一类二次KdV类型水波方程的行波解

应用平面动力系统理论研究了一类非线性KdV方程的行波解的动力学行为．在参数空间的不同区域内,给出了系统存在孤立波解,周期波解,扭子和反扭子波解的充分条件,并计算出所有可能的精确行波解的参数表示．     

M.Wadati可积非线性发展方程的精确行波解

用平面动力系统方法研究由M.Wadati提出的一类可积非线性发展方程的精确行波解,获得了该方程的扭波、反扭波解,周期波解和不可数无穷多光滑孤立波解的精确的参数表达式,以及上述解存在的参数条件．     

Jaulent-Miodek方程的行波解分支

利用平面动力系统分支理论研究了耦合的Jaulent-Miodek方程的孤立波及周期波的存在性,求出了分支参数集．在给定的参数条件下,得到了该方程光滑孤立波解及周期行波解的所有可能的显式表达式．     

一类广义四阶非线性Camassa-Holm方程的行波解

用动力系统的分支理论研究了一类广义四阶非线性Camassa-Holm方程的动力学行为和行波解,发现方程存在一些孤立波解,周期波解和一些诸如Compacton类型的非光滑行波解.在不同的参数条件下,给出了这些解存在的条件和一些特殊条件下的精确解.     

(2+1)-维广义Benney-Luke方程的精确行波解

用平面动力系统方法研究(2+1)-维广义Benney-Luke方程的精确行波解,获得了该方程的扭波解,不可数无穷多光滑周期波解和某些无界行波解的精确的参数表达式,以及上述解存在的参数条件．     

一类非线性方程的行波解分支

运用平面动力系统的分支方法,研究了一类非线性方程的行波解,画出了在不同参数条件下的相图,证明方程存在周期行波解和周期尖波解.给出了有界波的精确的参数表达式,指出了周期尖波是周期波的极限形式,同时指出了方程不存在圈孤子解.     

广义Drinfeld-Sokolov方程的行波解的分支

用Ansatz方法和动力系统理论研究了广义Drinfeld-Sokolov方程的行波解．在给定的两组参数条件下,得到了广义Drinfeld-Sokolov方程更多的孤立波解,扭子和反扭子波解及周期波解,并给出这些行波解精确的参数表示．     

广义特殊Tzitzeica-Dodd-Bullough类型方程的行波解

本文研究了广义特殊Tzitzeica-Dodd-Bullough类型方程,利用动力系统分支理论方法,证明该方程存在周期行波解,无界行波解和破切波解,并求出了一些用参数表示的显示精确行波解.     

耦合Higgs方程和Maccari系统的行波解分支

利用动力系统方法,对耦合Higgs方程和Maccari系统的定性行为和行波解进行了研究.基于这种方法,给出了系统在不同参数条件下的相图,得到了包括孤立波解和周期波解在内的行波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解进行了数值模拟.获得的结果完善了相关文献已有的研究成果.     

广义带导数的非线性Schrdinger方程的动态分析和精确解

利用动力系统方法,针对广义带导数的非线性Schrdinger方程的精确解问题进行研究分析.采用行波变换,将其化为常微分方程动力系统;计算出该方程动力系统的首次积分,讨论了系统在不同参数条件下的奇点与相图,得到对应的精确解,包括孤立波解、周期波解、扭结波解和反扭结波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解等进行了数值模拟.分析计算获得的结果完善了相关文献已有的研究成果.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM

Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.     

等离子声波方程的行波解的动力学行为

对等离子声波方程, 用平面动力系统理论得到了其光滑、非光滑孤立波解和不可数无穷多光滑、非光滑周期波解的存在性．进一步,在给定的参数条件下,得到了保证上述解存在的充分条件．     

由一类非线性常微分方程的抛物线解所确定的扭波

通过求解与平面动力系统的两个平衡点相连接的抛物线解,获得了6种非线性行波方程的扭波解存在条件,并给出了这些扭波解的参数表达式,以及上述解存在的参数条件．     

TRAVELINGWAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.     

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic travelingwaves for a class of nonlinear dispersive partial differential equations.By using the bifurcationtheory of dynamical systems to do qualitative analysis,all possible phase portraits in theparametric space for the traveling wave systems are obtained.It can be shown that the existenceof a singular straight line in the traveling wave system is the reason why smooth solitary wavesolutions converge to solitary cusp wave solution when parameters are varied.The differentparameter conditions for the existence of solitary and periodic wave solutions of different kindsare rigorously determined.     

Explicit Traveling Wave Solutions and Their Dynamical Behaviors for the Coupled Higgs Field Equation

In this paper, we focus on the traveling wave solutions of the coupled Higgs field equation from the perspective of dynamical systems. Through the phase portraits, in addition to periodic wave solutions and solitary wave solutions, we also obtain explicit periodic singular wave solutions, singular wave solutions and kink wave solutions, which were not found in the previous works. The dynamical behavior of these solutions and their internal relations are revealed through asymptotic analysis. The results will help supplement the study of field equation.