Nonlinear Schrödinger equation with time dependent potential

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.

Keywords

nonlinear Schrödinger equation, time dependent external potential, Avron–Herbst formula, lens transform, large time behavior

2010 Mathematics Subject Classification

35A01, 35B40, 35Q55

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Published 29 July 2011