scholarly journals The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

2003 ◽  
Vol 10 (sup1) ◽  
pp. 72-92 ◽  
Author(s):  
Robin S Johnson
Keyword(s):  
Water Waves ◽  
Classical Problem ◽  
Integrable Equations

scholarly journals Water waves and integrability

2007 ◽  
Vol 365 (1858) ◽  
pp. 2267-2280 ◽  
Author(s):  
Rossen I Ivanov
Keyword(s):  
Asymptotic Expansion ◽  
Shallow Water ◽  
Water Waves ◽  
Integrable Equation ◽  
Inviscid Fluid ◽  
Shallow Water Waves ◽  
Integrable Equations ◽  
Scale Parameters ◽  
Euler’S Equations ◽  
Euler's Equations

Euler's equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymptotic expansion of Euler's equations is considered (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modelling the motion of shallow water waves are reviewed in this contribution.

A note on the vertical distribution of momentum transport in water waves

2015 ◽  
Vol 44 (4) ◽  
Author(s):  
Jerzy Kołodko ◽  
Gabriela Gic-Grusza
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Continuous Distribution ◽  
Classical Problem ◽  
Ocean Dynamics ◽  
Momentum Transport ◽  
Generalized Function ◽  
Dirac Delta ◽  
The Mean ◽  
Transport Flux

AbstractIn this paper, the classical problem of horizontal waveinduced momentum transport is analyzed once again. A new analytical approach has been employed to reveal the vertical variation of this transport in the Eulerian description.In mathematical terms, this variation is shown to have (after “smoothing out” the surface corrugation) the character of a generalized function (distribution) and is described by a classical function in the water depths and by an additional Dirac-delta-function component on the averaged free surface.In terms of physics, the considered variation consists of two entities: (i) a continuous distribution of the mean momentum transport flux density (tensorial radiation pressure) over the entire water column, and (ii) an additional momentum transport flux concentrated on the mean free surface level (tensorial radiation surface pressure). Simple analytical formulae describing this variation have been derived.This allowed a conventional expression to be derived, describing the depth-integrated excess of horizontal momentum flux due to the presence of waves (the so-called “radiation stress”), confirming to some extent the correctness of the whole analysis carried out.The results obtained may be important to the ocean dynamics, especially in view of their possible application in the field of hydrodynamics of wave-dominated coastal zones.

Painlevé analysis for three integrable shallow water waves equations with time-dependent coefficients

2019 ◽  
Vol 30 (2) ◽  
pp. 996-1008 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Direct Method ◽  
Soliton Solutions ◽  
Handling Time ◽  
Time Dependent ◽  
Shallow Water Waves ◽  
Integrable Equations ◽  
Content Type ◽  
Multiple Complex Soliton Solutions

Purpose The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models. Design/methodology/approach The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions. Findings The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions. Research limitations/implications The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients. Practical implications This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions. Social implications The paper presents useful algorithms for finding integrable equations with time-dependent coefficients. Originality/value The paper presents an original work with a variety of useful findings.

scholarly journals On deep-sea water waves caused by a local disturbance on or beneath the surface

1915 ◽  
Vol 92 (635) ◽  
pp. 57-81 ◽  
Keyword(s):  
Free Surface ◽  
Deep Sea ◽  
Water Waves ◽  
Sea Water ◽  
Classical Problem ◽  
Finite Depth ◽  
Initial Disturbance ◽  
Integral Theorem ◽  
Local Disturbance ◽  
Deep Sea Water

The classical problem of waves produced in deep-sea water by a local disturbance of the free surface has been investigated by Prof. H. Lamb in a very able manner. He completed the theory of wave propagation in one dimension and in two horizontal dimensions when the initial disturbance is concentrated in the immediate neighbourhood of a line or a point, assuming that Fourier's double integral theorem can be applied in such a case. In this paper I venture to disturbance, followong Prof. Lamb's method, in cases where the initial disturbance is spread over a certain extent of the free surface; and, as an application of the general solution, I propose to treat the case in which the initial disturbing source is situated at a finite depth from the surface, i.e. where the surface wave is produced by an explosion like that of a mine under water.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

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. 

RESPONSE CHARACTERISTICS OF GRAVEL BEACH FROM THE VIEWPOINT OF SHALLOW WATER WAVES AND MOVEMENT OF GRAVELS

2020 ◽  
Vol 76 (2) ◽  
pp. I_565-I_570
Author(s):  
Shin-ichi AOKI ◽  
Tomoki HAMANO ◽  
Taishi NAKAYAMA ◽  
Eiichi OKETANI ◽  
Takahiro HIRAMATSU ◽  
...  
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Response Characteristics ◽  
Gravel Beach

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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