The solitary wave solutions of the nonlinear perturbed shallow water wave model

2020 ◽  
Vol 103 ◽  
pp. 106202 ◽  
Author(s):  
Jianjiang Ge ◽  
Zengji Du
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Water Wave ◽  
Wave Model ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

scholarly journals Dynamics of Bright Solitary-waves in a General Fifth-order Shallow Water-wave Model

2004 ◽  
Vol 59 (4-5) ◽  
pp. 257-265
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Water Wave ◽  
Wave Model ◽  
Wave Number ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Fifth Order

New analytic sech2-type traveling solitary-wave solutions, satisfying zero background at infinity, of a general fifth-order shallow water-wave model are found and compared with previously obtained non-zero background solutions. The allowed coefficient regions for the solitary-wave solutions are classified by requiring the wave number and angular frequency to be real. Detailed numerical simulations are performed to demonstrate the stability of the solitary-waves and to show the soliton-like behavior of two interacting solitary-waves. For a large nonlinear term we show the formation of a bounded state of two solitary-waves, called bion, which travels as a single coherent structure. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

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].

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.

Exact Traveling Wave Solutions and Bifurcations of Classical and Modified Serre Shallow Water Wave Equations

2019 ◽  
Vol 29 (12) ◽  
pp. 1950153
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Wave Equations ◽  
Wave Theory ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Shallow Water Wave Equations

Using the dynamical systems analysis and singular traveling wave theory developed by Li and Chen [2007] to the classical and modified Serre shallow water wave equations, it is shown that, in different regions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) can be obtained. More than 28 explicit and exact parametric representations are precisely derived. It is demonstrated that, more interestingly, the modified Serre equation has uncountably infinitely many smooth solitary wave solutions and uncountably infinitely many pseudo-peakon solutions. Moreover, it is found that, differing from the well-known peakon solution of the Camassa–Holm equation, the modified Serre equation has four new forms of peakon solutions.

Solitary wave solutions, fusionable wave solutions, periodic wave solutions and interactional solutions of the (3+1)-dimensional generalized shallow water wave equation

2021 ◽  
pp. 2150389
Author(s):  
Ai-Juan Zhou ◽  
Bing-Jie He
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

In this paper, we study exact solutions of the generalized shallow water wave equation. Based on the bilinear equation, we get [Formula: see text]-solitary wave solutions. For special parameters, we find [Formula: see text]-fusionable wave solutions. For complex parameters, periodic wave solutions and elastic interactional solutions of solitary waves with periodic waves are obtained. The properties of obtained exact solutions are also analyzed theoretically and graphically by using asymptotic analysis.

Exact Solitary-wave Solutions for the General Fifth-order Shallow Water-wave Models

1999 ◽  
Vol 54 (3-4) ◽  
pp. 272-274
Author(s):  
Woo-Pyo Hong ◽  
Young-Dae Jung
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Symbolic Computation ◽  
Water Wave ◽  
Model Parameters ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Special Cases ◽  
Wave Models ◽  
Fifth Order

We perform a computerized symbolic computation to find some general solitonic solutions for the general fifth-order shal-low water-wave models. Applying the tanh-typed method, we have found certain new exact solitary wave solutions. The pre-viously published solutions turn out to be special cases with restricted model parameters.

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.

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