EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR THE KUDRYASHOV–SINELSHCHIKOV EQUATION

2012 ◽  
Vol 22 (05) ◽  
pp. 1250118 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Parametric Plane

By using the approach of dynamical systems, the bifurcations of phase portraits for the traveling system of the Kudryashov–Sinelshchikov equation with ν = δ = 0 are studied, in different parametric regions of (α, c)-parametric plane. Corresponding to different phase orbits of the traveling system, more than 26 exact explicit traveling wave solutions are derived. The dynamics of singular nonlinear traveling system is completely determined.

BIFURCATIONS OF EXACT TRAVELING WAVE SOLUTIONS FOR THE (2 + 1)-DIMENSIONAL mKP EQUATION

2012 ◽  
Vol 22 (03) ◽  
pp. 1250051
Author(s):  
YUANFEN XU ◽  
ZHENXIANG DAI
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Wave System ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

For the (2 + 1)-dimensional mKP equation, what is the dynamical behavior of its traveling wave solutions and how does it depend on the parameters of the systems? This paper will answer these questions by using the methods of dynamical systems. Thirteen exact explicit parametric representations of the traveling wave solutions are given. Bifurcations of phase portraits of the corresponding singular traveling wave system are shown.

scholarly journals Exact Traveling Wave Solutions and Bifurcation of a Generalized (3+1)-Dimensional Time-Fractional Camassa-Holm-Kadomtsev-Petviashvili Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zhigang Liu ◽  
Kelei Zhang ◽  
Mengyuan Li
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Wave System ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Fractional Complex Transform ◽  
Wave Solutions ◽  
Petviashvili Equation

In this paper, we study the (3+1)-dimensional time-fractional Camassa-Holm-Kadomtsev-Petviashvili equation with a conformable fractional derivative. By the fractional complex transform and the bifurcation method for dynamical systems, we investigate the dynamical behavior and bifurcation of solutions of the traveling wave system and seek all possible exact traveling wave solutions of the equation. Furthermore, the phase portraits of the dynamical system and the remarkable features of the solutions are demonstrated via interesting figures.

Exact Cuspon and Compactons of the Novikov Equation

2014 ◽  
Vol 24 (03) ◽  
pp. 1450037 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Breaking Wave ◽  
Wave Solutions ◽  
Novikov Equation

In this paper, we apply the method of dynamical systems to the traveling wave solutions of the Novikov equation. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system and exact cuspon wave solution, as well as a family of breaking wave solutions (compactons). We find that the corresponding traveling system of Novikov equation has no one-peakon solution.

EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR THE GENERALIZED POCHHAMMER–CHREE EQUATIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250233 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the method of dynamical systems to study the generalized Pochhammer–Chree equations, the dynamics of traveling wave solutions are characterized under different parameter conditions. Some exact parametric representations of the traveling wave solutions are obtained. Thus, many results reported in the literature can be completed.

EXISTENCE OF EXACT FAMILIES OF TRAVELING WAVE SOLUTIONS FOR THE SIXTH-ORDER RAMANI EQUATION AND A COUPLED RAMANI EQUATION

2012 ◽  
Vol 22 (01) ◽  
pp. 1250002 ◽  
Author(s):  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Sixth Order ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the method of dynamical systems and the results in [Li & Zhang, 2011] to the sixth-order Ramani equation and a coupled Ramani equation, the families of exact traveling wave solutions can be obtained.

scholarly journals New groups of solutions to the Whitham-Broer-Kaup equation

2020 ◽  
Vol 41 (11) ◽  
pp. 1735-1746
Author(s):  
Yaji Wang ◽  
Hang Xu ◽  
Q. Sun
Keyword(s):  
Dynamical Systems ◽  
Riccati Equation ◽  
Bifurcation Analysis ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Tsunami Waves ◽  
Wave Solutions ◽  
Planar Dynamical Systems ◽  
Projective Riccati Equation Method

Abstract The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.

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.

On the Exact Traveling Wave Solutions to the van der Waals p-System

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
M. Bilal ◽  
M. Younis ◽  
H. Rezazadeh ◽  
T. A. Sulaiman ◽  
A. Yusuf ◽  
...  
Keyword(s):  
Traveling Wave ◽  
Van Der Waals ◽  
Traveling Wave Solutions ◽  
P System ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

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.

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