The periodic b-equation and Euler equations on the circle

Abstract

In this note we show that the periodic b-equation can only be realized as a Euler equation on the Lie group Diff∞(S1) 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∞(S1) is given by A=1−∂x2. In contrast, the Degasperis–Procesi equation, for which b=3, is not a Euler equation on Diff∞(S1) 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)].