scholarly journals Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Weiguo Rui
Keyword(s):  
Water Wave ◽  
Kdv Equation ◽  
Dynamic Properties ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Function Type ◽  
Bifurcation Method ◽  
Wave Solutions

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equationut+ux+νuxxt+βuxxx+αuux+1/3να(uuxxx+2uxuxx)+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx)+ν2μα2(ux2uxxx+2uxuxx2)=0is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.

More on Bifurcations and Dynamics of Traveling Wave Solutions for a Higher-Order Shallow Water Wave Equation

2019 ◽  
Vol 29 (01) ◽  
pp. 1950014
Author(s):  
Jibin Li ◽  
Guanrong Chen
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Periodic Wave ◽  
Wave System ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Straight Lines

This paper studies the dynamics of traveling wave solutions to a shallow water wave model with a large-amplitude regime in phase space. The corresponding traveling wave system is a singular planar dynamical system with two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are obtained. The existence of solitary wave solutions, periodic wave solutions, peakon, pseudo-peakon solution, periodic peakon solutions and compacton solutions are determined under different parameter conditions.

Dynamics of Traveling Wave Solutions to a New Highly Nonlinear Shallow Water Wave Equation

2017 ◽  
Vol 27 (03) ◽  
pp. 1750044 ◽  
Author(s):  
Jibin Li ◽  
Kit Ian Kou
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Periodic Wave ◽  
Wave System ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Highly Nonlinear

In this paper, the dynamics of traveling wave solutions in a shallow water wave model with a regime for large-amplitude is studied. The corresponding traveling wave system is a singular planar dynamical system with one or two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are presented. The existence of solitary wave solutions, periodic wave solutions, quasi-peakon solution, periodic peakon solutions and compacton solutions under different parameter conditions are determined.

scholarly journals A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Traveling Wave ◽  
Kdv Equation ◽  
Dynamic Properties ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Dark Soliton ◽  
Exact Traveling Wave Solutions ◽  
Generalized Kdv Equation ◽  
Wave Solutions

We study a generalized KdV equation of neglecting the highest order infinitesimal term, which is an important water wave model. Some exact traveling wave solutions such as singular solitary wave solutions, semiloop soliton solutions, dark soliton solutions, dark peakon solutions, dark loop-soliton solutions, broken loop-soliton solutions, broken wave solutions of U-form and C-form, periodic wave solutions of singular type, and broken wave solution of semiparabola form are obtained. By using mathematical softwareMaple, we show their profiles and discuss their dynamic properties. Investigating these properties, we find that the waveforms of some traveling wave solutions vary with changes of certain parameters.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

scholarly journals Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Point Of View ◽  
Bifurcation Method ◽  
Wave Solutions

The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

scholarly journals Traveling Wave Solutions for the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

scholarly journals Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Periodic Waves ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.

scholarly journals Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Shaoyong Li ◽  
Ming Song
Keyword(s):  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Wave Solutions ◽  
Two Component ◽  
New Phenomena ◽  
Whitham Equation

Using bifurcation method and numerical simulation approach of dynamical systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.

scholarly journals Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type forK(m,n)Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the integral bifurcation method, we study the nonlinearK(m,n)equation for all possible values ofmandn. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions ofK(n,n),K(2n−1,n),K(3n−2,n),K(4n−3,n), andK(m,1)equations are chosen to illustrate with the concrete features.

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