Explicit Symplectic Integrators for Massive Point Vortex Dynamics in Binary Mixture of Bose--Einstein Condensates

TL;DR

This paper develops explicit symplectic integrators of arbitrary even order for massive point vortex dynamics in Bose--Einstein condensates, outperforming standard methods in preserving Hamiltonian properties especially in oscillatory regimes.

Contribution

It introduces a family of explicit, high-order symplectic integrators that exactly preserve angular momentum and better maintain Hamiltonian invariants in complex vortex dynamics.

Findings

Integrators nearly preserve the Hamiltonian with minimal drift.

Standard Runge--Kutta methods perform poorly in highly oscillatory regimes.

Error estimates for the Hamiltonian are provided via asymptotic expansion.

Abstract

We construct explicit integrators of arbitrary even orders of accuracy for massive point vortex dynamics in binary mixture of Bose--Einstein condensates proposed by Richaud et al. The integrators are symplectic and preserve the angular momentum of the system exactly. Our main focus is the small-mass regime in which the minor component of the binary mixture comprises a very small fraction of the total mass. The solution behaviors in this regime change significantly depending on the initial momenta: they are highly oscillatory unless the momenta satisfy certain conditions. The standard Runge--Kutta method performs very poorly in preserving the Hamiltonian showing a significant drift in the long run, especially for highly oscillatory solutions. On the other hand, our integrators nearly preserve the Hamiltonian without drifts. We also give an estimate of the error in the Hamiltonian by finding an asymptotic expansion of the modified Hamiltonian for our second-order integrator.