On the role of quadratic oscillations in nonlinear Schrödinger equations II. The L²-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 L²-supercritical power-like nonlinearities and more general initial data. The present results concern the L²-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 make 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.