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arXiv · 1001.2987

The periodic b-equation and Euler equations on the circle

J. Escher·J. Seiler
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Abstract

In this note we show that the periodic b-equation can only be realized as an Euler equation on the Lie group 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-d^2/dx^2. In contrast, the Degasperis-Procesi equation, for which b=3, is not an Euler equation on Diff(S^1) for any inertia operator. Our result generalizes a recent result of B. Kolev.

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J. Escher, J. Seiler. 2010-01-18. The periodic b-equation and Euler equations on the circle. https://doi.org/10.1063/1.3405494

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