On the wellposedness of the defocusing mKdV equation below $L^{2}$

TL;DR

This paper establishes local well-posedness for the renormalized defocusing mKdV equation on Fourier Lebesgue spaces below L^2, revealing ill-posedness of the original equation in these spaces due to divergence of the phase factor.

Contribution

It introduces a novel representation of frequencies enabling analysis of the renormalized mKdV in Fourier Lebesgue spaces below L^2, linking integrability and well-posedness.

Findings

Proves local well-posedness of the renormalized mKdV on ${\\mathcal{F}\ell}^p$ for p>2.

Shows the original mKdV is ill-posed on these spaces due to phase divergence.

Develops a new frequency representation for the analysis of integrable PDEs.

Abstract

We prove that the renormalized defocusing mKdV equation on the circle is locally in time $C^{0}$-wellposed on the Fourier Lebesgue space ${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytically extend them to ${\mathcal{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.