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．     

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.     

Bifurcations of travelling wave solutions for a two-component camassa-holm equation

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of travelling wave solutions are listed.     

Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

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.     

Traveling wave solutions for Generalized Drinfeld–Sokolov equations

In this paper, solitary waves and periodic waves for Generalized Drinfeld–Sokolov equations are studied, by using the theory of dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained.     

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．     

The Exact Solutions of Generalized Zakharov Equations with High Order Singular Points and Arbitrary Power Nonlinearities

In this paper, we using bifurcation theory method of dynamical systems to find the exact solutions of generalized Zakharov equations with high order singular points and arbitrary power nonlinearities. Under different parameter conditions, we obtain exact solitary wave solutions, periodic wave solutions as well as kink and anti-kink wave solutions.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

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．     

一类变形的Boussinesq方程组的行波解分支

在Boussinesq方程组求解方面,用平面动力系统的分支理论研究了一类变形的Boussinesq方程组的行波解分支．得到了不同参数条件下的分支集、相图及所有孤立波和扭波的精确公式．     

K(n,2n,-n)方程行波解的分支

本文研究了K(n,2n,-n)方程行波解与参数a,b,c,g,n等的关系.利用动力系统分支理论,得到了孤立波、扭结和反扭结波解,以及不可数无穷多光滑周期波解的存在性.本文推广了文献[1]中的结果.     

Bifurcation of traveling wave solutions for the generalized compound KDV equations

By using the theory of bifurcations of dynamical systems to the generalized compound KDV equations, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By using the method of planar dynamical systems to an integrable nonlinear wave equation, the existence of periodic travelling wave, solitary wave and kink wave solutions is proved in the different parametric conditions. The phase portraits of the travelling wave system are given. It can be shown that the existence of singular curves in the travelling wave system is the reason why the travelling wave solutions lose their smoothness. Moreover, the so-called W/M-shaped solitary wave solutions are obtained.     

Solitary Waves, periodic peakons, pseudo-peakons and compactons given by three ion-acoustic wave models in electron plasmas

The nonlinear ion-acoustic oscillations models are governed by three partial differential equation systems. Their travelling wave equations are three first class singular traveling wave systems depending on different parameter groups, respectively. By using the method of dynamical system and the theory of singular traveling wave systems, in this paper, it is shown that there exist parameter groups such that these singular systems have solitary wave solutions, pseudo-peakons, periodic peakons and compactons as well as kink and anti-kink wave solutions. The results of this paper complete the studies of three papers [5,13] and [14].     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.     

Bifurcations of traveling wave solutions for the generalized coupled Hirota–Satsuma KdV system

In this paper, the bifurcations of solitary, kink and periodic waves for the generalized coupled Hirota–Satsuma KdV system are studied by using the bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas for solitary wave solutions, kink wave solutions and periodic wave solutions are obtained.     

Geometric Properties and Exact Travelling Wave Solutions for the Generalized Burger-Fisher Equation and the Sharma-Tasso-Olver Equation

In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained.     

Bifurcations of travelling wave solutions in the (N + 1)-dimensional sine–cosine-Gordon equations

In this paper, the (N + 1)-dimensional sine–cosine-Gordon equations are studied. The existence of solitary wave, kink and anti-kink wave, and periodic wave solutions are proved, by using the method of bifurcation theory of dynamical systems. All possible bounded exact explicit parametric representations of the above travelling solutions are obtained.     

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.