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.

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

Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques

2020 ◽  
Vol 34 (07) ◽  
pp. 2050045 ◽  
Author(s):  
Naila Nasreen ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Problems ◽  
Applied Physics ◽  
Kawahara Equation ◽  
Generalized Riccati Equation ◽  
All Solutions ◽  
Fifth Order

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

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.

Characteristic of the algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations

2019 ◽  
Vol 35 (07) ◽  
pp. 2050028 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Traveling Wave Solutions ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Restoring Forces ◽  
First Time

With the aid of the planar dynamical systems and invariant algebraic cure, all algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations, which can be used to model shallow water waves with weakly nonlinear restoring forces and to describe waves in ferromagnetic media, were obtained. Meanwhile, some new rational solutions are also yielded through an invariant algebraic cure with two different traveling wave transformations for the first time. These results are an effective complement to existing knowledge. It can help us understand the mechanism of shallow water waves more deeply.

Sign in / Sign up

Export Citation Format

Share Document