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.

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.

Smooth Exact Traveling Wave Solutions Determined by Singular Nonlinear Traveling Wave Systems: Two Models

2019 ◽  
Vol 29 (04) ◽  
pp. 1950047
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Shengfu Deng
Keyword(s):  
Traveling Wave ◽  
Singular System ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Regular System ◽  
Wave System ◽  
Optical Metamaterials ◽  
Wave Solutions ◽  
Straight Line ◽  
Kink Wave

For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED TWO-COMPONENT CAMASSA–HOLM EQUATION

2012 ◽  
Vol 22 (12) ◽  
pp. 1250305 ◽  
Author(s):  
JIBIN LI ◽  
ZHIJUN QIAO
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Breaking Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Kink Wave

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation

2020 ◽  
Vol 30 (07) ◽  
pp. 2050109
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Two Component System ◽  
Wave Solutions ◽  
Venant Equation

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.

New Non-traveling Solitary Wave Solutions for a Second-order Korteweg-de Vries Equation

1999 ◽  
Vol 54 (6-7) ◽  
pp. 375-378 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Young-Dae Jung
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Vries Equation ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Second Order ◽  
Solitary Wave Solutions ◽  
Current Interest ◽  
Wave Solutions

Abstract Modeling the propagation of two different wave modes simultaneously, the second-order KdV equation is of current interest. Applying a tanh-typed method with symbolic computation, we have found certain new analytic soliton-typed solutions which go beyond the the previously obtained traveling wave solutions.

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.

BIFURCATIONS OF TRAVELING WAVE SOLUTIONS IN A MICROSTRUCTURED SOLID MODEL

2013 ◽  
Vol 23 (01) ◽  
pp. 1350009 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Homoclinic Orbits ◽  
Phase Portraits ◽  
Solid Model ◽  
Wave System ◽  
Bounded Solutions ◽  
Wave Solutions ◽  
Kink Wave

The traveling wave system of a microstructured solid model belongs to the second class of singular traveling wave equations studied in [Li et al., 2009]. In this paper, by using methods from dynamical systems theory, bifurcations of phase portraits of such a traveling wave system are analyzed in its corresponding parameter space. The existence of kink wave solutions and uncountably infinitely many bounded solutions is proved. Moreover, the exact parametric representations of periodic solutions and homoclinic orbits are obtained.

scholarly journals Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Weiguo Rui
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.

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