  We consider an initial value problem for a quadratically nonlinear inviscid
Burgers-Hilbert equation that models the motion of vorticity discontinuities.
We use a normal form transformation, which is implemented by means of a
near-identity coordinate change of the independent spatial variable, to prove
the existence of small, smooth solutions over cubically nonlinear time-scales.
For vorticity discontinuities, this result means that there is a cubically
nonlinear time-scale before the onset of filamentation.

Analysis of PDEs