Critical nonlinear Schroedinger equations with and without harmonic potential

Abstract

We use a change of variables that turns the critical nonlinear Schroedinger equation into the critical nonlinear Schroedinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on $L^2$, and bijective on some time intervals. Using the known results for the critical nonlinear Schroedinger equation, this provides information for the properties of Bose-Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.