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PreprintCC BY 4.0arXiv

Nonlinear adiabatic theorem for the Gross-Pitaevskii equation

  • Sven Bachmann
  • Severin Schraven
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Abstract

We prove a nonlinear adiabatic theorem for the repulsive Gross-Pitaevskii (GP) equation with a time-dependent trapping potential V(x,t). If either (i) V(·,t) is smooth, subquadratic in space and with uniformly bounded spatial derivatives of order ≥ 2, or (ii) V(·,t) is a uniformly bounded perturbation of a fixed non-negative potential which is asymptotically |x|m, m≥ 2, then the GP equation has a global solution, which remains uniformly close to the instantaneous minimizer of the time-independent GP functional, provided the initial condition is a ground state. In fact, we show that the existence of a solution of the GP equation is the real bottleneck: If existence is assumed, then adiabaticity holds under very minimal assumptions.

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Cite

BibTeXRIS
BibTeX
@misc{bachmann2026nonlinear,
  title = {{Nonlinear adiabatic theorem for the Gross-Pitaevskii equation}},
  author = {Sven Bachmann and Severin Schraven},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02231},
  eprint = {2611.02231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02231},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nonlinear adiabatic theorem for the Gross-Pitaevskii equation
AU  - Sven Bachmann
AU  - Severin Schraven
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02231
UR  - https://arxiv.org/abs/2611.02231
AB  - We prove a nonlinear adiabatic theorem for the repulsive Gross-Pitaevskii (GP) equation with a time-dependent trapping potential V(x,t). If either (i) V(·,t) is smooth, subquadratic in space and with uniformly bounded spatial derivatives of order ≥ 2, or (ii) V(·,t) is a uniformly bounded perturbation of a fixed non-negative potential which is asymptotically |x|^m, m≥ 2, then the GP equation has a global solution, which remains uniformly close to the instantaneous minimizer of the time-independent GP functional, provided the initial condition is a ground state. In fact, we show that the existence of a solution of the GP equation is the real bottleneck: If existence is assumed, then adiabaticity holds under very minimal assumptions.
ER  -