Details
| Original language | English |
|---|---|
| Title of host publication | Nonlinear Water Waves |
| Publisher | Springer Nature |
| Pages | 83-119 |
| Number of pages | 37 |
| Volume | 2158 |
| Publication status | Published - 29 Jun 2016 |
Publication series
| Name | Lecture Notes in Mathematics |
|---|---|
| ISSN (Print) | 0075-8434 |
Abstract
A basic question in the theory of nonlinear partial differential equations is: when does a solution form a singularity and what is its nature? The aim of this lecture series is to offer an introduction into the analytic study of these questions for unidirectional shallow water waves models. Two particular models are investigated: the famous Korteweg–de Vries equation and the more recent Camassa–Holm equation. A rather classical approach to the Korteweg–de Vries equation is presented showing that this flow is globally well-posed for large classes of initial conditions. As a consequence wave breaking cannot be observed within the Korteweg–de Vries regime. In pronounced contrast to that a different picture may be seen in the case of the Camassa–Holm flow: while some solutions exist for ever other develop a wave breaking in finite time. Finally a geometric picture of the Camassa–Holm equation is presented as a geodesic flow on a Fréchet–Lie group consisting of smooth diffeomorphisms of the real line.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
Nonlinear Water Waves. Vol. 2158 Springer Nature, 2016. p. 83-119 (Lecture Notes in Mathematics).
Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review
}
TY - CHAP
T1 - Breaking Water Waves
AU - Escher, Joachim
PY - 2016/6/29
Y1 - 2016/6/29
N2 - A basic question in the theory of nonlinear partial differential equations is: when does a solution form a singularity and what is its nature? The aim of this lecture series is to offer an introduction into the analytic study of these questions for unidirectional shallow water waves models. Two particular models are investigated: the famous Korteweg–de Vries equation and the more recent Camassa–Holm equation. A rather classical approach to the Korteweg–de Vries equation is presented showing that this flow is globally well-posed for large classes of initial conditions. As a consequence wave breaking cannot be observed within the Korteweg–de Vries regime. In pronounced contrast to that a different picture may be seen in the case of the Camassa–Holm flow: while some solutions exist for ever other develop a wave breaking in finite time. Finally a geometric picture of the Camassa–Holm equation is presented as a geodesic flow on a Fréchet–Lie group consisting of smooth diffeomorphisms of the real line.
AB - A basic question in the theory of nonlinear partial differential equations is: when does a solution form a singularity and what is its nature? The aim of this lecture series is to offer an introduction into the analytic study of these questions for unidirectional shallow water waves models. Two particular models are investigated: the famous Korteweg–de Vries equation and the more recent Camassa–Holm equation. A rather classical approach to the Korteweg–de Vries equation is presented showing that this flow is globally well-posed for large classes of initial conditions. As a consequence wave breaking cannot be observed within the Korteweg–de Vries regime. In pronounced contrast to that a different picture may be seen in the case of the Camassa–Holm flow: while some solutions exist for ever other develop a wave breaking in finite time. Finally a geometric picture of the Camassa–Holm equation is presented as a geodesic flow on a Fréchet–Lie group consisting of smooth diffeomorphisms of the real line.
UR - http://www.scopus.com/inward/record.url?scp=84991817671&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-31462-4_2
DO - 10.1007/978-3-319-31462-4_2
M3 - Contribution to book/anthology
AN - SCOPUS:84991817671
VL - 2158
T3 - Lecture Notes in Mathematics
SP - 83
EP - 119
BT - Nonlinear Water Waves
PB - Springer Nature
ER -