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.
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- Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation
Journal of Statistical Physics
Volume 116, Issue 1-4 , pp 881-905
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- nonlinear Schrödinger equation
- Bose–Einstein condensate
- standing waves
- symmetry breaking
- linear instability
- Industry Sectors
- 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