Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

Reika Fukuizumi, Andrea Sacchetti

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)

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 languageEnglish
Pages (from-to)1546-1594
Number of pages49
JournalJournal of Statistical Physics
Volume145
Issue number6
DOIs
Publication statusPublished - 2011 Dec

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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