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.

New soliton solutions of the CH–DP equation using Lie symmetry method

2018 ◽  
Vol 32 (20) ◽  
pp. 1850234 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Symmetry Method

In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

EXACT TRAVELING WAVE SOLUTIONS OF A HIGHER-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

2010 ◽  
Vol 24 (10) ◽  
pp. 1011-1021 ◽  
Author(s):  
JONU LEE ◽  
RATHINASAMY SAKTHIVEL ◽  
LUWAI WAZZAN
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Higher Dimensional

The exact traveling wave solutions of (4 + 1)-dimensional nonlinear Fokas equation is obtained by using three distinct methods with symbolic computation. The modified tanh–coth method is implemented to obtain single soliton solutions whereas the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions for this higher-dimensional integrable equation. The Exp-function method gives generalized wave solutions with some free parameters. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

scholarly journals Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Can Chen ◽  
Weiguo Rui ◽  
Yao Long
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Dynamic Behaviors ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, by using the integral bifurcation method, we studied the Kudryashov-Sinelshchikov equation. In the special parametric conditions, some singular and nonsingular exact traveling wave solutions, such as periodic cusp-wave solutions, periodic loop-wave solutions, smooth loop-soliton solutions, smooth solitary wave solutions, periodic double wave solutions, periodic compacton solutions, and nonsmooth peakon solutions are obtained. Further more, the dynamic behaviors of these exact traveling wave solutions are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameters.

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.

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.

Bifurcations and Exact Solitary Wave, Compacton and Pseudo-Peakon Solutions in a Modified Generalized KdV Equation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Jianping Shi ◽  
Jibin Li
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Wave System ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Kink Wave ◽  
Straight Lines ◽  
The Given

A modified generalized KdV equation is considered in this paper. Under the given parameter conditions, the corresponding traveling wave system is a singular planar dynamical system with three singular straight lines. The bifurcations and traveling wave solutions of the system are investigated in the parameter space from the perspective of dynamical systems. The existence of solitary wave solutions, periodic peakon solutions, pseudo-peakon solutions, kink and anti-kink wave solutions and compactons is proved. Furthermore, possible exact explicit parametric representations of various solutions are given. Particularly, the model has uncountably infinite many solitary wave and pseudo-peakon solutions.

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