, Volume 116, Issue 1-4, pp 881-905

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

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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 2norm), \(N\) , 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.