Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces

TL;DR

This work analyzes the Cauchy problem for the complex-valued mKdV equation on $\mathbb{R}$ within the generalized Fourier-Lebesgue spaces $\widehat{M}^{s}_{r,q}(\mathbb{R})$, unifying modulation and Fourier-Lebesgue frameworks. By employing $X^{s,b}$-type spaces and a refined nonlinear analysis, the authors establish sharp local well-posedness for $1<r\le 2$, $r'\le q\le \infty$ with $s\ge s(r)=\frac{1}{2}-\frac{1}{2r}$, and prove that the data-to-solution map is not $C^{3}$ at the origin when $s<s(r)$. A novel improved bilinear Strichartz estimate (Lemma 'improve-bi') enhances control over high–low frequency interactions and underpins a detailed trilinear analysis that yields the main propagation bounds. The results identify the scaling-critical threshold $s(r)$ and extend prior work by Grünrock, Vega, and others, providing a robust low-regularity well-posedness theory for complex-valued mKdV in a broad family of Fourier-Lebesgue/modulation-type spaces. The ill-posedness result completes the picture by showing failure of $C^{3}$-continuity below the threshold.

Abstract

We study the \emph{complex-valued} solutions to the Cauchy problem of the modified Korteweg-de Vries equation on the real line. To study the low-regularity problems, we employ a generalized Fourier-Lebesgue space $\widehat{M}^{s}_{r,q}(\mathbb{R})$ that unifies the modulation spaces and the Fourier-Lebesgue spaces. We then prove sharp local well-posedness results in this space by perturbation arguments using $X^{s,b}$-type spaces. Our results improve the previous one in \cite{GV}.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1
• Lemma 2.1: Transference Principle
• proof
• Lemma 2.2: Linear estimate, Gr
• Proposition 2.3
• Remark 2.1
• Lemma 3.1: Corollary 3.6 and 3.7 Gr
• Remark 3.1
• Lemma 3.2: Bilinear estimates, Lemma 1 GV