On the extension of the frequency maps of the KdV and the KdV2 equations

TL;DR

This paper introduces a new method for representing frequencies of the KdV and KdV2 equations, enabling analysis in low-regularity spaces and providing insights into well-posedness and Hamiltonian convexity.

Contribution

It presents a novel frequency representation approach that extends to low-regularity spaces and analyzes the well-posedness of KdV2 in these spaces.

Findings

KdV2 is C^0-wellposed for s ≥ 0 in H^s spaces

KdV2 is ill-posed for s < 0 in a strong sense

New frequency extension method aids in studying asymptotics and convexity

Abstract

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on $H^{s}$ the KdV2 equation is $C^{0}$-wellposed if $s\ge 0$ and illposed (in a strong sense) if $s < 0$.