On the Role of Quadratic Oscillations in Nonlinear Schrödinger Equations Ii. the L-critical Case Rémi Carles and Sahbi Keraani

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-supercritical power-like nonlinearities and more general initial data. The present results concern the L-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.