Nonlinear Double-well Schrödinger Equations in the Semiclassical Limit

Sacchetti, Andrea

doi:10.1007/s10955-005-3766-x

Cited by 30 publications

(56 citation statements)

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“…They used sophisticated analysis based on Strichartz estimates and wave operators for the linear Schrödinger equations. Similar but more formal reduction to the two-mode equations was developed by Sacchetti [14] using the semi-classical analysis. In comparison with [9,11], Sacchetti [14] considered the defocussing version of the NLS equation, where the anti-symmetric stationary state undertakes a similar symmetry-breaking bifurcation.…”

Section: Theorem 2 (Seementioning

confidence: 96%

“…Similar but more formal reduction to the two-mode equations was developed by Sacchetti [14] using the semi-classical analysis. In comparison with [9,11], Sacchetti [14] considered the defocussing version of the NLS equation, where the anti-symmetric stationary state undertakes a similar symmetry-breaking bifurcation. Based on the two-mode reduction, Sacchetti [15] also reported the same threshold p * as in (1.9) that separates the supercritical and subcritical pitchfork bifurcations.…”

Section: Theorem 2 (Seementioning

confidence: 96%

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Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

Pelinovsky

Phan

2012

Journal of Differential Equations

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We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the timedependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.

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Section: Theorem 2 (Seementioning

confidence: 96%

Section: Theorem 2 (Seementioning

confidence: 96%

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

Pelinovsky

Phan

2012

Journal of Differential Equations

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“…In [23,38,39], a semiclassical regime is studied in the presence of an external potential and a nonlinearity. The potential is a double well potential, and the associated Hamiltonian has two eigenfunctions.…”

Section: Comparison With Related Workmentioning

confidence: 99%

Interaction of Coherent States for Hartree Equations

Carles

2012

Arch Rational Mech Anal

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We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.

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“…Mathematically the difficulty consists in proving that the dynamics of the complete NLS is close to the dynamics of the 2-dimensional reduced system. In [8], making use of semiclassical estimates and refined existence results for NLS this stability result has been obtained for times of the order of the beating period in dimensions d = 1, 2 and any σ ∈ R + . In the present paper we concentrate on the case of local nonlinearity (3) in any dimension d and for σ satisfying (4), in this framework we extend the previous result by [8] proving that, in the semiclassical limit, the 2-dimensional eigenspace is almost invariant for any time (Theorem 1).…”

Section: Introductionmentioning

confidence: 99%

“…In [8], making use of semiclassical estimates and refined existence results for NLS this stability result has been obtained for times of the order of the beating period in dimensions d = 1, 2 and any σ ∈ R + . In the present paper we concentrate on the case of local nonlinearity (3) in any dimension d and for σ satisfying (4), in this framework we extend the previous result by [8] proving that, in the semiclassical limit, the 2-dimensional eigenspace is almost invariant for any time (Theorem 1). However, due to the possible presence of positive Lyapunof exponents, our result does not allow to show that the 2-dimensional system describes the dynamics over time scales larger than −1 .…”

Section: Introductionmentioning

confidence: 99%