Semiclassical Wave Packet Dynamics For Nonlinear Schrödinger Equations with Harmonic Potentials

Abstract

We propose an approximate Hamiltonian dynamics for the Gross--Pitaevskii equation $i\hbarψ_t = -\frac{\hbar^2}{2m}Δψ+ V(x)ψ+ \hbar^2 λ|ψ|^2ψ$ with harmonic potential $V(x)$ and $x\in\mathbb{R}^d$. 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.