Scattering for the quartic generalised Korteweg–de Vries equation

We show that the quartic generalised KdV equationut+uxxx+(u4)x=0 is globally well posed for data in the critical (scale-invariant) space with small norm (and locally well posed for large norm), improving a result of Grünrock [A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations 18 (12) (2005) 1333–1339]. As an application we obtain scattering results in for the radiation component of a perturbed soliton for this equation, improving the asymptotic stability results of Martel and Merle [Y. Martel, F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 (3) (2001) 219–254].