Semiclassical Wave Packet Dynamics For Nonlinear Schrödinger Equations with Harmonic Potentials
Abstract
We propose an approximate Hamiltonian dynamics for the Gross--Pitaevskii equation iℏψt = −ℏ2/2mΔψ+ V(x)ψ+ ℏ2 λ|ψ|2ψ with harmonic potential V(x) and x∈ℝd. We do this by extending the Gaussian ansatz of Perez-Garcia et al.~(Phys.~Rev.~A 56, 1424 (1997)) to the semiclassical wave packets of Hagedorn. We derive the evolution equations for the wave packets from the Hamiltonian/symplectic perspective, rather than the Lagrangian/variational one taken by Perez-Garcia et al. We first show that these two perspectives yield identical evolution equations in the Gaussian case. We then derive the evolution equations for an arbitrary element of the semiclassical wave packets. Finally, we demonstrate the efficacy of the approximate dynamics with numerical experiments.
Related
- Pullback dynamics and statistical solutions for semi-dissipative nonlinear Schrödinger equations
- Normalized solutions to mass supercritical Schrödinger equations with radial potentials
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Stochastic Homogenization of Schrödinger Equations with Exponentially Mixing Sub-Gaussian Potentials
- Standing waves for Schrodinger equations with Kato class potentials and L∞-bounded nonlinearities
BibTeX
@misc{smith2026semiclassical,
title = {{Semiclassical Wave Packet Dynamics For Nonlinear Schrödinger Equations with Harmonic Potentials}},
author = {Jared Smith and Tomoki Ohsawa},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01108},
eprint = {2611.01108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01108},
}
RIS
TY - UNPB
TI - Semiclassical Wave Packet Dynamics For Nonlinear Schrödinger Equations with Harmonic Potentials
AU - Jared Smith
AU - Tomoki Ohsawa
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01108
UR - https://arxiv.org/abs/2611.01108
AB - We propose an approximate Hamiltonian dynamics for the Gross--Pitaevskii equation iℏψ_t = −ℏ^2/2mΔψ+ V(x)ψ+ ℏ^2 λ|ψ|^2ψ with harmonic potential V(x) and x∈ℝ^d. We do this by extending the Gaussian ansatz of Perez-Garcia et al.~(Phys.~Rev.~A 56, 1424 (1997)) to the semiclassical wave packets of Hagedorn. We derive the evolution equations for the wave packets from the Hamiltonian/symplectic perspective, rather than the Lagrangian/variational one taken by Perez-Garcia et al. We first show that these two perspectives yield identical evolution equations in the Gaussian case. We then derive the evolution equations for an arbitrary element of the semiclassical wave packets. Finally, we demonstrate the efficacy of the approximate dynamics with numerical experiments.
ER -