  We consider a nonlinear semi-classical Schroedinger 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 scaling. As an
application, we precise a result by F. Merle and L. Vega concerning finite time
blow up for the critical Schroedinger equation. The proof relies on linear and
nonlinear profile decompositions.
