Sharp well-posedness for a coupled system of mKdV type equations

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $$\partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=\phi(x),$$ $$\partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) =0,\ \ w(x,0)=\psi(x),$$ and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\textrm{I}\!\textrm{R})\times H^{k}(\textrm{I}\!\textrm{R})$, $s,k> -\frac12$ and $|s-k|\leq 1/2$, for $\alpha\neq 0,1$. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $C^3$ at the origin, when $s<-1/2$ or $k<-1/2$ or $|s-k|>2$; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1/2$ (b) $k-2s>1$ or $s<-1/2$; (c) $s=k=-1/2$; (III) the local well-posedness result is sharp in a sense that we can not reduce the proof of the trilinear estimates, proving some related bilinear estimates (as in Tao [19]).