scholarly journals On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhixi Shen ◽  
Yujuan Wang ◽  
Hamid Reza Karimi ◽  
Yongduan Song
Keyword(s):  
Differential Equations ◽  
Energy Loss ◽  
Shallow Water ◽  
Water Waves ◽  
Wave Breaking ◽  
Water Wave ◽  
Water System ◽  
Shallow Water Waves ◽  
Two Component ◽  
Shallow Water System

This paper investigates the multipeakon dissipative behavior of the modified coupled two-component Camassa-Holm system arisen from shallow water waves moving. To tackle this problem, we convert the original partial differential equations into a set of new differential equations by using skillfully defined characteristic and variables. Such treatment allows for the construction of the multipeakon solutions for the system. The peakon-antipeakon collisions as well as the dissipative behavior (energy loss) after wave breaking are closely examined. The results obtained herein are deemed valuable for understanding the inherent dynamic behavior of shallow water wave breaking.

scholarly journals On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Yujuan Wang ◽  
Yongduan Song ◽  
Hamid Reza Karimi
Keyword(s):  
Shallow Water ◽  
Wave Breaking ◽  
Water System ◽  
Original System ◽  
Collision Time ◽  
Two Component ◽  
Shallow Water System

The global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm shallow water system after wave breaking was studied in this paper. The underlying approach is based on a skillfully defined characteristic and a set of newly introduced variables which transform the original system into a Lagrangian semilinear system. It is the transformation, together with the associated properties, that allows for the continuity of the solution beyond collision time to be established, leading to a uniquely global dissipative solution, which constructs a semigroup, and the multipeakon dissipative solution.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

scholarly journals X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

2021 ◽  
Author(s):  
Yuan Shen ◽  
Bo Tian ◽  
Tian-Yu Zhou ◽  
Xiao-Tian Gao
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Water Waves ◽  
Water Wave ◽  
Soliton Solutions ◽  
Hirota Method ◽  
Shallow Water Waves ◽  
Soliton Resonance ◽  
Hybrid Solutions

Abstract Water waves are observed in the rivers, lakes, oceans, etc. Under investigation in this paper is a (2+1)-dimensional Hirota-Satsuma-Ito system arising in the shallow water waves. Via the Hirota method and symbolic computation, we derive some X-type and resonance Y-type soliton solutions. We also work out some hybrid solutions consisting of the resonance Y-type solitons, solitons, breathers and lumps. Graphics we present reveal that the hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps describe the interactions between the resonance Y-type solitons and solitons/breathers/lumps, respectively. The obtained results rely on the water-wave coefficient in that system.

scholarly journals A fractal variational theory of the Broer-Kaup system in shallow water waves

2021 ◽  
pp. 87-87
Author(s):  
Wei-Wei Ling ◽  
Pin-Xia Wu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Theory ◽  
Shallow Water Waves ◽  
Solution Structures ◽  
Fractal Space ◽  
Variational Formulations

The Broer-Kaup equation is one of many equations describing some phenomena of shallow water wave. There are many errors in scientific research because of the existence of the non-smooth boundaries. In this paper, we generalize the Broer-Kaup equation to the fractal space and establish fractal variational formulations through the semi-inverse method. The acquired fractal variational formulation reveals conservation laws in an energy form in the fractal space and suggests possible solution structures of the morphology of the solitary waves.

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