On the Well-posedness of the defocusing mKdV equation below $L^{2}$

Abstract

We prove that the renormalized defocusing modified KdV (mKdV) equation on the circle is locally in time $C^{0}$-well-posed on the Fourier Lebesgue space ${\mathscr{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is ill-posed 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 nonlinear Schrödinger hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing us to analytically extend them to ${\mathscr{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.