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Published February 28, 2017 | Submitted
Journal Article Open

Adiabatic theorem for the Gross–Pitaevskii equation

Abstract

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.

Additional Information

© 2017 Taylor & Francis. Partly supported by NSF grant DMS-1308985 and DMS-1443225.

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Created:
August 19, 2023
Modified:
October 25, 2023