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Global Bifurcation of Water Waves over Constant Vorticity Flows with Critical Layers

Thylefors, Erik LU (2025) In Master's Theses in Mathematical Sciences FMAM05 20251
Mathematics (Faculty of Engineering)
Abstract
We consider two-dimensional travelling water waves over flows with constant vorticity and critical layers. We use global analytic bifurcation theory to show the existence of a global curve of solutions which either approaches a solitary wave or forms a critical point along the surface. In contrast to most of the previous studies, we fix the Bernoulli constant and use the wavenumber as a bifurcation parameter. We also derive several new bounds for Stokes waves with critical layers and constant vorticity.
Popular Abstract
Waves are ubiquitous in nature, arising as sound waves, light waves and wave functions of
particles. However, water waves, perhaps the most common occurrence of waves and the most
tangible ones in everyday life, are very different from the above waves and the study of them
is generally more difficult. In particular, water waves are scale dependent and water waves of
small- and large-amplitude may have very different qualitative and quantitative properties. One
active area of research is to investigate extreme behaviour of water waves when the amplitude
becomes large under different assumptions on the underlying water flow.

This thesis investigates two-dimensional travelling waves with counter-currents (critical lay-
ers) and... (More)
Waves are ubiquitous in nature, arising as sound waves, light waves and wave functions of
particles. However, water waves, perhaps the most common occurrence of waves and the most
tangible ones in everyday life, are very different from the above waves and the study of them
is generally more difficult. In particular, water waves are scale dependent and water waves of
small- and large-amplitude may have very different qualitative and quantitative properties. One
active area of research is to investigate extreme behaviour of water waves when the amplitude
becomes large under different assumptions on the underlying water flow.

This thesis investigates two-dimensional travelling waves with counter-currents (critical lay-
ers) and local rotation (vorticity) in the flow. The presence of critical layers and vorticity
presents additional mathematical difficulties and has received much attention in recent years.
In the thesis, a sequence of water waves is constructed that either approaches a solitary wave
or begins to form a critical point along its surface.

A solitary wave, as the name implies, is a wave that is isolated from other waves and con-
sists of a single peak travelling along some (almost flat) surface. While rare, such waves do
occur in nature under the right circumstances.

The other case of a critical point is less intuitive and means that there is some point along
the surface of the fluid that travels with the same speed as the wave itself in the horizontal
direction. The existence of such points indicates that the solution is becoming less smooth,
which could be due to several possible causes. One possibility is that the sequence of waves
could approach a wave of the highest possible height. These waves were conjectured by Stokes
to have a sharp crest with an interior angle of 120◦. This conjecture has been proved for some
cases, for example when there is no vorticity or no critical layers. The sequence could also be
approaching an overhanging or breaking wave.

This work is a step in proving the existence of the limiting waves which would then show
that there are travelling water waves under the assumptions of critical layers and vorticity that
are either solitary waves or have a critical point along the surface. As an example, it would
be of great interest to show that the Stokes conjecture holds for rotational waves with counter
currents. (Less)
Please use this url to cite or link to this publication:
author
Thylefors, Erik LU
supervisor
organization
course
FMAM05 20251
year
type
H1 - Master's Degree (One Year)
subject
keywords
Water waves, global analytic bifurcation theory, Euler equations, gravity waves
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3597-2025
ISSN
1404-6342
other publication id
2025:E83
language
English
id
9206388
date added to LUP
2025-09-15 11:09:52
date last changed
2025-09-15 11:09:52
@misc{9206388,
  abstract     = {{We consider two-dimensional travelling water waves over flows with constant vorticity and critical layers. We use global analytic bifurcation theory to show the existence of a global curve of solutions which either approaches a solitary wave or forms a critical point along the surface. In contrast to most of the previous studies, we fix the Bernoulli constant and use the wavenumber as a bifurcation parameter. We also derive several new bounds for Stokes waves with critical layers and constant vorticity.}},
  author       = {{Thylefors, Erik}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Global Bifurcation of Water Waves over Constant Vorticity Flows with Critical Layers}},
  year         = {{2025}},
}