Two remarks on the generalised Korteweg de-Vries equation

Abstract

We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = μ(|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering result for the $L^2$-critical nonlinear Schrödinger equation $iu_t + u_{xx} = μ|u|^4 u$. Secondly, in the defocusing case $μ> 0$ we present a new dispersion estimate which asserts, roughly speaking, that energy moves to the left faster than the mass, and hence strongly localised soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times.