Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation
 R. K. Jackson,
 M. I. Weinstein
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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 3dimensional models, that for large boson number (squared L ^{2}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 1dimensional model Gross–Pitaevskii equation, in which the external potential (boson trap) is an attractive doublewell, 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.
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 Title
 Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation
 Journal

Journal of Statistical Physics
Volume 116, Issue 14 , pp 881905
 Cover Date
 20040801
 DOI
 10.1023/B:JOSS.0000037238.94034.75
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 nonlinear Schrödinger equation
 Bose–Einstein condensate
 standing waves
 symmetry breaking
 linear instability
 Industry Sectors
 Authors

 R. K. Jackson ^{(1)}
 M. I. Weinstein ^{(2)} ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Boston University, USA
 2. Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York, USA
 3. Fundamental Mathematics Research, Bell Laboratories, Murray Hill, New Jersey, USA