Dynamical understanding of loop soliton solution for several nonlinear wave equations

It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.     

Existence of Travelling Wave Solution of Nonlinear Equations with Nonlocal Advection

In this paper, the existence of travelling wave solution for nonlinear equation wiili non local advection ρ\frac{∂}{∂t}(\frac{u^m}{m}) = \frac{∂²u}{∂x²}-\frac{∂}{∂x}[φ(k∗u)u]+u^nf(u) is studied in the case of m ≥ 1, n ≥ 1. When ε,φ, f, m and n satisfy some determinate conditions, there exists the travelling wave solution.     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

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.     

The bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation

By using methods from dynamical systems theory, this paper researches the bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation. Implicit exact parametric representations of all travelling wave solutions as well as some explicit analytic solutions are given. Specially, breaking wave solutions are obtained, which KdV equation does not include.     

Bifurcations of exact travelling wave solutions for the generalized R-K-L equation

Using the method of dynamical systems for the the generalized Radhakrishnan, Kundu, Lakshmanan equation, the existence of soliton solutions, uncountably infinite many periodic wave solutions and unbounded wave solution are obtained. Exact explicit parametric representations of the above travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

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.     

Limit periodic travelling wave solution of a model for biological invasions

In this paper, we consider a class of biological invasion model with density-dependent migrations and Allee effect, which is reduced to one ordinary differential form via the travelling wave solution ansatz. For the corresponding planar system, we firstly obtain the first several weak focal values of its one equilibrium by computing the singular point quantities, then determine the existence of one stable limit cycle from its Hopf bifurcation. Thus a special periodic travelling wave solution which is isolate as a limit is obtained, and it corresponds to the particular real patterns of spread during biological invasions, which is an interesting discovery.     

一个曲面波方程的精确解

Abstract. Some exact travelling wave solutions and rational travelling wave solutions of a sur-face wave equation in a convecting fluid are given in this paper.     

BIFURCATIONS AND NEW EXACT TRAVELLING WAVE SOLUTIONS OF THE COUPLED NONLINEAR SCHRDINGER-KdV EQUATIONS

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrdinger-KdV equations are studied. Based on this method, all phase portraits of the system in the parametric space are given. All possible bounded travelling wave solutions such as solitary wave solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, the numerical simulations are conducted for solitary wave solutions and periodic travelling wave solutions to the coupled nonlinear Schrdinger-KdV equations. The results show that the presented findings improve the related previous conclusions.     

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.     

双函数法及一类非线性发展方程的精确行波解

给出一种求解非线性发展方程精确行波解的新方法：双函数法。使用此方法，获得了一类非线性发展方程的许多精确行波解，其中包括孤波解和周期解，推广了文献用其它方法取得的结果，同时还获得了许多新的弧波解和周期解，借助于Mathemat-ica，此方法能部分地在计算机上实现。     

广义变形的DP-CH方程的显式精确解

通过构造一个辅助方程,利用动力系统方法,Jacobi椭圆函数,来讨论广义变形的DP-CH方程,得到了广义变形的DP-CH方程更丰富的行波解的显示精确表示.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

Travelling wave solutions of the Fornberg-Whitham equation

Zhou and Tian [J.B. Zhou, L.X. Tian, A type of bounded travelling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261] successfully found a type of bounded travelling wave solutions of the Fornberg-Whitham equation. In this paper, we improve the previous result by using the phase portrait analytical technology. Moreover, some smooth periodic wave, smooth solitary wave, periodic cusp wave and loop-soliton solutions are given, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.     

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.     

Long-time behaviour of solutions of a class of nonlinear parabolic equations

In this paper, the long-time behaviour of solutions of a class of nonlinear parabolic equations is studied. It is shown that the solutions of initial-boundary value problem to the equations converge to a travelling wave solution of the equation or a self-similar solution of a Hamilton–Jacobi equation under certain conditions on initial and boundary values of the solutions.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

This paper deals with the construction of a nonstandard numerical method to compute the travelling wave solutions of nonlinear reaction diffusion equations at high wave speeds. Related general properties are studied using the perturbation approximation. At high wave speed the perturbation parameter approaches to zero and the problem exhibits a multiscale character. That is, there are thin layers where the solution varies rapidly, while away from these layers the solution behaves regularly and varies slowly. Most of the conventional methods fail to capture this layer behavior. Thus, the quest for some new numerical techniques that may handle the travelling wave solutions at high wave speeds earns relevance. In this paper, one such parameter robust nonstandard numerical scheme is constructed, in the sense that its numerical solution converges in the maximum norm to the exact solution uniformly well for all finite wave speeds. To overcome the difficulty due to the nonlinearity, the problem is linearized using the quasilinearization process followed by nonstandard finite difference discretization. An extensive amount of analysis is carried out which uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiment is carried out in support of the predicted theory that validates computationally the theoretical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007