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Two remarks on the generalised Korteweg de-Vries equation

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  • We make two observations concerning the generalised Korteweg de Vriesequation $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 scatteringresult for the $L^2$-critical nonlinear Schrödinger equation $iu_t + $uxx$ = \mu |u|^4 u$. Secondly, in thedefocusing 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 longtimes.
    Mathematics Subject Classification: Primary: 35Q53.

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