Compacton and generalized kink wave solutions of the CH-DP equation

The CH-DP equation is investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of compactons and generalized kink waves are simulated by using mathematical software Maple. Exact explicit parameter expressions of compactons and implicit expressions of generalized kink wave solutions are given, and the dynamic characters of these solutions are investigated.     

Bifurcations of traveling wave solutions in a compound KdV-type equation

By using the theory of planar dynamical systems to a compound KdV-type nonlinear wave equation, the bifurcation boundaries of the system are obtained in this paper. These bifurcation sets divide the parameter space into different regions, which correspond to qualitatively different phase portraits and therefore different types of the solutions may exist in different regions. The parameter conditions for the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions are therefore obtained.     

Exact compacton and generalized kink wave solutions of the extended reduced Ostrovsky equation

The extended reduced Ostrovsky equation (EX-ROE) are investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of the compactons and the generalized kink waves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink wave solutions are given. The dynamic behavior of these solutions are also investigated.     

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.     

带色散项的Degasperis-Procesi方程的孤立尖波解

用动力系统的定性分析理论研究了带有色散项的Degasperis-Procesi方程的孤立尖波解.在一定的参数条件下,利用Degasperis-Procesi方程对应行波系统的相图分支从两种不同方式给出了孤立尖波解的表达式.     

Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the nonlinear dispersive Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.     

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.     

On the solitary wave solutions of the CQNLS

In this paper, coexistence and simplified formulations of the solitary waves of the cubic–quintic non-linear Schrödinger equation (CQNLS) are investigated by analyzing the steady bifurcation and the energy integral of the conservative dynamical system satisfied by the wave packet. It is found that the bright solitary waves can coexist with kinks and anti-kinks in a range of the bifurcation control parameter. There exists a critical parameter value at which the dark solitary waves are distinguished from the bright solitary waves, kinks and anti-kinks. All of the simplified solitary wave solutions, kinks and anti-kinks are obtained by using our previously developed approximate method.     

Bifurcations and parametric representations of traveling wave solutions for the (2+1)‐dimensional Boiti–Leon–Pempinelle system

By using the method of bifurcation theory of planar dynamical systems to the traveling wave system of the (2+1)‐dimensional Boiti–Leon–Pempinelle system, exact explicit parametric representations of the traveling wave solutions are obtained in different parameter regions. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Exact solitary wave solutions of the generalized (2 + 1) dimensional Boussinesq equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the generalized (2 + 1) dimensional Boussinesq equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Several types of periodic wave solutions and their relations of a Fujimoto--Watanabe equation

In this paper, we study periodic wave solutions of a Fujimoto--Watanabe equation by exploiting the bifurcation method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space, and then give the sufficient conditions to guarantee the existence of several types of periodic wave solutions. What"s more, we present their exact expressions and reveal their inside relations as well as their relations with solitary wave solutions.     

New exact solutions for the nonlinear wave equation with fifth-order nonlinear term

The purpose of this paper is to reveal the dynamical behavior of the nonlinear wave equation with fifth-order nonlinear term, and provides its bounded traveling wave solutions. Applying the bifurcation theory of planar dynamical systems, we depict phase portraits of the traveling wave system corresponding to this equation under various parameter conditions. Through discussing the bifurcation of phase portraits, we obtain all explicit expressions of solitary wave solutions and kink wave solutions. Further, we investigate the relation between the bounded orbit of the traveling wave system and the energy level h. By analyzing the energy level constant h, we get all possible periodic wave solutions.     

Sub-manifold and traveling wave solutions of Ito's 5th-order mKdV equation

In this paper, we study Ito''s 5th-order mKdV equation with the aid of symbolic computation system and by qualitative analysis of planar dynamical systems. We show that the corresponding higher-order ordinary differential equation of Ito''s 5th-order mKdV equation, for some particular values of the parameter, possesses some sub-manifolds defined by planar dynamical systems. Some solitary wave solutions, kink and periodic wave solutions of the Ito''s 5th-order mKdV equation for these particular values of the parameter are obtained by studying the bifurcation and solutions of the corresponding planar dynamical systems.     

Planar bifurcation method of dynamical system for investigating different kinds of bounded travelling wave solutions of a generalized Camassa-Holm equation

In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated.     

Cantor families of periodic solutions for wave equations via a variational principle

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities.     

Exact traveling wave solutions and bifurcations for the Dullin-Gottwald-Holm equation

Utilizing the methods of dynamical system theory, the Dullin-Gottwald-Holm equation is studied in this paper. The dynamical behaviors of the traveling wave solutions and their bifurcations are presented in different parameter regions. Furthermore, the exact explicit forms of all possible bounded solutions, such as solitary wave solutions, periodic wave solutions and breaking loop wave solutions are obtained.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.