Understanding Peakons, Periodic Peakons and Compactons via a Shallow Water Wave Equation

2016 ◽  
Vol 26 (12) ◽  
pp. 1650207 ◽  
Author(s):  
Jibin Li ◽  
Wenjing Zhu ◽  
Guanrong Chen
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Wave Solution ◽  
Water Wave ◽  
Solitary Wave Solution ◽  
Shallow Water Equation ◽  
Wave Model ◽  
Periodic Wave Solution ◽  
Wave System ◽  
Shallow Water Wave

In this paper, a shallow water wave model is used to introduce the concepts of peakon, periodic peakon and compacton. Traveling wave solutions of the shallow water equation are presented. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.

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.

Completing the Study of Traveling Wave Solutions for Three Two-Component Shallow Water Wave Models

2020 ◽  
Vol 30 (03) ◽  
pp. 2050036 ◽  
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Two Component ◽  
Wave Models

For three two-component shallow water wave models, from the approach of dynamical systems and the singular traveling wave theory developed in [Li & Chen, 2007], under different parameter conditions, all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are derived. More than 19 explicit exact parametric representations are obtained. Of more interest is that, for the integrable two-component generalization of the Camassa–Holm equation, it is found that its [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions. In addition, its [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions. The new results complete a recent study by Dutykh and Ionescu-Kruse [2016].

scholarly journals Approximate traveling wave solution for shallow water wave equation

2012 ◽  
Vol 36 (4) ◽  
pp. 1550-1557 ◽  
Author(s):  
A.A. Imani ◽  
D.D. Ganji ◽  
Houman B. Rokni ◽  
H. Latifizadeh ◽  
Esmail Hesameddini ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Traveling Wave ◽  
Wave Solution ◽  
Water Wave ◽  
Traveling Wave Solution ◽  
Shallow Water Wave

Bifurcations and Exact Solutions of Generalized Two-Component Peakon Type Dual Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950128
Author(s):  
Jianli Liang ◽  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solution ◽  
Wave System ◽  
Dual Systems ◽  
Two Component

This paper investigates two generalized two-component peakon type dual systems, which can be reduced to the same planar dynamical systems via the dynamical system approach and the theory of singular traveling wave systems, where one of them contains the two-component Camassa–Holm system. By bifurcation analysis on the corresponding traveling wave system, we obtain the phase portraits and derive possible exact traveling wave solutions that include solitary wave solution, peakon and anti-peakon, pseudo-peakon, periodic peakon, compacton and periodic wave solution. Our results are also applicable to the two-component Camassa–Holm equation.

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