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.

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.

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].

Dynamic interaction of behaviors of time-fractional shallow water wave equation system

2021 ◽  
pp. 2150353
Author(s):  
Serbay Duran
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Equation System ◽  
Wave Transformation ◽  
Trigonometric Functions ◽  
Shallow Water Wave ◽  
Wave Solutions

In this study, the traveling wave solutions for the time-fractional shallow water wave equation system, whose physical application is defined as the dynamics of water bodies in the ocean or seas, are investigated by [Formula: see text]-expansion method. The nonlinear fractional partial differential equation is transformed to the non-fractional ordinary differential equation with the use of a special wave transformation. In this special wave transformation, we consider the conformable fractional derivative operator to which the chain rule is applied. We obtain complex hyperbolic and complex trigonometric functions for the time-fractional shallow water wave equation system with the help of this technique. New traveling wave solutions are obtained for the special values given to the parameters in these complex hyperbolic and complex trigonometric functions, and the behavior of these solutions is examined with the help of 3D and 2D graphics.

EXTENDED TANH-METHODS AND SOLITONIC SOLUTIONS OF THE GENERAL SHALLOW WATER WAVE MODELS

2004 ◽  
Vol 15 (03) ◽  
pp. 363-370 ◽  
Author(s):  
WOO-PYO HONG ◽  
SEOUNG-HWAN PARK
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Wave Models

Based on the two recent extended tanh-function methods, we find traveling-wave solutions for the general shallow water wave models. The obtained solutions include periodical, singular and solitary-wave solutions.

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.

Exact Traveling Wave Solutions and Bifurcations for a Shallow Water Equation Modeling Surface Waves of Moderate Amplitude

2016 ◽  
Vol 26 (10) ◽  
pp. 1650172
Author(s):  
Wenjing Zhu ◽  
Jibin Li
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Shallow Water Equation ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Equation Modeling ◽  
Wave System ◽  
Wave Solutions ◽  
Straight Line

In this paper, we consider the traveling wave solutions for a shallow water equation. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions, as well as periodic peakon solutions. We show the existence of compacton solutions of the equation under different parameter conditions.

Exploration of the algebraic traveling wave solutions of a higher order model

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jian-Gen Liu ◽  
Yi-Ying Feng ◽  
Hong-Yi Zhang
Keyword(s):  
Dynamical Systems ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Wave Phenomena ◽  
Planar Dynamical Systems

Purpose The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena. Design/methodology/approach The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions. Findings As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections. Research limitations/implications The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure. Practical implications By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections. Originality/value To the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.

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.

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