scholarly journals Traveling Wave Solutions of a Generalized Camassa-Holm Equation: A Dynamical System Approach

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Lina Zhang ◽  
Tao Song
Keyword(s):  
Differential System ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Single Peak ◽  
Analytical Technique ◽  
Polynomial Differential System ◽  
Holm Equation ◽  
Kink Wave ◽  
Transformation Of Variables ◽  
Planar Polynomial Differential System

We investigate a generalized Camassa-Holm equationC(3,2,2):ut+kux+γ1uxxt+γ2(u3)x+γ3ux(u2)xx+γ3u(u2)xxx=0. We show that theC(3,2,2)equation can be reduced to a planar polynomial differential system by transformation of variables. We treat the planar polynomial differential system by the dynamical systems theory and present a phase space analysis of their singular points. Two singular straight lines are found in the associated topological vector field. Moreover, the peakon, peakon-like, cuspon, smooth soliton solutions of the generalized Camassa-Holm equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for single peak soliton, kink wave, and kink compacton solutions of theC(3,2,2)equation.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA–HOLM EQUATION

2013 ◽  
Vol 23 (03) ◽  
pp. 1350057 ◽  
Author(s):  
JIBIN LI ◽  
ZHIJUN QIAO
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Breaking Wave ◽  
Wave System ◽  
Wave Solutions ◽  
Holm Equation ◽  
Kink Wave ◽  
Quadratic And Cubic Nonlinearities

In this paper, we study all possible traveling wave solutions of an integrable system with both quadratic and cubic nonlinearities: [Formula: see text], m = u-uxx, where b, k1 and k2 are arbitrary constants. We call this model a generalized Camassa–Holm equation since it is kind of a cubic generalization of the Camassa–Holm (CH) equation: mt + mxu + 2mux = 0. In the paper, we show that the traveling wave system of this generalized Camassa–Holm equation is actually a singular dynamical system of the second class. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. Some exact solutions such as smooth soliton solutions, kink and anti-kink wave solutions, M-shape and W-shape wave profiles of the breaking wave solutions are obtained. To guarantee the existence of those solutions, some constraint parameter conditions are given.

New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

2021 ◽  
Author(s):  
Xiao Hong ◽  
A. G. Davodi ◽  
S. M. Mirhosseini-Alizamini ◽  
M. M. A. Khater ◽  
Mustafa Inc
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Fractional Heat Equation ◽  
Multiple Soliton Solutions ◽  
Holm Equation ◽  
Hyperbolic Method

Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated [Formula: see text]-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the [Formula: see text]-fractional heat equation is found. Graphical representations are also given.

Various Exact Soliton Solutions and Bifurcations of a Generalized Dullin–Gottwald–Holm Equation with a Power Law Nonlinearity

2017 ◽  
Vol 27 (08) ◽  
pp. 1750129 ◽  
Author(s):  
Temesgen Desta Leta ◽  
Jibin Li
Keyword(s):  
Power Law ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Wave System ◽  
Wave Solutions ◽  
Holm Equation ◽  
Exact Soliton Solutions ◽  
Planar Dynamical Systems

In this paper, we study a model of generalized Dullin–Gottwald–Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010].

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

2016 ◽  
Vol 8 (6) ◽  
pp. 1084-1098
Author(s):  
Wei Wang ◽  
Chunhai Li ◽  
Wenjing Zhu
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Single Peak ◽  
Dynamical System Theory ◽  
Solitary Wave Solutions ◽  
Analytical Technique ◽  
Wave Solutions

AbstractDynamical system theory is applied to the integrable nonlinear wave equation ut±(u3–u2)x+(u3)xxx=0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

A (2 + 1)-dimensional extension of the Benjamin-Ono equation

2018 ◽  
Vol 28 (11) ◽  
pp. 2681-2687 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Content Type ◽  
Multiple Soliton Solutions ◽  
Proposed Model ◽  
New Findings ◽  
Propagation Of Waves ◽  
Model Features ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

Purpose The purpose of this paper is concerned with developing a (2 + 1)-dimensional Benjamin–Ono equation. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for this equation. Design/methodology/approach The proposed model has been handled by using the Hirota’s method. Other techniques were used to obtain traveling wave solutions. Findings The examined extension of the Benjamin–Ono model features interesting results in propagation of waves and fluid flow. Research limitations/implications The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple soliton solutions. Practical implications This work is entirely new and provides new findings, where although the new model gives multiple soliton solutions, it is nonintegrable. Originality/value The work develops two complete sets of multiple soliton solutions, the first set is real solitons, whereas the second set is complex solitons.

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