### Abstract

We study the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude ε tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of ε^{-2}, the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as ε goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.

Original language | English |
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Pages (from-to) | 189-235 |

Number of pages | 47 |

Journal | Asymptotic Analysis |

Volume | 63 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

### Keywords

- Expected mode powers
- Harmonic potential
- Nonlinear Schrödinger equation
- Standing waves
- Stochastic partial differential equations
- White noise

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Asymptotic Analysis*,

*63*(4), 189-235. https://doi.org/10.3233/ASY-2008-0931