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 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.

8 pages

References in corpus (3)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Water Waves and Integrability
• A New Integrable Equation with Peakon Solutions

Cited by in corpus (5)

• The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations
• The Degasperis-Procesi equation as a non-metric Euler equation
• The geometry of a vorticity model equation
• The periodic $\mu$-$b$-equation and Euler equations on the circle
• The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus