Two remarks on the generalised Korteweg deVries equation
Abstract
We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (u^{p1} u)_x$. 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 $iu_t + u_{xx} = \mu u^4 u$. 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 solitonlike behaviour at a fixed scale cannot persist for arbitrarily long times.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606236
 Bibcode:
 2006math......6236T
 Keywords:

 Mathematics  Analysis of PDEs;
 35Q53
 EPrint:
 16 pages, no figures. A footnote is corrected