Following recent experiments at the Ecole normale superieure,
we are interested in the vortex configurations for
a Bose-Einstein condensate placed in a rotating trap.
Experimentally, the condensate has a cigar shape and is rotated along the $z$ axis and the first vortex line is not straight along the $z$ axis but bending
.
We give a simplified expression of the
energy, which only depends on the number and shape of the vortex lines.
Then we study this simplified energy in the case of a single vortex line
in order to determine its shape according to
the rotational velocity and the elongation of the condensate. We prove that if the condensate has the shape of the a cigar, the vortex is bending while in the case of a
pancake, the vortex stays straight along the $z$ axi. We study the
local stability of the straight vortex and find an estimate for the
critical angular speed at which bent vortices are nucleated.
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