Two remarks on the generalised Korteweg de-Vries equation

We make two observations concerning the generalised Korteweg de Vriesequation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scalingargument that shows, roughly speaking, that any quantitative scattering resultfor $L^2$-critical equation ($p=5$) automatically implies an analogousscattering result for the $L^2$-critical nonlinear Schr\"odinger equation $iu_t+ u_{xx} = \mu |u|^4 u$. Secondly, in the defocusing case $\mu > 0$ we presenta new dispersion estimate which asserts, roughly speaking, that energy moves tothe left faster than the mass, and hence strongly localised soliton-likebehaviour at a fixed scale cannot persist for arbitrarily long times.