Structure-Preserving Integration for Magnetic Gaussian Wave Packet Dynamics

Abstract

We develop structure-preserving time integration schemes for Gaussian wave packet dynamics associated with the magnetic Schrödinger equation. The variational Dirac--Frenkel formulation yields a finite-dimensional Hamiltonian system for the wave packet parameters, where the presence of a magnetic vector potential leads to a non-separable structure and a modified symplectic geometry. By introducing kinetic momenta through a minimal substitution, we reformulate the averaged dynamics as a Poisson system that closely parallels the classical equations of charged particle motion. This representation enables the construction of Boris-type integrators adapted to the variational setting. In addition, we propose explicit high-order symplectic schemes based on splitting methods and partitioned Runge--Kutta integrators. The proposed methods conserve the quadratic invariants characterizing the Hagedorn parametrization, preserve linear and angular momentum under symmetry assumptions, and exhibit near-conservation of the averaged Hamiltonian over long time intervals. Rigorous error estimates are derived for both the wave packet parameters and observable quantities, with bounds uniform in the semiclassical parameter. Numerical experiments demonstrate the favorable long-time behavior and structure preservation of the integrators.

Figures (5)

• Figure 1: Exemplary visualization of the induction step.
• Figure 1: The symplectic and the Boris splitting integrator are applied to system \ref{['equation: non-lin ex']} with $\alpha = 1/2$ for step size $\tau = 0.01$; the plots on the left show the deviation from symplecticity over time, the plot on the right illustrates that a nonlinear vector potential destroys the invariance proven in \ref{['proposition: modified magnetic invariant']}; the $y$-axis is scaled logarithmically.
• Figure 2: The symplectic and the Boris splitting integrator are applied to system \ref{['equation: non-lin ex']} with $\alpha = 0$ for step size $\tau = 0.01$; on the left, we plot the relative energy error of both integrators over time; the $y$-axis is scaled logarithmically. On the right, we plot the relative error of the energy of the symplectic integrator at time $10$ for different step sizes against a dashed reference line $\mathop{\mathrm{\tau}}\nolimits\mapsto\mathop{\mathrm{\tau}}\nolimits^2$; both axes are scaled logarithmically.
• Figure 3: The Boris and the symplectic splitting integrator is applied to the Penning trap with step size $\mathop{\mathrm{\tau}}\nolimits = 0.001$. The upper left plot shows the Boris error in the invariant $Y_t^\top\Omega Y_t - \Omega$. Below, we compare the modified invariant $Y_t^\top \Omega_B(\mathop{\mathrm{\tau}}\nolimits) Y_t - Y_0^\top\Omega_B(\mathop{\mathrm{\tau}}\nolimits)Y_0$ with the symplectic invariant of the symplectic integrator. On the right, we compare the semiclassical angular-momentum error in the $x_1$- $x_2$ plane for both integrators. In both plots, the $y$-axis is scaled logarithmically.
• Figure 4: We apply the symplectic integrator with step size $\mathop{\mathrm{\tau}}\nolimits = 0.01$ to the two symmetric systems described above. On the left, we plot the error of the total linear momentum for the first system; on the right, we plot the error of the semiclassical angular momentum for the second system, here the $y$-axis is scaled logarithmically.

Theorems & Definitions (42)

• Theorem 3.1: Derivative formulas
• Proof 1
• Corollary 3.3: Equations of motion
• Proof 2
• Remark 3.4
• Corollary 3.5: Well-posedness
• Proof 3
• Corollary 3.6: SBHL2025, Theorem 4.1
• Proof 4
• Lemma 3.7: Invariants for kinetic momenta