Self-similar dynamics for the modified Korteweg-de Vries equation

We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and construct solutions with a prescribed blow up behavior as t $\rightarrow$ 0.