Global well-posedness and equicontinuity for mKdV in modulation spaces

TL;DR

This paper proves global well-posedness and equicontinuity for the complex-valued mKdV equations in modulation spaces, extending the understanding of solution behavior for all p in [1,∞) and s in [0, 3/2 - 1/p].

Contribution

It establishes the first global well-posedness results for mKdV in a broad class of modulation spaces, including bounds and equicontinuity properties of solutions.

Findings

Solutions have global-in-time bounds in modulation spaces.

Equicontinuous initial data lead to equicontinuous solution ensembles.

The results apply to both focusing and defocusing mKdV equations.

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