January  2007, 18(1): 1-14. doi: 10.3934/dcds.2007.18.1

Two remarks on the generalised Korteweg de-Vries equation

1. 

Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095, United States

Received  June 2006 Revised  December 2006 Published  February 2007

We make two observations concerning the generalised Korteweg de Vries equation $u_t + $uxxx$ = \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 $iu_t + $uxx$ = \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.
Citation: Terence Tao. Two remarks on the generalised Korteweg de-Vries equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 1-14. doi: 10.3934/dcds.2007.18.1
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