scholarly journals Periodic Wave Solutions and Their Limit Forms of the Modified Novikov Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical System ◽  
Nonlinear Waves ◽  
Bifurcation Theory ◽  
Dynamic Properties ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Novikov Equation

The modified Novikov equationut-utxx+(b+1)u2ux=buuxuxx+u2uxxxis studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. The existences, dynamic properties, and limit forms of periodic wave solutions forbbeing a negative even are investigated. All possible exact parametric representations of the different kinds of nonlinear waves also are presented.

scholarly journals Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Masoud Mosaddeghi
Keyword(s):  
Numerical Simulation ◽  
Bifurcation Theory ◽  
Parametric Representation ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Zakharov Equation ◽  
Periodic Wave Solutions ◽  
Wave Solutions

By using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized Zakharov equation are classified in to different parametric regions. In each of these parametric regions the exact explicit parametric representation of all solitary, kink (antikink), and periodic wave solutions as well as their numerical simulation and their corresponding phase portraits are obtained.

Bifurcations and Exact Traveling Wave Solutions of the Celebrated Green–Naghdi Equations

2017 ◽  
Vol 27 (07) ◽  
pp. 1750114 ◽  
Author(s):  
Zhenshu Wen
Keyword(s):  
Traveling Wave ◽  
Bifurcation Theory ◽  
Blow Up ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, we study the bifurcations and exact traveling wave solutions of the celebrated Green–Naghdi equations by using the qualitative theory of differential equations and the bifurcation theory of dynamical systems. We obtain all possible phase portraits of bifurcations of the system under various conditions about the parameters associated with the planar dynamical system. Then we show the existence of traveling wave solutions including solitary wave solutions, blow-up solutions, periodic wave solutions and periodic blow-up solutions, and give their exact explicit expressions. These results can help to understand the dynamical behavior of the traveling wave solutions of the system.

scholarly journals Bounded Traveling Waves of the (2+1)-Dimensional Zoomeron Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Yuqian Zhou ◽  
Shanshan Cai ◽  
Qian Liu
Keyword(s):  
Dynamical System ◽  
Traveling Waves ◽  
Simulation Method ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Parameter Bifurcation

The bifurcation method of dynamical system and numerical simulation method of differential equation are employed to investigate the (2+1)-dimensional Zoomeron equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. According to these phase portraits, all bounded traveling waves are identified and simulated, including solitary wave solutions, shock wave solutions, and periodic wave solutions. Furthermore, all exact expressions of these bounded traveling waves are given. Among them, the elliptic function periodic wave solutions are new solutions.

Bifurcations and Exact Solutions of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (II)

2015 ◽  
Vol 25 (03) ◽  
pp. 1550045 ◽  
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Transmission Line ◽  
Singular Systems ◽  
Dynamical Behavior ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Electrical Transmission ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electrical Transmission Line ◽  
Straight Lines

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

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 Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation:mt=-3mux-mxu+kρρx,  ρt=-ρxu+2ρux,the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.

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 Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical System ◽  
Numerical Simulation ◽  
Solitary Wave ◽  
Planar Graphs ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Periodic Cusp Waves

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.

scholarly journals Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Anti-cubic Nonlinearity

2021 ◽  
Author(s):  
Guoan Xu ◽  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Dynamical System ◽  
Optical Waveguides ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Wave Transformation ◽  
Cubic Nonlinearity ◽  
Planar Dynamical System ◽  
Soliton Model ◽  
Wave Solutions

Abstract This paper investigates Raman soliton model in optical metamaterials, having anti-cubic nonlinearity. By travelling wave transformation, the model is transformed into a singular planar dynamical system having three singular straight lines. Using the bifurcation theory method of dynamical systems, under different parameter conditions, bifurcations of phase portraits are studied. More than 30 exact explicit solutions of planar dynamical system are derived, such as exact periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic peakons and peakons as well as compacton solutions. In more general parametric conditions, all possible solutions are found.

scholarly journals Periodic Loop Solutions and Their Limit Forms for the Kudryashov-Sinelshchikov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang ◽  
Yao Long
Keyword(s):  
Dynamical Systems ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. We show that the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. Also, some new exact travelling wave solutions are presented through some special phase orbits.

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