We show that the periodic b-equation can only be realized as an Euler equation on the Lie goup Diff(S^1) of all smooth and orientiation preserving diffeomorphisms on the cirlce 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-\partial_{x}^2. In contrast, the Degasperis-Procesi equation, for which b=3, is not an Euler equation on Diff^1(S) for any inertia operator. Our result generalizes a recent result of Kolev.
The periodic b-equation and Euler equations on the circle
SEILER, JOERG
2010-01-01
Abstract
We show that the periodic b-equation can only be realized as an Euler equation on the Lie goup Diff(S^1) of all smooth and orientiation preserving diffeomorphisms on the cirlce 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-\partial_{x}^2. In contrast, the Degasperis-Procesi equation, for which b=3, is not an Euler equation on Diff^1(S) for any inertia operator. Our result generalizes a recent result of Kolev.
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