  This paper proves that the motion of small-slope vorticity fronts in the
two-dimensional incompressible Euler equations is approximated on cubically
nonlinear timescales by a Burgers-Hilbert equation derived by Biello and Hunter
(2010) using formal asymptotic expansions. The proof uses a modified energy
method to show that the contour dynamics equations for vorticity fronts in the
Euler equations and the Burgers-Hilbert equation are both approximated by the
same cubically nonlinear asymptotic equation. The contour dynamics equations
for Euler vorticity fronts are also derived.
