Global well-posedness and equicontinuity for mKdV in modulation spaces

We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg–de Vries equations on the real line in modulation spaces $M_p^{s,2}(\mathbb{R})$, for all $1\le p<\mathrm{\infty }$ and $0\le s<3\mathrm{/}2-1\mathrm{/}p$. We will also show that such solutions admit global-in-time bounds in these spaces and that equicontinuous sets of initial data lead to equicontinuous ensembles of orbits. Indeed, such information forms a crucial part of our well-posedness argument.