## Abstract

We consider the stationary solutions for a class of Schrödinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the nonlinearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.

Original language | English |
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Pages (from-to) | 1546-1594 |

Number of pages | 49 |

Journal | Journal of Statistical Physics |

Volume | 145 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2011 Dec |

## Keywords

- Nonlinear Schrödinger equation
- Orbital stability
- Spontaneous symmetry breaking bifurcation

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics