The periodic b -equation and Euler equations on the circle

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 o...

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Veröffentlicht in:Journal of mathematical physics 2010-05, Vol.51 (5), p.053101-053101-6
Hauptverfasser: Escher, Joachim, Seiler, Jörg
Format: Artikel
Sprache:eng
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Zusammenfassung: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 Motion 46, 412–419 (2009)].
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3405494