On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case.

We consider a nonlinear semi-classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian-Kammerer and I. Gallagher for $L^2$-supercritical power-like nonlinearities and more general initial data. The present results concern the $L^2$-critical case, in space dimensions 1 and 2; we describe the set of non-linearizable data, which is larger, due to the conformal invariance. As an application, we precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrödinger equation. The proof relies on linear and nonlinear profile decompositions