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 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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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)]. |
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ISSN: | 0022-2488 1089-7658 |
DOI: | 10.1063/1.3405494 |