Solitary wave and chaotic behavior of traveling wave solutions for the coupled KdV equations

2011 ◽  
Vol 218 (5) ◽  
pp. 1794-1797 ◽  
Author(s):  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Chaotic Behavior ◽  
Traveling Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Coupled Kdv Equations

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 Water Wave Solutions of the Coupled System Zakharov-Kuznetsov and Generalized Coupled KdV Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
A. R. Seadawy ◽  
K. El-Rashidy
Keyword(s):  
Solitary Wave ◽  
Traveling Wave Solutions ◽  
Coupled System ◽  
Solitary Wave Solutions ◽  
Coupled Partial Differential Equations ◽  
Extended Tanh Method ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Tanh Method ◽  
Coupled Kdv Equations

An analytic study was conducted on coupled partial differential equations. We formally derived new solitary wave solutions of generalized coupled system of Zakharov-Kuznetsov (ZK) and KdV equations by using modified extended tanh method. The traveling wave solutions for each generalized coupled system of ZK and KdV equations are shown in form of periodic, dark, and bright solitary wave solutions. The structures of the obtained solutions are distinct and stable.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS (II): AN EXAMPLE OF GCKdV EQUATIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 1995-2007 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
XIAOHUA ZHAO
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Coupled Kdv Equations

By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

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.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

2018 ◽  
Vol 15 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Aly R. Seadawy
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

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.

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