Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system

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].

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Localized Excitations in a Dispersive Long Water-Wave System via an Extended Projective Approach

Zeitschrift für Naturforschung A ◽

10.1515/zna-2007-3-404 ◽

2007 ◽

Vol 62 (3-4) ◽

pp. 140-146 ◽

Cited By ~ 4

Author(s):

Jin-Xi Fei ◽

Chun-Long Zheng

Keyword(s):

Coherent Structures ◽

Water Wave ◽

Wave System ◽

Variable Separation ◽

Localized Excitations ◽

Complex Wave ◽

New Type ◽

03.65 Ge

By means of an extended projective approach, a new type of variable separation excitation with arbitrary functions of the (2+1)-dimensional dispersive long water-wave (DLW) system is derived. Based on the derived variable separation excitation, abundant localized coherent structures such as single-valued localized excitations, multiple-valued localized excitations and complex wave excitations are revealed by prescribing appropriate functions. - PACS numbers: 03.65.Ge, 05.45.Yv

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Quasi-periodic waves and their interactions in the (2+1)-dimensional modified dispersive water-wave system

Physica Scripta ◽

10.1088/0031-8949/80/01/015006 ◽

2009 ◽

Vol 80 (1) ◽

pp. 015006 ◽

Cited By ~ 9

Author(s):

M F El-Sabbagh ◽

M M Hassan ◽

E A-B Abdel-Salam

Keyword(s):

Water Wave ◽

Wave System ◽

Periodic Waves

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The soliton-like solutions to the (2+1)-dimensional modified dispersive water-wave system

Chinese Physics ◽

10.1088/1009-1963/13/7/002 ◽

2004 ◽

Vol 13 (7) ◽

pp. 984-987 ◽

Cited By ~ 3

Author(s):

Li De-Sheng ◽

Zhang Hong-Qing

Keyword(s):

Water Wave ◽

Wave System

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Diversity of Interaction Solutions of a Shallow Water Wave Equation

Complexity ◽

10.1155/2019/5874904 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6

Author(s):

Jian-Ping Yu ◽

Wen-Xiu Ma ◽

Bo Ren ◽

Yong-Li Sun ◽

Chaudry Masood Khalique

Keyword(s):

Fluid Dynamics ◽

Wave Equation ◽

General Theory ◽

Shallow Water ◽

Water Wave ◽

Direct Method ◽

Soliton Solutions ◽

Interaction Solutions ◽

Shallow Water Wave ◽

Wave Solutions

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito (gHSI) equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics.

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