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

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 ${\hat{M}}_{r,q}^s(\mathbb{R})$ 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}.

References in corpus (5)

• Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers
• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
• Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces
• On global well-posedness of the modified KdV equation in modulation spaces
• A remark on norm inflation for nonlinear wave equations