  We study the nonlinear dynamics of binary black hole systems with scalar
charge by numerically evolving the full equations of motion for shift-symmetric
Einstein scalar Gauss-Bonnet gravity. We consider quasi-circular binaries with
different mass-ratios, varying the Gauss-Bonnet coupling and quantifying its
impact on the emitted scalar and gravitational waves. We compare our numerical
results to post-Newtonian calculations of the radiation emitted during the
inspiral. We demonstrate the accuracy of the leading-order terms in
post-Newtonian theory in modeling the amplitude of the scalar waveform, but
find that, at least for the last few orbits before merger, the currently
available post-Newtonian theory is not sufficient to model the dephasing of the
gravitational wave signal in this theory. We further find that there is
non-negligible nonlinear enhancement in the scalar field at merger, but that
the effect on the peak gravitational wave emission is small.
