Nonlinear Schrodinger equations with repulsive harmonic potential and applications

Details Nonlinear Schrodinger equations with repulsive harmonic potential and applications Nonlinear Schrodinger equations with repulsive harmonic potential and applications

2002-10-31

arxiv.org/abs/math/0210481v3

SIAM J. Math. Anal., 35, #4 (2003), 823-843.

Mathematics

Analysis of PDEs

Final version, to appear in SIAM J. Math. Anal

Scientific paper

We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity.

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