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Nonlinear Schrodinger equations with repulsive harmonic potential and applications

Remi Carles
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Abstract

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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Remi Carles. 2003-03-14. Nonlinear Schrodinger equations with repulsive harmonic potential and applications. https://arxiv.org/abs/math/0210481

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