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Steady waves in local and nonlocal models for water waves

Truong, Tien LU (2022) In Doctoral Theses in Mathematical Sciences
Abstract
We study the steady Euler equations for inviscid, incompressible, and irrotational water waves of constant density. The thesis consists of three papers. The first paper approaches the Euler equations through a famous nonlocal model equation for gravity waves, namely the Whitham equation. We prove the existence of a highest gravity solitary wave which reaches the largest amplitude and forms a $C^{1/2}$ cusp at its crest. This confirms a 50-year-old conjecture by Whitham in the case of solitary waves, that the full linear dispersion in the Whitham equation would allow for high-frequency phenomena such as highest waves. In the second paper, we use a recently developed center manifold theorem for nonlocal and nonlinear equations to study... (More)
We study the steady Euler equations for inviscid, incompressible, and irrotational water waves of constant density. The thesis consists of three papers. The first paper approaches the Euler equations through a famous nonlocal model equation for gravity waves, namely the Whitham equation. We prove the existence of a highest gravity solitary wave which reaches the largest amplitude and forms a $C^{1/2}$ cusp at its crest. This confirms a 50-year-old conjecture by Whitham in the case of solitary waves, that the full linear dispersion in the Whitham equation would allow for high-frequency phenomena such as highest waves. In the second paper, we use a recently developed center manifold theorem for nonlocal and nonlinear equations to study small-amplitude gravity--capillary generalized and modulated solitary waves in a Whitham equation with small surface tension. The last paper treats the steady Euler equations directly. Here, the gravity and capillary coefficients are fixed but arbitrary, and for simplicity we place a non-resonance condition on the problem. We address the transverse dynamics of two-dimensional gravity--capillary periodic waves using a spatial dynamics technique, followed by a perturbation argument. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Scheel, Arnd, University of Minnesota
organization
alternative title
Steady waves in local and nonlocal models for water waves
publishing date
type
Thesis
publication status
published
subject
keywords
The steady Euler equations, the Whitham equation, solitary waves, highest waves, periodic waves, transverse dynamics, The steady Euler equations, the Whitham equation, solitary waves, highest waves, periodic waves, transverse dynamics
in
Doctoral Theses in Mathematical Sciences
issue
2022:2
pages
180 pages
publisher
Lunds universitet, Media-Tryck
defense location
MH: Hörmander, Lund. Join via zoom: https://lu-se.zoom.us/j/68344752115
defense date
2022-03-11 13:15:00
ISSN
1404-0034
1404-0034
ISBN
978-91-8039-164-1
978-91-8039-163-4
language
English
LU publication?
yes
id
fe787f2f-46ab-4f72-9544-1a23bd07934b
date added to LUP
2022-02-02 10:21:02
date last changed
2022-02-14 11:38:51
@phdthesis{fe787f2f-46ab-4f72-9544-1a23bd07934b,
  abstract     = {{We study the steady Euler equations for inviscid, incompressible, and irrotational water waves of constant density. The thesis consists of three papers. The first paper approaches the Euler equations through a famous nonlocal model equation for gravity waves, namely the Whitham equation. We prove the existence of a highest gravity solitary wave which reaches the largest amplitude and forms a $C^{1/2}$ cusp at its crest. This confirms a 50-year-old conjecture by Whitham in the case of solitary waves, that the full linear dispersion in the Whitham equation would allow for high-frequency phenomena such as highest waves. In the second paper, we use a recently developed center manifold theorem for nonlocal and nonlinear equations to study small-amplitude gravity--capillary generalized and modulated solitary waves in a Whitham equation with small surface tension. The last paper treats the steady Euler equations directly. Here, the gravity and capillary coefficients are fixed but arbitrary, and for simplicity we place a non-resonance condition on the problem. We address the transverse dynamics of two-dimensional gravity--capillary periodic waves using a spatial dynamics technique, followed by a perturbation argument.}},
  author       = {{Truong, Tien}},
  isbn         = {{978-91-8039-164-1}},
  issn         = {{1404-0034}},
  keywords     = {{The steady Euler equations; the Whitham equation; solitary waves; highest waves; periodic waves; transverse dynamics; The steady Euler equations; the Whitham equation; solitary waves; highest waves; periodic waves; transverse dynamics}},
  language     = {{eng}},
  month        = {{02}},
  number       = {{2022:2}},
  publisher    = {{Lunds universitet, Media-Tryck}},
  school       = {{Lund University}},
  series       = {{Doctoral Theses in Mathematical Sciences}},
  title        = {{Steady waves in local and nonlocal models for water waves}},
  url          = {{https://lup.lub.lu.se/search/files/114057654/Avh_Tien_Truong_hela.pdf}},
  year         = {{2022}},
}