Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation

Anne de Bouard (1) , Reika Fukuizumi (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Department of Mathematics

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.

Mots clés: nonlinear Schrödinger equation; stochastic partial differential equations; white noise; harmonic potential; standing waves

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Anne de Bouard, Reika Fukuizumi. Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation. Asymptotic Analysis, 2009, 63 (4), pp.189-235. ⟨10.3233/ASY-2008-0931⟩. ⟨hal-00804572⟩