Boussinesq-Burgers方程的分岔及一些有界行波解

利用平面动力系统理论研究了Boussinesq-Burgers方程,讨论了方程在行波变换后所对应的平面动力系统的分岔行为,并基于相平面上特定的相轨道求出了该方程的扭结波、孤立波及周期波的解析表达式.数值模拟进一步验证了所得结论的正确性.     

Konopelchenko-Dubrovsky方程的周期解

By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.     

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.     

Bifurcations and exact traveling wave solutions of the equivalent complex short-pulse equations

In this paper, we study the traveling wave solutions for a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrodinger (NLS) equation in the ultrashort-pulse regime. The corresponding traveling wave systems of the equivalent complex short-pulse equations are two singular planar dynamical systems with four singular straight lines. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.     

On the new solutions of the generalized Burgers–Huxley equation

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

(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．     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.     

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.     

Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems北大核心CSCD

该文研究了一类格竞争系统的双稳周期行波解的存在性.首先,将两种群竞争系统转化为合作系统;其次,构造合作系统的上下解,并建立比较原理,得到当初始函数满足一定条件时,解在无穷远处是收敛的;最后,利用黏性消去法证明系统连接两个稳定周期平衡点的行波解的存在性.     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

使用解析方法获得FitzHugh-Nagumo方程新的peakon解

依据对FitzHugh-Nagumo方程的研究,通过微分变化法近似分析出FitzHugh-Nagumo方程,获得了这个方程的尖峰孤立波（peakon soliton）的解,从而获得了更多形式的peakon解,同时也分析了微分变换法（differential transform method, DTM）收敛区域和收敛速度．构建的微分变换法,结合帕德(Padé)逼近,构建一个明确的,完全解析,对FitzHugh-Nagumo方程全部有意义的尖波解．其主要思想是限制边界条件而令导数在孤立波不存在峰值,但导数的孤立波在两侧存在．结果表明,微分变换法在参数很小的情况下可以避免摄动的限制．表明这种方法提供了一种强大而有效地获得FitzHugh-Nagumo方程新的peakon解的数学方法．     

Existence of kink and unbounded traveling wave solutions of the Casimir equation for the Ito system

This paper study the traveling wave solutions of the Casimir equation for the Ito system. Since the derivative function of the wave function is a solution of a planar dynamical system, from which the exact parametric representations of solutions and bifurcations of phase portraits can be obtained. Thus, we show that corresponding to the compacton solutions of the derivative function system, there exist uncountably infinite kink wave solutions of the wave equation. Corresponding to the positive or negative periodic solutions and homoclinic solutions of the derivative function system, there exist unbounded wave solutions of the wave function equation.     

Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation

In this paper, the traveling wave solutions for a generalized Camassa-Holm equation $u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}$ are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized Camassa-Holm equation are given for $p$ either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.     

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}.     

Soliton and Periodic Wave Solutions of the Nonlinear Loaded (3+1)-Dimensional Version of the Benjamin-Ono Equation by Functional Variable Method

In this article, we establish new travelling wave solutions for the nonlinear loaded (3+1)-dimensional version of the Benjamin-Ono equation by the functional variable method. The performance of this method is reliable and effective and the method provides the exact solitary wave solutions and periodic wave solutions. The solution procedure is very simple and the traveling wave solutions are expressed by hyperbolic functions and trigonometric functions. After visualizing the graphs of the soliton solutions and the periodic wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It has been shown that the method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.     

广义Pochhammer-Chree方程的新孤波解和余弦周期波解

本文利用假设待定法求出了具5阶非线性项的广义Pochhammer-Chree方程具双曲正割函数分式形式的2个新孤波解和6个余弦函数周期波解,并分别给出了它们的有界性条件.揭示了行波波速v的改变与钟状孤波解和余弦周期波解波形变化的相关性.     

具有局部时间分数阶导数的非线性Zoomeron方程的精确解

In this paper, we are concerned with the nonlinear Zoomeron equation with local conformable time-fractional derivative. The concept of local conformable fractional derivative was newly proposed by R. Khalil et al. The bifurcation and phase portrait analysis of traveling wave solutions of the nonlinear Zoomeron equation are investigated. Moreover, by utilizing the exp(-?(ε))-expansion method and the first integral method, we obtained various exact analytical traveling wave solutions to the Zoomeron equation such as solitary wave, breaking wave and periodic wave.     

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.     

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.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.