Two remarks on the generalised Korteweg de-Vries equation

We make two observations concerning the generalised Korteweg de Vriesequation uₜ +uxxx= μ ( |u|ᵖ⁻¹ u )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 scatteringresult for the L²-critical nonlinear Schrödinger equation iuₜ +uxx= μ |u|⁴ u. Secondly, in thedefocusing 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 longtimes.