Two remarks on the generalised Korteweg de-Vries equation

Abstract

We make two observations concerning the generalised Korteweg de Vries equation $u_t+u_{xxx}=\mu (|u{|}^{p-1}u{)}_x$. 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 $i{u}_t+u_{xx}=\mu |u{|}^{4}u$. Secondly, in the defocusing case $\mu >0$ 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.

16 pages, no figures. A footnote is corrected

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• Global behaviour of nonlinear dispersive and wave equations