Two remarks on the generalised Korteweg de-Vries equation

TL;DR

This paper explores the generalized Korteweg de Vries equation, establishing a scaling connection to nonlinear Schrödinger equations and introducing a dispersion estimate that limits soliton persistence in the defocusing case.

Contribution

It provides a scaling argument linking scattering results between the gKdV and NLS equations and introduces a new dispersion estimate for the defocusing gKdV.

Findings

Scaling argument relating gKdV and NLS scattering results

New dispersion estimate showing energy moves faster than mass in defocusing case

Implication that localized solitons cannot persist indefinitely in defocusing gKdV

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 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.