The periodic b-equation and Euler equations on the circle

TL;DR

This paper demonstrates that the periodic b-equation corresponds to an Euler equation on the circle's diffeomorphism group only when b=2, linking it specifically to the Camassa-Holm equation, and shows that for b=3, it does not arise as such an Euler equation.

Contribution

It establishes a precise condition under which the periodic b-equation is an Euler equation on Diff(S^1), extending previous results by B. Kolev.

Findings

b=2 corresponds to the Camassa-Holm equation as an Euler equation

b=3 (Degasperis-Procesi) is not an Euler equation on Diff(S^1)

The inertia operator for b=2 is A=1-d^2/dx^2

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.