小展弦比波的Green-Naghdi渐进模型的行波解（英文）

本文研究了小展弦比波的Green-Naghdi渐进模型.利用平面自治系统的稳定性分析方法,在不同的参数条件下,讨论了它的行波系统的分岔并且给出了对应的相图,得到了光滑周期波解,广义扭波解,广义反扭波解,广义紧波解,周期尖波解,孤波解和孤立尖波解的精确表达式.进一步,通过数学软件Maple模拟了这些解.     

广义Gilson-Pickering方程的分支解

本文利用动力系统方法和奇行波方程理论研究广义Gilson-Pickering方程的动力学行为和行波解.利用软件画出了给定参数条件下系统的相图分支,得到了孤立波解、扭结波解和反扭结波解、不可数无穷多破缺波解、光滑周期波解和非光滑周期尖波解、尖孤子解的存在性.在β≠1,p=2时,对于广义Gilson-Pickering方程不同的参数条件下,给出了保证上述解存在的条件及参数表示.     

一类变式Boussinesq系统的行波解

本文研究—类变式Boussinesq系统ηt+((1+αη)w)x-β/6wxxx=0, wt+αwwx+ηx-β/2wxxt=0,其中α和β都是正常数.许多逼近模型都能从此系统中被推导出,比如Boussinesq系统和两分量Camassa-Holm系统等.本文利用平面动力系统方法研究它的行波解及相图,得到了孤立波解,广义扭波解,广义反扭波解,紧孤立波解和周期波解,并给出了这些解的数值模拟.     

广义Camassa-Holm方程的显示尖波解

通过构造辅助微分方程,求得了广义Camassa-Holm方程的尖波解,此解包含了由椭圆函数表达的周期尖波解,推广了有关文献的结果.     

含频散项的K(2,3)方程的精确行波解（英文）

本文研究了包含频散项的K(2,3)方程u_t+(u~2)_x-(u~3)_(xxx)=0的分支问题.利用动力系统的定性分析,并且借助Maple软件进行数值模拟得到行波解系统相应的相图,然后通过积分计算得到周期尖波解、类扭波和类反扭波的精确解的函数表达式,以及孤立波精确解的隐函数表达式.     

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Camassa-Holm方程的行波解

运用平面动力系统理论和方法给出了广义Camassa-Holm方程在各种参数条件下的相图与分支,分析了奇线对其行波解的影响,获得了广义Camassa-Holm方程光滑、非光滑孤立波解和周期波解的存在性及个数,求出了它的两组新周期尖波解的显式表达式.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

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.     

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.     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

Bifurcation and the exact smooth,cusp solitary and periodic wave solutions of the generalized Kudryashov–Sinelshchikov equation

The main aim of this paper is to study the exact traveling wave solutions of the generalized Kudryashov–Sinelshchikov equation by using the auxiliary equation method based on the conclusion of qualitative analysis. The advantage of this method is to choose the effective and proper auxiliary equation on the base of the behaviors and traits of solutions revealed by analysis of phase portraits to study the solution of differential equations. By applying the proposed approach to the generalized Kudryashov–Sinelshchikov equation, the number, behavior and existence of smooth and non-smooth traveling wave solutions are gained, at the same time, the new exact smooth solitary, periodic wave solutions and cusp solitary, periodic wave solutions are obtained. From the dynamic point of view, the behavior of traveling wave solutions is analyzed. The profile,type and the form of exact expression of traveling wave solutions are influenced by the order of nonlinear term and nonlinear terms.

     

Smooth and non-smooth traveling wave solutions of the Fornberg-Whitham equation with linear dispersion term

In this paper, the Fornberg-Whitham equation with linear dispersion term is investigated by employing the bifurcation method of dynamical systems. As a result, the existence of smooth and non-smooth traveling wave solutions is obtained. And the analytic expressions of solitary wave solutions, periodic cusp wave solutions and peakons are given under some parameter conditions.     

Bifurcations and New Traveling Wave Solutions for the Nonlinear Dispersion Drinfel'd-Sokolov ($D(m,n)$) System

In this paper, we employ the theory of the planar dynamical system to investigate the dynamical behavior and bifurcations of solutions of the traveling systems of the $D(m,n)$ equation. On the basis of the previous work of the reference \cite{zhang}, we obtain the solitary cusp waves solutions (peakons and valleyons), breaking wave solutions (compactons) and other periodic cusp wave solutions. Morever, we make a summary of exact traveling wave solutions to the $D(m,n)$ system including all the solutions which have been found from the references \cite{Deng,Xie,zhang}.     

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.     

Symbolic computation of exact solutions for a nonlinear evolution equation

In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here.     

Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation

The bounded traveling wave solutions of a generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.