We consider the stationary solutions for a class of Schrödinger equations witha symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassicallimit 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 ata given value for the strength of the nonlinear term. The kind of bifurcation picture onlydepends on the nonlinearity power. We then discuss the stability/instability properties ofeach branch of the stationary solutions. Finally, we consider an explicit one-dimensional toymodel where the double well potential is given by means of a couple of attractive Dirac’sdelta pointwise interactions.

Bifurcation and Stability for Nonlinear SchrödingerEquations with DoubleWell Potential in the SemiclassicalLimit / R., Fukuizumi; Sacchetti, Andrea. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 145:6(2011), pp. 1546-1594. [10.1007/s10955-011-0356-y]

Bifurcation and Stability for Nonlinear SchrödingerEquations with DoubleWell Potential in the SemiclassicalLimit

SACCHETTI, Andrea
2011

Abstract

We consider the stationary solutions for a class of Schrödinger equations witha symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassicallimit 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 ata given value for the strength of the nonlinear term. The kind of bifurcation picture onlydepends on the nonlinearity power. We then discuss the stability/instability properties ofeach branch of the stationary solutions. Finally, we consider an explicit one-dimensional toymodel where the double well potential is given by means of a couple of attractive Dirac’sdelta pointwise interactions.
2011
145
6
1546
1594
Bifurcation and Stability for Nonlinear SchrödingerEquations with DoubleWell Potential in the SemiclassicalLimit / R., Fukuizumi; Sacchetti, Andrea. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 145:6(2011), pp. 1546-1594. [10.1007/s10955-011-0356-y]
R., Fukuizumi; Sacchetti, Andrea
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/727658
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