On global well-posedness of the modified KdV equation in modulation spaces

Abstract

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M_s^{2,p}(\mathbb{R})$ for $s\ge \frac{1}{4}$ and $2\le p<\infty$. For $s<\frac{1}{4}$, we show that the solution map for mKdV is not locally uniformly continuous in $M_s^{2,p}(\mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M_s^{2,p}(\mathbb{R})$ for $s\ge \frac{1}{4}$ and $2\le p<\infty$.

23 pages. We added Remark 1.6

References in corpus (4)

• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
• Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces
• Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces
• Local Well-Posedness for the Derivative Nonlinear Schrödinger Equations with $L^2$ Subcritical Data

Cited by in corpus (3)

• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
• Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$
• Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces