利用Lie群方法求Burgers-Huxley方程行波类首次积分

本文运用Lie群理论,证明了Burgers-Huxley方程的行波解所满足的二阶非线性方程在参数满足一定关系时,在经典意义下接受一个两参数Lie群,此时可用积分法求其首次积分.     

时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先,利用含时滞的动力学Hamilton原理,建立了含时滞的非保守系统的分段Lagrange运动方程;其次,利用微分方程容许Lie群理论,得到系统的Lie对称确定方程;然后,根据对称性与守恒量之间的关系,通过构造结构方程,得到含时滞的非保守系统的Lie定理;最后,给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性,相应的Lie对称性确定方程的个数应是自由度数目的2倍,这对生成元函数提出了更高的限制,同时,守恒量呈现依赖速度项的分段表达。     

几类非线性差分方程的对称和精确解

本文将微分方程的Lie变换群方法推广到差分方程,给出了三类非线性差分方程的不变变换,利用这种变换由差分方程的平凡解得到非平凡的单参数解族。     

变质量非Четаев型非完整系统在相空间的Lie对称性与守恒量

在相空间引入无限小群变换,研究变质量非Четаев型非完整系统的Lie对称和守恒量.利用系统运动微分方程在无限小群变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量,并举例说明结果的应用.     

GdKP方程的最优系统和群不变解

利用经典李群方法对Gd KP方程进行Lie对称分析,求得该方程的Lie对称代数,及其相应的约化方程和最优系统.更进一步,作者求出了d KP方程的部分群不变解.该方法在物理中有广泛的应用.     

利用单参数Lie群组的一种可解性求自治系统首次积分的方法

讨论了自治系统接受的单参数Lie群组具有一种可解性的情况下求系统的一个首次积分的具体方法.对于n阶自治系统,给出相应参数的一组确定取值,求得系统首次积分；对于三阶自治系统,当系统接受的单参数Lie群组可解时,验证求得首次积分的条件一定成立.     

基于Lie群的刚体动力学建模及数值计算方法研究

基于Lie群和Lie代数之间的指数映射等价关系,推导了基于Lie群的自由刚体连续动力学方程.结合离散变分原理,推导了其Lie群离散变分积分子.通过证明可知连续和离散动力学系统都具有动量守恒性.对连续动力学方程进行同维化处理,使其变为常规非线性方程组的形式,利用Runge-Kutta法进行求解;基于Runge-Kutta基本理论,推导了直接用于Lie群的Runge-Kutta法,从而使Runge-Kutta法可用于求解变维非线性方程组;通过Lie代数变换,利用Kelly变换和Newton迭代对Lie群离散变分积分子进行求解.仿真对比结果表明,3种算法下的计算结果高度吻合,且能高精度地保持系统的结构守恒和动量守恒性.     

耦合Burgers方程的Painleve分析和相关性质

为研究耦合Burgers方程的可积性,利用WTC测试方法,给出了第一类Burgers方程的Painleve性质和第二类Burgers方程的条件Painleve性质．进而得到了第一类方程的变量分离解和第二类方程的（N2＋3N＋6/2）-参数Lie点对称群．     

耦合Burgers方程的Painleve分析和相关性质

为研究耦合Burgers方程的可积性，利用WTC测试方法，给出了第一类Burgers方程的Painleve性质和第二类Burgers方程的条件Painleve性质．进而得到了第一类方程的变量分离解和第二类方程的（N2＋3N＋6/2）-参数Lie点对称群．     

用两个单参数Lie群求三阶自治系统的积分因子

给出并证明了利用三阶自治系统接受的两个单参数Lie群的生成元求系统积分因子的力法.     

The lie group and integrability of the Fisher type travelling wave equation

For the Fisher-type wave equation, which has two stable states and one unstable state, it is proved that only in two particular cases, the corresponding travelling wave equation admits a double parameter Lie group, and based on a method different to the traditional one, its two independent first integrals are given. It is proved further that in the two integrable cases, the different bounded and non-trivial travelling wave solutions, which are corresponding the invariant manifolds of the corresponding equation under the Lie transformation, can be expressed with elementary functions although they cannot be obtained directly from the two independent first integrals.     

Classification of the travelling waves in the nonlinear dispersive KdV equation

In this paper, we classify the travelling wave solutions to the nonlinear dispersive KdV equation (called K(2, 2) equation). The parameter region is specified and the parameter dependence of its solitary waves is described. Besides the previously known compacton solutions, the equation is shown to admit more new solutions such as cuspons, peakons, loopons, stumpons and fractal-like waves. Furthermore, by the qualitative results, we give some new explicit travelling wave solutions.     

非线性发展方程行波解的符号计算

Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.     

On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.     

应用(1/G)-展开法求解五阶KdV方程

本篇论文首次提出(1/G) -展开法,用于求解非线性演化方程的行波解.将该法应用于五阶KdV方程的求解,当参数满足一定条件时,该方程可化为Sawada-Kotera (SK)方程、Caudrey-Dodd-Gibbon(CDG)方程、Kaup-Kupershmidt (KK)方程、Lax方程和Ito方程.其解可被表示为...     

Bifurcation of travelling wave solutions for the modified dispersive water wave equation

The method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. In different regions, different types of travelling solutions including solitary wave solutions, shock wave solutions and periodic wave solutions are simulated. Furthermore, with a generalized projective Riccati equation method, several new explicit exact solutions are obtained.