<p><span style="background-color: rgb(255, 255, 255); color: rgb(36, 36, 36);">Scattering theory has long been investigated for non linear PDE’s and in particular non linear Schrôdinger equations. In many instances it is possible to prove that solutions associated to some initial data $u_0$ scatter (i.e. are asymptotically close to linear solutions $e^{it \Delta} u_\pm$ as $t \rightarrow \pm \infty$). In this context, the maps $u_0 \mapsto u_\pm$ encode the large time behavior of the non linear system and the scattering operator $S: u_- \mapsto S(u_-)= u_+$ encodes the large time influence of the non linear interaction. Surprisingly, though the existence (and invertibility) of these operators has been the topic of many researches, il this non linear context, very little was known about the structure of these scattering operators. Inspired by some recent results on random data scattering theories, I will present in this talk a step toward the description of these operators. Namely, some recent results showing that in the high frequency limit, these maps are actually close to the identity. This is based on joint works with H. Koch (Bonn, N. Tzvetkov (ENS Lyon) and N. Visciglia (Pisa).</span></p><p><br></p>
