Pseudo-Peakon, Periodic Peakons and Compactons on a Shallow Water Model with Coriolis Effect

2021 ◽  
Vol 31 (08) ◽  
pp. 2150144
Author(s):  
Zhenshu Wen ◽  
Guanrong Chen ◽  
Jibin Li
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Solitary Wave Solution ◽  
Wave Theory ◽  
Water Model ◽  
Wave System ◽  
Coriolis Effect ◽  
Bounded Solutions ◽  
Shallow Water Model

For a shallow water model with Coriolis effect, by applying the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to its traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solution, pseudo-peakon and periodic peakons as well as compactons) are obtained. Some exact explicit parametric representations are presented.

Bifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component Systems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150001
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Yan Zhou
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Wave Theory ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Two Component

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its [Formula: see text]-traveling wave system has a pseudo-peakon wave solution. In addition, its [Formula: see text]-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].

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].

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 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.

On the Cauchy problem for the shallow-water model with the Coriolis effect

2019 ◽  
Vol 267 (11) ◽  
pp. 6370-6408
Author(s):  
Yongsheng Mi ◽  
Yue Liu ◽  
Ting Luo ◽  
Boling Guo
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Water Model ◽  
Coriolis Effect ◽  
Shallow Water Model ◽  
The Cauchy Problem

On a shallow-water model with the Coriolis effect

2019 ◽  
Vol 267 (5) ◽  
pp. 3232-3270 ◽  
Author(s):  
Ting Luo ◽  
Yue Liu ◽  
Yongsheng Mi ◽  
Byungsoo Moon
Keyword(s):  
Shallow Water ◽  
Water Model ◽  
Coriolis Effect ◽  
Shallow Water Model

The instability of vertically curved fronts in a two-layer shallow water model

2020 ◽  
Vol 32 (12) ◽  
pp. 124117
Author(s):  
M. W. Harris ◽  
F. J. Poulin ◽  
K. G. Lamb
Keyword(s):  
Shallow Water ◽  
Water Model ◽  
Shallow Water Model

scholarly journals Modelling Weirs in Two-Dimensional Shallow Water Models

Water ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2152
Author(s):  
Gonzalo García-Alén ◽  
Olalla García-Fonte ◽  
Luis Cea ◽  
Luís Pena ◽  
Jerónimo Puertas
Keyword(s):  
Experimental Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Experimental Dataset ◽  
River Hydraulics ◽  
Shallow Water Models ◽  
Water Models

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.

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