Details
| Originalsprache | Englisch |
|---|---|
| Fachzeitschrift | Journal of Mathematical Physics |
| Jahrgang | 51 |
| Ausgabenummer | 5 |
| Publikationsstatus | Veröffentlicht - 13 Mai 2010 |
Abstract
In this note we show that the periodic b-equation can only be realized as a Euler equation on the Lie group Diff ∞(S 1) of all smooth and orientation preserving diffeomorphisms on the circle if b=2, i.e., for the Camassa-Holm equation. In this case the inertia operator generating the metric on Diff ∞(S 1) is given by A=1-∂ x 2. In contrast, the Degasperis-Procesi equation, for which b=3, is not a Euler equation on Diff ∞(S 1) for any inertia operator. Our result generalizes a recent result of Kolev ["Some geometric investigations on the Degasperis-Procesi shallow water equation," Wave Motion46, 412-419 (2009)].
ASJC Scopus Sachgebiete
- Physik und Astronomie (insg.)
- Statistische und nichtlineare Physik
- Mathematik (insg.)
- Mathematische Physik
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in: Journal of Mathematical Physics, Jahrgang 51, Nr. 5, 13.05.2010.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - The periodic b-equation and Euler equations on the circle
AU - Escher, Joachim
AU - Seiler, Jörg
PY - 2010/5/13
Y1 - 2010/5/13
N2 - In this note we show that the periodic b-equation can only be realized as a Euler equation on the Lie group Diff ∞(S 1) of all smooth and orientation preserving diffeomorphisms on the circle if b=2, i.e., for the Camassa-Holm equation. In this case the inertia operator generating the metric on Diff ∞(S 1) is given by A=1-∂ x 2. In contrast, the Degasperis-Procesi equation, for which b=3, is not a Euler equation on Diff ∞(S 1) for any inertia operator. Our result generalizes a recent result of Kolev ["Some geometric investigations on the Degasperis-Procesi shallow water equation," Wave Motion46, 412-419 (2009)].
AB - In this note we show that the periodic b-equation can only be realized as a Euler equation on the Lie group Diff ∞(S 1) of all smooth and orientation preserving diffeomorphisms on the circle if b=2, i.e., for the Camassa-Holm equation. In this case the inertia operator generating the metric on Diff ∞(S 1) is given by A=1-∂ x 2. In contrast, the Degasperis-Procesi equation, for which b=3, is not a Euler equation on Diff ∞(S 1) for any inertia operator. Our result generalizes a recent result of Kolev ["Some geometric investigations on the Degasperis-Procesi shallow water equation," Wave Motion46, 412-419 (2009)].
UR - http://www.scopus.com/inward/record.url?scp=77955234722&partnerID=8YFLogxK
U2 - 10.1063/1.3405494
DO - 10.1063/1.3405494
M3 - Article
AN - SCOPUS:77955234722
VL - 51
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 5
ER -