Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross–Pitaevskii Equation

Gross–Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schrödinger type that play an important role in the theory of Bose–Einstein condensation. Recent results of Aschbacher et al. (3) demonstrate, for a class of 3-dimensional models, that for large boson number (squared L norm), N, the ground state does not have the symmetry properties of 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.