scholarly journals Global bifurcation of irrotational water waves using a flow force formulation

2021 ◽  
Vol 301 ◽  
pp. 73-96
Author(s):  
Biswajit Basu ◽  
Florian Kogelbauer
Keyword(s):  
Water Waves ◽  
Global Bifurcation

scholarly journals Large-amplitude solitary water waves with discontinuous vorticity

2017 ◽  
Author(s):  
◽  
Adelaide Akers
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Elliptic Equations ◽  
Large Amplitude ◽  
Global Bifurcation ◽  
Constant Density ◽  
Solitary Wave Solutions ◽  
Continuous Curve ◽  
Wave Problem ◽  
Water Wave Problem

Consider a two-dimensional body of water with constant density which lies below a vacuum. The ocean bed is assumed to be impenetrable, while the boundary which separates the uid and the vacuum is assumed to be a free boundary. Under the assumption that the vorticity is only bounded and measurable, we prove that for any upstream velocity field, there exists a continuous curve of large-amplitude solitary wave solutions. This is achieved via a local and global bifurcation construction of weak solutions to the elliptic equations which constitute the steady water wave problem. We also show that such solutions possess a number of qualitative features; most significantly that each solitary wave is a symmetric, monotone wave of elevation.

scholarly journals A global investigation of solitary-wave solutions to a two-parameter model for water waves

1997 ◽  
Vol 342 ◽  
pp. 199-229 ◽  
Author(s):  
ALAN R. CHAMPNEYS ◽  
MARK D. GROVES
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Model Equation ◽  
Global Bifurcation ◽  
Solitary Wave Solutions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Wave Solutions

The model equationformula herearises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravity–capillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails.

scholarly journals Global bifurcation of steady gravity water waves with critical layers

2016 ◽  
Vol 217 (2) ◽  
pp. 195-262 ◽  
Author(s):  
Adrian Constantin ◽  
Walter Strauss ◽  
Eugen Vărvărucă
Keyword(s):  
Water Waves ◽  
Global Bifurcation ◽  
Critical Layers

Global bifurcation of capillary–gravity-stratified water waves

2014 ◽  
Vol 144 (4) ◽  
pp. 775-786 ◽  
Author(s):  
David Henry ◽  
Anca-Vocihita Matioc
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Governing Equations ◽  
Stagnation Points ◽  
Global Bifurcation Theory

We study steady periodic water waves with variable vorticity and density, where we allow for stagnation points in the flow, and where we admit the capillarity effects of surface tension. Using global bifurcation theory, we extend a local curve of non-trivial solutions of the governing equations to a global continuum of solutions. Furthermore, we obtain a description of the behaviour of the solutions along this continuum.

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

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