Diophantine conditions in well-posedness theory for a coupled modulated Korteweg-de Vries system

We study the well-posedness theory of a coupled modulated Korteweg-de Vries (KdV) system on the circle with a time non-homogeneous modulation acting on the linear dispersion term. When the coupling parameter is equal to one, it has been recently proved that given any $s\in \mathbb{R}$, the resulting modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$, with a sufficiently irregular modulation. For couplings different from one, we use Diophantine conditions to characterize the resonances and prove that (under further restrictions on the coupling constant) for any $s\in \mathbb{R}$ the coupled modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$. This result differs from its unmodulated counterpart where it is known that global well-posedness holds for $s\ge s_{\ast }\in (5\mathrm{/}7,1]$.