ABSTRACT:
Wigner transforms (WTs) and Wigner measures (WMs) have been successfully used as a non-parametric tool to provide homogenised descriptions of wave problems. Notable applications are the efficient simulation of large linear wave fields, and the painless resolution of linear caustics. However, their application to non-linear problems has been extremely limited to date. A brief discussion of the state of the art of bilinear microlocal methods, such as WTs and WMs, will be given. The role of smoothness of the underlying linear flow will be specifically highlighted. The main focus of the talk is the presentation of recent results by the author. More specifically, semiclassical asymptotics for a wide family of nonlinear Schrödinger equations (including 3D Bose-Einstein condensates and the 1D Schrödinger-Poisson equation) are shown. These results substantially extend what can be found in the literature; in particular they apply to what is called the “supercritical” nonlinear regime.