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

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.

Soliton Solutions, Bäcklund Transformation and Wronskian Solutions for the (2+1)-Dimensional Variable-Coefficient Konopelchenko–Dubrovsky Equations in Fluid Mechanics

2012 ◽  
Vol 67 (3-4) ◽  
pp. 132-140
Author(s):  
Peng-Bo Xu ◽  
Yi-Tian Gao
Keyword(s):  
Shear Flow ◽  
Fluid Mechanics ◽  
Shallow Water ◽  
Bilinear Form ◽  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Shallow Water Waves ◽  
Stratified Shear Flow ◽  
Dimensional Variable

This paper is to investigate the (2+1)-dimensional variable-coefficient Konopelchenko- Dubrovsky equations, which can be applied to the phenomena in stratified shear flow, internal and shallow-water waves, plasmas, and other fields. The bilinear-form equations are transformed from the original equations, and soliton solutions are derived via symbolic computation. Soliton solutions and collisions are illustrated. The bilinear-form B¨acklund transformation and another soliton solution are obtained. Wronskian solutions are constructed via the B¨acklund transformation and solution

scholarly journals On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 828-842
Author(s):  
Aly R. Seadawy ◽  
Shafiq U. Rehman ◽  
Muhammad Younis ◽  
Syed T. R. Rizvi ◽  
Ali Althobaiti
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Sound Propagation ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Capillary Waves ◽  
Nonlinear Phenomena ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Fifth Order

Abstract This article studies the fifth-order KdV (5KdV) hierarchy integrable equation, which arises naturally in the modeling of numerous wave phenomena such as the propagation of shallow water waves over a flat surface, gravity–capillary waves, and magneto-sound propagation in plasma. Two innovative integration norms, namely, the G ′ G 2 \left(\frac{{G}^{^{\prime} }}{{G}^{2}}\right) -expansion and ansatz approaches, are used to secure the exact soliton solutions of the 5KdV type equations in the shapes of hyperbolic, singular, singular periodic, shock, shock-singular, solitary wave, and rational solutions. The constraint conditions of the achieved solutions are also presented. Besides, by selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in three-dimensional, two-dimensional, and contour graphs. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex nonlinear phenomena.

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