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 \begin{equation*} \begin{cases} \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), \end{cases} \end{equation*} and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$, $s> -\frac12$, for $0<\alpha<1$. Our result covers the whole scaling sub-critical range of Sobolev regularity contrary to the case $\alpha =1$, where the local well-posedness holds only for $s\geq \frac14$. We also prove that the local well-posedness result is sharp in two different ways, viz., for $s<-\frac12$ the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow-map that takes initial data to the solution fails to be $C^3$ at the origin. These results hold for $\alpha>1$ as well.