Two remarks on the generalised Korteweg de-Vries equation

Abstract

We make two observations concerning the generalised Korteweg de Vries equation Ut + uxxx = μ(|u|P-1u)x. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for L2-critical equation (p = 5) automatically implies an analogous scattering result for the L 2-critical nonlinear Schrödinger equation iut+u xx = μ|u|4u. 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.