Bifurcations of Solitary Waves of a Simple Equation

2020 ◽  
Vol 30 (09) ◽  
pp. 2050138 ◽  
Author(s):  
Jiaopeng Yang ◽  
Rui Liu ◽  
Yiren Chen
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Wave ◽  
Wave Speed ◽  
Blow Up ◽  
Periodic Wave ◽  
Simple Equation ◽  
Singular Line ◽  
Wave System ◽  
Bifurcation Method

In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line. Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties: (i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation. Secondly, we perform numerical simulations to visualize the above properties. Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

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.

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.

scholarly journals Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Blow Up ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Bbm Equation

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary 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.

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.

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 Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Weiguo Rui
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.

scholarly journals Application of Bifurcation Method to the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Ming Song
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
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 blow-up wave solutions and solitary wave solutions.

scholarly journals New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yiren Chen ◽  
Shaoyong Li
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Bifurcation Method ◽  
Bifurcation Phenomenon ◽  
Wave Solutions ◽  
Bifurcation Phenomena

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

scholarly journals Some Further Results on Traveling Wave Solutions for the ZK-BBM() Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Bifurcation Method ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions ◽  
Limit Form

We investigate the traveling wave solutions for the ZK-BBM() equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.

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