Two remarks on the generalised Korteweg de-Vries equation

Details Two remarks on the generalised Korteweg de-Vries equation Two remarks on the generalised Korteweg de-Vries equation

2006-06-09

arxiv.org/abs/math/0606236v2

Discrete Cont. Dynam. Systems 18 (2007), 1-14

Mathematics

Analysis of PDEs

16 pages, no figures. A footnote is corrected

Scientific paper

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 Schr\"odinger 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 soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times.

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