Large Time Behavior in Nonlinear Schrödinger Equation with Time Dependent Potential Rémi Carles and Jorge
نویسنده
چکیده
We consider the large time behavior of solutions to defocusing nonlinear Schrödinger 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.
منابع مشابه
On the Role of Quadratic Oscillations in Nonlinear Schrödinger Equations Ii. the L-critical Case Rémi Carles and Sahbi Keraani
We consider a nonlinear semi–classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian–Kammerer and I. Gallagher for L-supercritical power-like nonlinearities and more general initial data. The present results concern the L-critical case, ...
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