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arXiv · 1305.4089

Large time behavior in nonlinear Schrodinger equation with time dependent potential

Rémi Carles·Jorge Drumond Silva
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Abstract

We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation 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.

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Rémi Carles, Jorge Drumond Silva. 2013-05-17. Large time behavior in nonlinear Schrodinger equation with time dependent potential. https://arxiv.org/abs/1305.4089

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