Splitting methods for the Gross-Pitaevskii equation on the full space and vortex nucleation

TL;DR

This paper proves convergence and conservation properties of splitting schemes for the Gross-Pitaevskii equation, with numerical tests and applications to vortex nucleation in quantum fluids.

Contribution

It provides the first rigorous convergence analysis of Lie-Trotter and Strang splitting methods for the Gross-Pitaevskii equation with time-dependent potentials.

Findings

Convergence in Zhidkov spaces for splitting schemes

Near-preservation of Ginzburg-Landau energy law

Numerical validation on dark soliton and vortex nucleation

Abstract

We prove the convergence in Zhidkov spaces of the first-order Lie-Trotter and the second-order Strang splitting schemes for the time integration of the Gross-Pitaesvkii equation with a time-dependent potential and non-zero boundary conditions at infinity. We also show the conservation of the generalized mass and the near-preservation of the Ginzburg-Landau energy balance law. Numerical accuracy tests performed on a one-dimensional dark soliton corroborate our theoretical findings. We finally investigate the nucleation of quantum vortices in two experimentally relevant settings.