scholarly journals Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Planar Dynamical System ◽  
Wave Solutions

We employ the bifurcation theory of planar dynamical system to investigate the traveling-wave solutions of the generalized Zakharov-Kuznetsov equation. Four important types of traveling wave solutions are obtained, which include the solitary wave solutions, periodic solutions, kink solutions, and antikink solutions.

scholarly journals Periodic and Solitary-Wave Solutions for a Variant of theK(3,2)Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jiangbo Zhou ◽  
Lixin Tian
Keyword(s):  
Solitary Wave ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Polynomial Differential Systems ◽  
Wave Solutions ◽  
Paper Supplement ◽  
Planar Dynamical Systems

We employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to derive new bounded traveling-wave solutions for a variant of theK(3,2)equation. For the focusing branch, we obtain hump-shaped and valley-shaped solitary-wave solutions and some periodic solutions. For the defocusing branch, the nonexistence of solitary traveling wave solutions is shown. Meanwhile, some periodic solutions are also obtained. The results presented in this paper supplement the previous results.

scholarly journals Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Zhao Li ◽  
Peng Li ◽  
Tianyong Han
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Solitary Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Coupled Kdv Equations

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

scholarly journals Traveling Wave Solutions of the Nonlinear -Dimensional Kadomtsev-Petviashvili Equation Using the Two Variables -Expansion Method

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
E. M. E. Zayed ◽  
S. A. Hoda Ibrahim ◽  
M. A. M. Abdelaziz
Keyword(s):  
Solitary Wave ◽  
Periodic Solutions ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
Special Values ◽  
Wave Solutions ◽  
Petviashvili Equation

The two variables -expansion method is proposed in this paper to construct new exact traveling wave solutions with parameters of the nonlinear -dimensional Kadomtsev-Petviashvili equation. This method can be considered as an extension of the basic -expansion method obtained recently by Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions and the trigonometric periodic solutions of this equation were rediscovered from the traveling waves.

scholarly journals Bifurcation of Traveling Wave Solutions of the Dual Ito Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Xinghua Fan ◽  
Shasha Li
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Ito Equation

The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

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 The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Norhashidah Hj. Mohd. Ali
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Simple Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Simple Equation Method ◽  
Bbm Equation

The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.

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 Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

2014 ◽  
Vol 6 (2) ◽  
pp. 273-284 ◽  
Author(s):  
K. Khan ◽  
M. A. Akbar
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Long Wave

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.  Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)  

scholarly journals DISPERSIVE SOLITARY WAVE SOLUTIONS OF COUPLING BOITI-LEON-PEMPINELLI SYSTEM USING TWO DIFFERENT METHODS

2021 ◽  
Vol 21 (1) ◽  
pp. 91-104
Author(s):  
MAHA S.M. SHEHATA ◽  
HADI REZAZADEH ◽  
EMAD H.M. ZAHRAN ◽  
MOSTAFA ESLAMI ◽  
AHMET BEKIR
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
The Other ◽  
Solitary Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Special Value ◽  
Ode Method

In this paper, new exact traveling wave solutions for the coupling Boiti-Leon-Pempinelli system are obtained by using two important different methods. The first is the modified extended tanh function methods which depend on the balance rule and the second is the Ricatti-Bernoulli Sub-ODE method which doesn’t depend on the balance rule. The solitary waves solutions can be derived from the exact wave solutions by give the parameters a special value. The consistent and inconsistent of the obtained solutions are studied not only between these two methods but also with that relisted by the other methods.

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.

Sign in / Sign up

Export Citation Format

Share Document