Abstract

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 $M_p^{s,2}(ℝ)$, for all $1\le p<\infty$ and $0\le s<\frac{3}{2}-\frac{1}{p}$. 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.

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