The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves

2008 ◽  
Vol 201 (1-2) ◽  
pp. 489-503 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Direct Method ◽  
Soliton Solutions ◽  
Shallow Water Waves ◽  
Multiple Soliton Solutions ◽  
Model Equations

Painlevé analysis for three integrable shallow water waves equations with time-dependent coefficients

2019 ◽  
Vol 30 (2) ◽  
pp. 996-1008 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Direct Method ◽  
Soliton Solutions ◽  
Handling Time ◽  
Time Dependent ◽  
Shallow Water Waves ◽  
Integrable Equations ◽  
Content Type ◽  
Multiple Complex Soliton Solutions

Purpose The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models. Design/methodology/approach The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions. Findings The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions. Research limitations/implications The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients. Practical implications This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions. Social implications The paper presents useful algorithms for finding integrable equations with time-dependent coefficients. Originality/value The paper presents an original work with a variety of useful findings.

The Novel Multi-Soliton Solutions of Equation for Shallow Water Waves

2003 ◽  
Vol 72 (3) ◽  
pp. 763-764 ◽  
Author(s):  
Yi Zhang ◽  
Shu-fang Deng ◽  
Deng-yuan Chen
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Soliton Solutions ◽  
The Novel ◽  
Shallow Water Waves

A multiple exp-function method for the three model equations of shallow water waves

2017 ◽  
Vol 89 (3) ◽  
pp. 2291-2297 ◽  
Author(s):  
Yakup Yildirim ◽  
Emrullah Yasar ◽  
Abdullahi Rashid Adem
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Function Method ◽  
Shallow Water Waves ◽  
Model Equations

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.

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