Dispersion dynamics for the defocusing generalized Korteweg-de Vries equation

TL;DR

This paper investigates the dispersion behavior of the defocusing generalized Korteweg-de Vries (gKdV) equation, demonstrating that the $L^2$-mass cannot concentrate excessively over time, using a variance functional and Tao's monotonicity formula.

Contribution

It introduces a variance-type functional to quantify dispersion in the defocusing gKdV equation, extending previous methods with new sublevel estimates.

Findings

The $L^2$-mass dispersion is quantitatively characterized.

The variance functional growth is controlled by sublevel estimates.

The approach extends Tao's monotonicity techniques to gKdV.

Abstract

We study dispersion for the defocusing gKdV equation. It is expected that it is not possible for the bulk of the $L^2-$mass to concentrate in a small interval for a long time. We study a variance-type functional exploiting Tao's monotonicity formula in the spirit of earlier work by Tao as well as Kwon & Shao and quantify its growth in terms of sublevel estimates.