Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation $$u_t+u_{xxx}+(u^4{)}_x=0$$ is globally wellposed for data in the critical (scale-invariant) space ${\dot{H}}_x^{-1\mathrm{/}6}(\mathbb{R})$ with small norm (and locally wellposed for large norm), improving a result of Gr"unrock. As an application we obtain scattering results in $H_x^1(\mathbb{R})\cap {\dot{H}}_x^{-1\mathrm{/}6}(\mathbb{R})$ for the radiation component of a perturbed soliton for this equation, improving the asymptotic stability results of Martel and Merle.