Solitary Waves, Shock Waves and Singular Solitons of Gardner's Equation for Shallow Water Dynamics

Daoui, Abdel Kader; Triki, Houria; Mirzazadeh, Mohammad; Biswas, A.

doi:10.5506/aphyspolb.45.1135

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…There are several models that describe the shallow water waves along lakeshores and beaches. These are Korteweg-de Vries (KdV) equation, 1 Gardner's equation, 2 3 Peregrine equation, Benjamin-Bona-Mahoney equation, 4 regularized long wave (RLW) equation, Kawahara equation, 5 Boussinesq equation 6 and others. However, special forms of water waves known as dispersive shallow water waves is receiving quite a bit of attention during the past few years.…”

Section: Introductionmentioning

confidence: 98%

Numerical Simulation of Dispersive Shallow Water Waves with an Efficient Method

Karakoç

Biswas

2015

j comput theor nanosci

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The main purpose of this paper has been to compare the numerical results when the septic B-spline is used in the collocation method, and performance of the splitting technique in the numerical methods has been investigated. So numerical solutions of the Rosenau-KdV equation have been constructed by using the collocation method with septic B-splines as interpolation functions. The Rosenau-KdV equation is split both in space and in time. Those coupled systems of differential equations are also solved by way of the septic B-spline collocation method over uniform finite intervals.

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Section: Introductionmentioning

confidence: 98%

Numerical Simulation of Dispersive Shallow Water Waves with an Efficient Method

Karakoç

Biswas

2015

j comput theor nanosci

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“…The Gardner equation adopts the same type of behaviors as the standard KdV equation; however, the former claim the validity to the wider parametric domain for internal wave motion in a particular environment. The extension of the parametric domain for modeling of internal wave motion is found in [10][11][12][13][14][15]. But the KdV as well as the Gardner model can be used to study the theory of soliton in one dimension only.…”

Section: Introductionmentioning

confidence: 99%

The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method

Sarkar

Raut²,

Mali

2022

Computational and Mathematical Methods

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The purpose of this article is to explore different types of solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which is termed as KP-Gardner equation, extensively used to model strong nonlinear internal waves in ( 1 + 2 )-dimensions on the stratified ocean shelf. This evolution equation is also used to describe weakly nonlinear shallow-water wave and dispersive interracial waves traveling in a mildly rotating channel with slowly varying topography. Introducing Liu’s approach regarding the complete discrimination system for polynomial and the trial equation technique, a set of new solutions to the KP-mKP equation containing Jacobi elliptic function have been derived. It is found that these analytical solutions numerically exhibit different nonlinear structures such as solitary waves, shock waves, and periodic wave profiles. The reliability and effectiveness are confirmed from the numerical graphs of the solutions. Finally, the existence and validity of the various topological structures of the solutions are confirmed from the phase portrait of the dynamical system. Based on this investigation, it is confirmed that the method is not only suited for obtaining the classification of the solutions but also for qualitative analysis, which means that it can also be extended to other fields of application.

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“…It is an expansion of the KdV equation and has identical features with KdV equation, but also expands the order of availability. Unlike the KdV equation which represents shallow water waves, Gardner equation construct a model for deep ocean waves [4], [9], [24]. The equation has crucial applications in different fields such as quantum field theory, fluid dynamics, solid state physics, and plasma physics [5], [10].…”

Section: Introductionmentioning

confidence: 99%

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

Odabaşı

2021

Mathematical Sciences and Applications E-Notes

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This study investigates exact analytical solutions of some nonlinear partial differential equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation, the ZK-BBM equation and the Gardner equation have been considered. With the implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling wave solutions of the considered equations have been attained. The solutions have been checked and graphs have been given via package programs to see the behavior of the waves.

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Solitary Waves, Shock Waves and Singular Solitons of Gardner's Equation for Shallow Water Dynamics

Cited by 7 publications

References 14 publications

Numerical Simulation of Dispersive Shallow Water Waves with an Efficient Method

Numerical Simulation of Dispersive Shallow Water Waves with an Efficient Method

The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

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