A new class of critical solutions for 1D cubic NLS

The aim of this article is to prove the existence of a new class of solutions of 1D cubic NLS with an initial data related to a sum of Dirac masses, of critical regularity $F(L^\infty)$, and belonging to $\dot H^s$ for any $s <-1/2$. This problem is motivated by the lack of result for critical regularity initial condition. Our result is based on a scattering approach, after performing a pseudo-conformal transformation, and on fine estimations of oscillatory integrals.