  Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear
Schroedinger type, which play an important role in the theory of Bose-Einstein
condensation. Recent results of Aschenbacher et. al. [AFGST] demonstrate, for a
class of 3- dimensional models, that for large boson number (squared L^2 norm),
N, the ground state does not have the symmetry properties as the ground state
at small N. We present a detailed global study of the symmetry breaking
bifurcation for a 1-dimensional model Gross-Pitaevskii equation, in which the
external potential (boson trap) is an attractive double-well, consisting of two
attractive Dirac delta functions concentrated at distinct points. Using
dynamical systems methods, we present a geometric analysis of the symmetry
breaking bifurcation of an asymmetric ground state and the exchange of
dynamical stability from the symmetric branch to the asymmetric branch at the
bifurcation point.
