

非紧空间上折现Hamilton-Jacobi方程粘性解的存在性讨论*

当底空间紧时, 初始函数为连续函数的Lax-Oleinik型粘性解是局部半凹的,所以是相应的Hamilton-Jacobi\ (以下简称为H-J) 演化方程(简称为接触H-J方程)的粘性解.当底空间非紧时, 对于H-J方程和接触H-J方程, 其Lax-Oleinik型解的下确界未必能取到.文章将探讨在非紧空间上, 折现H-J方程粘性解有限性的条件, 并给出了在此假设下粘性解的表达式.     

非齐次Burgers方程解的渐近性行为

构造了非齐次Burgers方程的解,方程服从有界和紧致的初始曲线[Kloosterziel RC.J Engrg Math,1990,24(3):213-236],作了一个有趣的探索.将热方程初值问题(L2(R,ex2/2)中有初值)的解,表示为该热方程自相似解的一个级数,Kloosterziel方法立即显示出该初值问题解的渐近性行为.受Kloosterziel方法的启发,根据热方程的自相似解,来表示非齐次Burgers方程的解.最后得到该非齐次Burgers方程解的渐近性特征.     

矩阵方程X-A*X~(-1)A+B*X~(-2)B=I正定解存在的充分条件

通过构造单调有界迭代序列,研究矩阵方程X-A~*X~(-1)A+B~*X~(-2)B=I的艾米特正定解.给出了方程正定解存在的充分条件及正定解的范围.     

Whitham-Broer-Kaup浅水波方程的Backlund变换和精确解

用一种新的方法并借助Mathematica,求出了Whitham-Broer-Kaup(简记WBK)方程的一种Backlund变换,并建立了WBK方程与热传导方程及Burgers方程的联系.利用这种关系得到了WBK方程的三组精确解,其中一组为孤波解.     

借“1／x＋a＋1／x＋b=1／x”发挥

题中方程的实数解 ,众所周知 ,仅当 ab>0时 ,x=± ab.如果 a,b∈ N ,a≠ b,且只需解这方程的正整数解 ,那么必须满足 ab=k2 ( k∈N) .本文仅借方程的构形 ,发挥它正整数解的功能 ,来揭示“一个单位分数表示为两个不同单位分数的和”的实质 . (注 1)由上得 :1k=1k a 1k b ( 1)令 k2 ( =ab) =pe11 · pe22 … peii … pett.( pi 为素数 ) .不难推出 ( 1)的不同表示法的种数 =d( k2 ) -12 .(其中 d( k2 ) =( e1 1) ( e2 1)… ( ei 1)… ( et 1) ) .令 ( 1)中 a=k2 ,b=1.经移项整理得1k( k 1) =1k-1k 1( 2 )由 ( 2 )可求出 ( n-1)个形如它左…     

一类带强色散项DGH方程的精确解

利用改进的F-展开法,求出了一类带强色散项DGH方程的一系列类孤子解,三角函数周期解和有理数解,方程结合了KdV方程的线性色散项和C-H方程的非线性色散项.而且改进的F-展开法在借助于计算机符号系统Mathematica(Maple)下,操作方便,适用于大量的非线性偏微分方程(组),并有助于发现新解.     

Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method

We extended the (G′/G)-expansion method to two well-known nonlinear differential-difference equations, the discrete nonlinear Schrödinger equation and the Toda lattice equation, for constructing traveling wave solutions. Discrete soliton and periodic wave solutions with more arbitrary parameters, as well as discrete rational wave solutions, are revealed. It seems that the utilized method can provide highly accurate discrete exact solutions to NDDEs arising in applied mathematical and physical sciences.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

A note on “New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation”

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.     

The Riccati Equation Method Combined with the Generalized Extended $(G''/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

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．     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

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.     

KK方程和改进的Boussinesq方程的新精确解

应用改进的Fan's代数方法,得到了KK方程和改进的Boussinesq方程的一系列新精确解,包括孤立波解、类孤立波解、纽结波解、奇异纽结波解和三角函数周期解.     

The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.