scholarly journals Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Point Of View ◽  
Bifurcation Method ◽  
Wave Solutions

The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

scholarly journals Traveling Wave Solutions for the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. 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.

Bifurcations of Exact Traveling Wave Solutions of Maccari's Equations

2013 ◽  
Vol 328 ◽  
pp. 580-584
Author(s):  
Hong Xian Zhou
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Qualitative Theory ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave

For the Maccari's equations, the objective of this paper is to investigate the dynamical behavior of its traveling wave solutions by using the bifurcation method and qualitative theory of dynamical systems. All exact explicit parametric respresentations of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained under given parameter conditions, and the dynamics characters of these solutions are also analyzed.

Traveling Wave Solutions of a Two-Component Dullin–Gottwald–Holm System

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Jiyu Zhong ◽  
Shengfu Deng
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Planar Systems

In this paper, we investigate the traveling wave solutions of a two-component Dullin–Gottwald–Holm (DGH) system. By qualitative analysis methods of planar systems, we investigate completely the topological behavior of the solutions of the traveling wave system, which is derived from the two-component Dullin–Gottwald–Holm system, and show the corresponding phase portraits. We prove the topological types of degenerate equilibria by the technique of desingularization. According to the dynamical behaviors of the solutions, we give all the bounded exact traveling wave solutions of the system, including solitary wave solutions, periodic wave solutions, cusp solitary wave solutions, periodic cusp wave solutions, compactonlike wave solutions, and kinklike and antikinklike wave solutions. Furthermore, to verify the correctness of our results, we simulate these bounded wave solutions using the software maple version 18.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED TWO-COMPONENT CAMASSA–HOLM EQUATION

2012 ◽  
Vol 22 (12) ◽  
pp. 1250305 ◽  
Author(s):  
JIBIN LI ◽  
ZHIJUN QIAO
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Breaking Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Kink Wave

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation

2020 ◽  
Vol 30 (07) ◽  
pp. 2050109
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Two Component System ◽  
Wave Solutions ◽  
Venant Equation

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.

scholarly journals Bifurcations of Traveling Wave Solutions for the Coupled Higgs Field Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Shengqiang Tang ◽  
Shu Xia
Keyword(s):  
Field Equation ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Higgs Field ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions

By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the above waves are determined.

scholarly journals Peakon, Solitary and Smooth Periodic Wave Solutions for a Modification of theK(2,2)Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Point Of View ◽  
Phase Portraits ◽  
Periodic Waves ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We consider a modification of theK(2,2)equationut=2uuxxx+2kuxuxx+2uuxusing the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, some peakons, solitary, and smooth periodic waves are found and their exact parametric representations are presented. Also, the coexistence of peakon and solitary wave solutions is investigated.

scholarly journals Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Periodic Waves ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.

scholarly journals Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Can Chen ◽  
Weiguo Rui ◽  
Yao Long
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Dynamic Behaviors ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, by using the integral bifurcation method, we studied the Kudryashov-Sinelshchikov equation. In the special parametric conditions, some singular and nonsingular exact traveling wave solutions, such as periodic cusp-wave solutions, periodic loop-wave solutions, smooth loop-soliton solutions, smooth solitary wave solutions, periodic double wave solutions, periodic compacton solutions, and nonsmooth peakon solutions are obtained. Further more, the dynamic behaviors of these exact traveling wave solutions are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameters.

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