225 research outputs found
Nonlinear Schrodinger equation with time dependent potential
We prove a global well-posedness result for defocusing nonlinear Schrodinger
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
here is the growth of high Sobolev norms of the solution.Comment: 27 pages. Some typos fixe
Sharp weights in the Cauchy problem for nonlinear Schrodinger equations with potential
We review different properties related to the Cauchy problem for the
(nonlinear) Schrodinger equation with a smooth potential. For
energy-subcritical nonlinearities and at most quadratic potentials, we
investigate the necessary decay in space in order for the Cauchy problem to be
locally (and globally) well-posed. The characterization of the minimal decay is
different in the case of super-quadratic potentials.Comment: 8 pages, Corollary 3.5 is now a bit more genera
On semi-classical limit of nonlinear quantum scattering
We consider the nonlinear Schr{\"o}dinger equation with a short-range
external potential, in a semi-classical scaling. We show that for fixed Planck
constant, a com-plete scattering theory is available, showing that both the
potential and the nonlinearity are asymptotically negligible for large time.
Then, for data under the form of coherent state, we show that a scattering
theory is also available for the approximate envelope of the propagated
coherent state, which is given by a nonlinear equation. In the semi-classical
limit, these two scattering operators can be compared in terms of classical
scattering the-ory, thanks to a uniform in time error estimate. Finally, we
infer a large time decoupling phenomenon in the case of finitely many initial
coherent states.Comment: 41 page
On the instability for the cubic nonlinear Schrodinger equation
We study the flow map associated to the cubic Schrodinger equation in space
dimension at least three. We consider initial data of arbitrary size in ,
where , the critical index, and perturbations in H^\si, where
\si is independent of . We show an instability mechanism in some
Sobolev spaces of order smaller than . The analysis relies on two features
of super-critical geometric optics: creation of oscillation, and ghost effect.Comment: 4 page
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