In this work, we consider the initial value problem (IVP) for a system of modified Korteweg-de Vries (mKdV) equations \(\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \partial _x^3 v+ \partial _x (v w^2) = 0, \hspace{0.98 cm} v(x,0)=\psi (x),\\ \partial _t w + \alpha \partial _x^3 w+\partial _x (v^2 w) = 0,\hspace{0.5 cm} w(x,0)=\phi (x). \end{array}\right. } \end{aligned}\) The main interest is in addressing the well-posedness issues of the IVP when the initial data are considered in the modulation space \(M_s^{2,p}(\mathbb R)\) , \(p\ge 2\) . In the case when \(0<\alpha \ne 1\) , we derive new trilinear estimates in these spaces and prove that the IVP is locally well-posed for data in \(M_s^{2,p}(\mathbb R)\) whenever \(s> \frac{1}{4}-\frac{1}{p}\) and \(p\ge 2\) . In deriving the trilinear estimate, the fact that the Fourier supports of the solution components v and w lie on distinct cubic curves, namely \(\tau = \xi ^3\) and \(\tau = \alpha \xi ^3\) , introduces additional difficulties in handling the resonant case. This makes the analysis substantially different from what one encounters in the single-equation setting. To overcome the difficulties arising in the resonant case, it was necessary to impose the more restrictive condition \(s> \frac{1}{4}-\frac{1}{p}\) on the trilinear estimate, rather than the natural threshold \(s> \frac{1}{4}-\frac{3}{2p}\) , which would otherwise yield sharp local well-posedness for \(s>-\frac{1}{2}\) when \(p=2\) .
