Nonlinear Schrödinger equation with time dependent potential

Abstract

We prove a global well-posedness result for defocusing nonlinear Schrödinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by physics (Bose-Einstein condensation), and appears also as a preparation for the analysis of the propagation of wave packets in a nonlinear context. The main aspect considered in this paper is the growth of high Sobolev norms of the solution when the potential is exactly quadratic in space. Such a control is needed to study the large time propagation of nonlinear coherent states. © 2011 International Press.