Russell K. Jackson and Michael I. Weinstein
Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross-Pitaevskii equation
(438K, Postscript )
ABSTRACT. Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schr\"odinger type, which play an important role in the theory of Bose-Einstein condensation. Recent results of Aschenbacher {\it et al} \cite{AFGST} demonstrate, for a class of $3-$ dimensional models, that for large boson number (squared $L^2$ norm),\ $\cN$,
the ground state does not have the symmetry properties as the ground state at small $\cN$. 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.