34,049 research outputs found
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
Remarks on nonlinear Schroedinger equations with harmonic potential
Bose-Einstein condensation is usually modeled by nonlinear Schroedinger
equations with harmonic potential. We study the Cauchy problem for these
equations. We show that the local problem can be treated as in the case with no
potential. For the global problem, we establish an evolution law, which is the
analogue of the pseudo-conformal conservation law for the nonlinear
Schroedinger equation. With this evolution law, we give wave collapse criteria,
as well as an upper bound for the blow up time. Taking the physical scales into
account, we finally give a lower bound for the blow up time.Comment: 16 pages, no figur
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