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Adiabatic theorem for the Gross–Pitaevskii equation

Gang, Zhou and Grech, Philip (2017) Adiabatic theorem for the Gross–Pitaevskii equation. Communications in Partial Differential Equations, 42 (5). pp. 731-756. ISSN 0360-5302.

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We prove an adiabatic theorem for the nonautonomous semilinear Gross–Pitaevskii equation. More precisely, we assume that the external potential decays suitably at infinity and the linear Schrödinger operator −Δ+V_s admits exactly one bound state, which is ground state, for any s∈[0,1]. In the nonlinear setting, the ground state bifurcates into a manifold of (small) ground state solutions. We show that, if the initial condition is at the ground state manifold, bifurcated from the ground state of −Δ+V_0, then, for any fixed s∈[0,1], as ε→0, the solution will converge to the ground state manifold bifurcated from the ground state of −Δ+V_s. Moreover, the limit is of the same mass to the initial condition.

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Additional Information:© 2017 Taylor & Francis. Partly supported by NSF grant DMS-1308985 and DMS-1443225.
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Subject Keywords:Adiabatic theory, non-autonomous dynamical systems, non-linear Schrödinger equation, soliton
Issue or Number:5
Classification Code:2010 Mathematics Subject Classification: 81V70, 35Q55, 82C10, 35Q41
Record Number:CaltechAUTHORS:20170525-080748355
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77748
Deposited By: Tony Diaz
Deposited On:25 May 2017 15:50
Last Modified:03 Oct 2019 18:01

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