scholarly journals On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

2017 ◽  
Vol 37 (3) ◽  
pp. 1509-1537 ◽  
Author(s):  
Huijun He ◽  
◽  
Zhaoyang Yin ◽  
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Water Wave ◽  
Higher Order ◽  
Wave System ◽  
Shallow Water Wave ◽  
Two Component ◽  
The Cauchy Problem

Non-uniform dependence on initial data for the two-component fractional shallow water wave system

2020 ◽  
Vol 192 ◽  
pp. 111714
Author(s):  
Shouming Zhou ◽  
Shihang Pan ◽  
Chunlai Mu ◽  
Honglin Luo
Keyword(s):  
Initial Data ◽  
Shallow Water ◽  
Water Wave ◽  
Wave System ◽  
Shallow Water Wave ◽  
Two Component

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 Camassa-Holm equation on shallow water wave equation

2017 ◽  
Vol 5 (12) ◽  
pp. 7758-7764
Author(s):  
Sh. Hajrulla, L. Bezati, F. Hoxha
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Weak Solution ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Kdv Equation ◽  
Shallow Water Wave ◽  
Holm Equation ◽  
The Cauchy Problem

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption  Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution  to the Cauchy problem for  Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given

scholarly journals The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ying Wang ◽  
YunXi Guo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Sufficient Conditions ◽  
Global Weak Solution ◽  
Dissipative Term ◽  
Shallow Water Wave ◽  
Global Strong Solution ◽  
Special Cases ◽  
The Cauchy Problem

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that(1-∂x2)u0∈M+(R),u0∈H1(R),andu0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.

Study of the Variable Separation Solutions and Chaotic Behaviors for the Extended (2+1)-Dimensional Shallow Water Wave System

2013 ◽  
Vol 329 ◽  
pp. 144-147
Author(s):  
Xiao Xin Zhu ◽  
Song Hua Ma ◽  
Qing Bao Ren
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Water Wave ◽  
Mapping Method ◽  
Separation Method ◽  
Wave System ◽  
Variable Separation ◽  
Wave Excitation ◽  
Shallow Water Wave ◽  
Variable Separation Method

With the mapping method and a variable separation method, a series of variable separation solutions to the extended (2+1)-dimensional shallow water wave (ESWW) system is derived. Based on the derived solitary wave excitation, some chaotic behaviors are investigated.

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