# American Institute of Mathematical Sciences

November  2015, 20(9): 2793-2817. doi: 10.3934/dcdsb.2015.20.2793

## Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time

 1 Centre de Mathématiques Appliquées, CNRS et Ecole Polytechnique, 91128 Palaiseau cedex, France 2 Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan 3 Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau cedex, France

Received  September 2014 Revised  March 2015 Published  November 2015

The aim of this paper is to perform a theoretical and numerical study on the dynamics of vortices in Bose-Einstein condensation in the case where the trapping potential varies randomly in time. We take a deterministic vortex solution as an initial condition for the stochastically fluctuated Gross-Pitaevskii equation, and we observe the influence of the stochastic perturbation on the evolution. We theoretically prove that up to times of the order of $\epsilon^{-2}$, the solution having the same symmetry properties as the vortex decomposes into the sum of a randomly modulated vortex solution and a small remainder, and we derive the equations for the modulation parameter. In addition, we show that the first order of the remainder, as $\epsilon$ goes to zero, converges to a Gaussian process. Finally, some numerical simulations on the dynamics of the vortex solution in the presence of noise are presented.
Citation: Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793
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