Unconditional uniqueness for the modified Korteweg-de Vries equation on the line

We prove that the modified Korteweg- de Vries equation (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s> \frac 13$. Our method of proof combines the improvement of the energy method introduced recently by the first and third authors with the construction of a modified energy. Our approach also yields \textit{a priori} estimates for the solutions of mKdV in $H^s(\mathbb R)$, for $s>0$, and enables us to construct weak solutions at this level of regularity.