A stochastic NLS equation arising in Bose

A stochastic NLS equation arising in Bose-Einstein
condensation
A. de Bouard, R. Fukuizumi
Laboratoire de Math´ematiques
CNRS et Universit´e de Paris-Sud
Orsay
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Optically confined Bose-Einstein condensates
Stamper-Kurn et al., Phys. Rev. Lett, 1998
Advantages
Obtain different geometrical configurations
study magnetic properties of atoms (trapping not limited to specific
magnetic states)
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Drawbacks
ex : fluctuations of the laser intensity
introduce stochasticity in the dynamical behavior of the condensate,
which has to be taken into acccount in real situations
Mean field theory : fluctuations of laser field intensity regarded as
modulations of the harmonic trap
Dynamics of BEC under regular variations of trap parameters :
widely studied
Castin and Dum, Phys. Rev. Lett. 1996
Kagan, et. al ; Phys. Rev. A, 1996
Ripoll, Perez-Garcia, Phys. Rev. A, 1999, etc...
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Randomly varying optical trap potential
Abdullaev, Baizakov, Konotop, in Nonlinearity and disorder, 2001
2D radially symmetric Gross-Pitaevskii equation :
i∂ψ
∂t=−1 r
∂
∂rr∂ψ ∂r+ (1 + ε(t))r2ψ+χ|ψ|2ψ−iγψ
in dimensionless variables, with
χ=±1 (related to the sign of the s-wave scattering length aof atoms)
γdamping coefficient (thermal cloud)
ε(t) = E(t)−E0
E0with E(t) = laser field intensity, mean value E0.
Assume ε(t) is δ-correlated with zero mean : hε(t)i= 0,
hε(t)ε(t0)i=σ2δ(t−t0)
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Our aim : study the preceding model from a mathematical point of view
in the attractive case (negative scattering length) in space dimensions 1, 2
or 3 (no result in 3-D for the moment)
Content :
Physical motivations of the model
Mathematical description of the equation
Local existence of solutions in 1-D or 2-D
Global existence (no collapse) in 1-D, or 2-D with small total number
of atoms
Collapse in 2-D (without damping)
Conclusion and open problems
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