Large time behavior in nonlinear Schrodinger equation with time dependent potential

Rémi Carles (1) , Jorge Drumond Silva (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 CAMGSD - Center for Mathematical Analysis, Geometry and Dynamical Systems

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.

Domaines: Equations aux dérivées partielles [math.AP]

Rémi Carles, Jorge Drumond Silva. Large time behavior in nonlinear Schrodinger equation with time dependent potential. Communications in Mathematical Sciences, 2015, 13 (2), pp.443-460. ⟨10.4310/CMS.2015.v13.n2.a9⟩. ⟨hal-00823573⟩