Steady waves in local and nonlocal models for water waves
(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:
https://lup.lub.lu.se/record/fe787f2f-46ab-4f72-9544-1a23bd07934b
- author
- Truong, Tien LU
- supervisor
-
- Erik Wahlén LU
- Nils Dencker LU
- opponent
-
- Professor Scheel, Arnd, University of Minnesota
- organization
- alternative title
- Steady waves in local and nonlocal models for water waves
- publishing date
- 2022-02-15
- 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
- 978-91-8039-163-4
- 978-91-8039-164-1
- 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 = {{978-91-8039-163-4}}, 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}}, }