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

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

2018 ◽  
Vol 32 (08) ◽  
pp. 1750268 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Yong-Jiang Guo ◽  
Hui-Min Li
Keyword(s):  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Linear Superposition ◽  
Elastic Interactions ◽  
The One ◽  
Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

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

On Lagrangian drift in shallow-water waves on moderate shear

2010 ◽  
Vol 660 ◽  
pp. 221-239 ◽  
Author(s):  
W. R. C. PHILLIPS ◽  
A. DAI ◽  
K. K. TJAN
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Shallow Water ◽  
Water Waves ◽  
Stokes Drift ◽  
Shallow Water Waves ◽  
Sea Floor ◽  
Langmuir Circulations ◽  
A Minor ◽  
The Waves

The Lagrangian drift in anO(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity isO(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.

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.

Bilinear form, bilinear Bäcklund transformation and dynamic features of the soliton solutions for a variable-coefficient (3+1)-dimensional generalized shallow water wave equation

2017 ◽  
Vol 31 (22) ◽  
pp. 1750126 ◽  
Author(s):  
Qian-Min Huang ◽  
Yi-Tian Gao
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Bilinear Form ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Backlund Transformation ◽  
Spectral Parameters ◽  
Shallow Water Wave

Under investigation in this letter is a variable-coefficient (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave equation. Bilinear form and Bäcklund transformation are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Stability of the solitons is studied numerically. Soliton amplitude is determined by the spectral parameters. Soliton velocity is not only related to the spectral parameters, but also to the variable coefficients. Phase shifts are the only difference between the two-soliton solutions and the superposition of the two relevant one-soliton solutions. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

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