Travelling Wave Solutions to Stretched Beam's Equation: Phase Portraits Survey

2011 ◽  
Vol 55 (4) ◽  
pp. 605-608 ◽  
Author(s):  
Gambo Betchewe ◽  
Kuetche Kamgang Victor ◽  
Bouetou Bouetou Thomas ◽  
Timoleon Crepin Kofane
Keyword(s):  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Wave Solutions

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 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.

Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

2010 ◽  
Vol 140 (2) ◽  
pp. 241-257
Author(s):  
Weiguo Zhang ◽  
Lanyun Bian ◽  
Yan Zhao
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Travelling Wave ◽  
Oscillatory Solution ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Oscillatory Solutions ◽  
Wave Solutions ◽  
Damped Oscillatory Solution

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ < λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ > λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

scholarly journals Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hang Zheng ◽  
Yonghui Xia ◽  
Yuzhen Bai ◽  
Guo Lei
Keyword(s):  
Dynamical System ◽  
Equilibrium Points ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
System Method ◽  
Dynamical System Method ◽  
Nonzero Equilibrium ◽  
Derivative Order

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

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.

scholarly journals Nonlinear Dynamics and Exact Traveling Wave Solutions of the Higher-Order Nonlinear Schrödinger Equation with Derivative Non-Kerr Nonlinear Terms

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Heng Wang ◽  
Longwei Chen ◽  
Hongjiang Liu ◽  
Shuhua Zheng
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Travelling Wave ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

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