  We define a right-invariant Riemannian metric on the group of
contactomorphisms and study its Euler-Arnold equation. If the metric is
associated to the contact form, the Euler-Arnold equation reduces to $m_t +
u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$,
the contact vector field $u$ defined by $f$, and the momentum $m = f - \Delta
f$. Here the equation is considered on a compact manifold $M$ of dimension
$2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize
the analogy with the higher-order equation.
  We use the usual momentum conservation law for Euler-Arnold equations to
rewrite the geodesic equation as a smooth first-order equation on the
contactomorphism group of Sobolev class $H^s$, and thus obtain local existence
in time of solutions which depend smoothly on initial data. In addition we
prove a global existence criterion analogous to the Beale-Kato-Majda criterion
in fluid mechanics, and show how this criterion is automatically satisfied on
the totally geodesic subgroup of quantomorphisms. Finally we briefly discuss
singular solutions and conservation laws of the Euler-Arnold equation.

Analysis of PDEs