Small data scattering and soliton stability in $\dot{H}^{-\frac16}$ for the quartic KdV Equation

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.