Controlling nonlinear water waves: boundary stabilization of the Korteweg-de Vries-Burgers equation

1999 ◽  
Keyword(s):  
Water Waves ◽  
Burgers Equation ◽  
Boundary Stabilization ◽  
Nonlinear Water Waves ◽  
De Vries

scholarly journals Shallow water equations for equatorial tsunami waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170100 ◽  
Author(s):  
Anna Geyer ◽  
Ronald Quirchmayr
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Water Model ◽  
Shallow Water Model ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Tsunami Waves ◽  
De Vries ◽  
Model Equations

We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear water waves’.

On the excitation of long nonlinear water waves by a moving pressure distribution

1984 ◽  
Vol 141 ◽  
pp. 455-466 ◽  
Author(s):  
T. R. Akylas
Keyword(s):  
Pressure Distribution ◽  
Water Waves ◽  
Vries Equation ◽  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Phase Speed ◽  
Finite Depth ◽  
Nonlinear Water Waves ◽  
De Vries

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.

scholarly journals Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
N. Smaoui ◽  
B. Chentouf ◽  
A. Alalabi
Keyword(s):  
Nonlinear Boundary ◽  
Burgers Equation ◽  
Stabilization Problem ◽  
Boundary Stabilization ◽  
Wave Approximation ◽  
Long Wave ◽  
Control Schemes ◽  
De Vries ◽  
Long Wave Approximation ◽  
Stability Of The Solution

Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is $[0,1]$ [ 0 , 1 ] . First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $L^{2} $ L 2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

scholarly journals Nonlinear water waves (KdV) equation and Painleve technique

2015 ◽  
Vol 4 (2) ◽  
pp. 216
Author(s):  
Attia Mostafa
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Nonlinear Pde ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
The Third ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Study on the Interaction of Nonlinear Water Waves considering Random Seas

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Marten Hollm ◽  
Leo Dostal ◽  
Hendrik Fischer ◽  
Robert Seifried
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Random Seas

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

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