TY - JOUR

T1 - Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

AU - Fukuizumi, Reika

AU - Sacchetti, Andrea

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/12

Y1 - 2011/12

N2 - 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.

AB - 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.

KW - Nonlinear Schrödinger equation

KW - Orbital stability

KW - Spontaneous symmetry breaking bifurcation

UR - http://www.scopus.com/inward/record.url?scp=82255175818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82255175818&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0356-y

DO - 10.1007/s10955-011-0356-y

M3 - Article

AN - SCOPUS:82255175818

VL - 145

SP - 1546

EP - 1594

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -