Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Large time behavior in nonlinear Schrödinger equations with time dependent potential

Pages: 443 – 460

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n2.a9

Authors

Rémi Carles (Institut National des Sciences Mathématiques, CNRS, Université Montpellier, France)

Jorge Drumond Silva (Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Abstract

We consider the large time behavior of solutions to defocusing nonlinear Schrödinger equations in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev norms are bounded, and momenta grow at most polynomially in time, because the potential becomes negligible for large time: there is scattering, even though the potential is unbounded in space for fixed time.

Keywords

nonlinear Schrödinger equation, time dependent potential, norm growth, lens transform, large time properties

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 05B40, 35Q40, 35Q41, 81Q05.

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Published 3 December 2014