Abstract.
Bose-Einstein condensation is usually modeled by nonlinear Schrödinger 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 Schrödinger 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 breaking time. This study relies on two explicit operators, suited to nonlinear Schrödinger equations with harmonic potential, already known in the linear setting.
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Submitted 04/12/01, accepted 21/05/02
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Carles, R. Remarks on Nonlinear Schrödinger Equations with Harmonic Potential. Ann. Henri Poincaré 3, 757–772 (2002). https://doi.org/10.1007/s00023-002-8635-4
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DOI: https://doi.org/10.1007/s00023-002-8635-4