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

scholarly journals Global Bifurcation of Waves with Multiple Critical Layers

2020 ◽  
Vol 52 (5) ◽  
pp. 5066-5089
Author(s):  
Kristoffer Varholm
Keyword(s):  
Global Bifurcation ◽  
Critical Layers

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 Stratified Large-Amplitude Steady Periodic Water Waves with Critical Layers

2020 ◽  
Author(s):  
Susanna V. Haziot
Keyword(s):  
Conformal Mapping ◽  
Density Function ◽  
Water Waves ◽  
Large Amplitude ◽  
Bifurcation Theory ◽  
Critical Layers

Abstract By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles. We also provide certain conditions for which these waves cannot overturn.

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 Small-amplitude steady water waves with critical layers: Non-symmetric waves

2019 ◽  
Vol 267 (7) ◽  
pp. 4170-4191 ◽  
Author(s):  
V. Kozlov ◽  
E. Lokharu
Keyword(s):  
Water Waves ◽  
Small Amplitude ◽  
Critical Layers

scholarly journals Steady Water Waves with Multiple Critical Layers: Interior Dynamics

2011 ◽  
Vol 14 (3) ◽  
pp. 407-419 ◽  
Author(s):  
Mats Ehrnström ◽  
Joachim Escher ◽  
Gabriele Villari
Keyword(s):  
Water Waves ◽  
Critical Layers

scholarly journals Steady Water Waves with Multiple Critical Layers

2011 ◽  
Vol 43 (3) ◽  
pp. 1436-1456 ◽  
Author(s):  
Mats Ehrnström ◽  
Joachim Escher ◽  
Erik Wahlén
Keyword(s):  
Water Waves ◽  
Critical Layers
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