New lower bounds on the radius of spatial analyticity for the KdV equation

TL;DR

This paper establishes a lower bound on the decay rate of the spatial analyticity radius for KdV solutions, showing it does not decay faster than t^{-1/4}, improving previous bounds using a Gevrey space I-method approach.

Contribution

It introduces a higher order almost conservation law in Gevrey spaces to derive sharper lower bounds on the analyticity radius decay for KdV solutions.

Findings

Analyticity radius decays no faster than t^{-1/4} as t→∞

Improves previous bounds by Selberg, da Silva, and Tesfahun

Uses a Gevrey space I-method for analysis

Abstract

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{-1/4}$ as time $t$ goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincar\'{e}, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the $I-$method.