  In this note we prove scattering for perturbations of solitons in the scaling
space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$.
The article relies strongly on refined estimates for a KdV equation linearized
at the soliton. In contrast to the work of Tao (2006), we are able to work
purely in the scaling space without additional regularity assumptions, allowing
us to prove some results on the existence of inverse wave operators.
