### Abstract

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

Original language | English (US) |
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Pages (from-to) | 795-844 |

Number of pages | 50 |

Journal | Communications in Mathematical Physics |

Volume | 308 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2011 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*308*(3), 795-844. https://doi.org/10.1007/s00220-011-1361-3