Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical simulation of the Gross-Pitaevskii equation via vortex tracking

Thiago Carvalho Corso, Gaspard Kemlin, Christof Melcher, Benjamin Stamm

TL;DR

This work develops a reduced-order, vortex-tracking approach to the two-dimensional Gross–Pitaevskii equation in the small-$\varepsilon$ regime, leveraging the point-vortex dynamics governed by the renormalized energy $W$ and canonical harmonic maps. The authors construct well-prepared initial data, solve a one-dimensional ODE for vortex trajectories, and reconstruct GP solutions by smoothing the canonical map with a radial profile $f_{\varepsilon}$, achieving substantial computational savings while maintaining asymptotic accuracy as $\varepsilon\to0$. They provide rigorous error controls for the supercurrents and demonstrate convergence through numerical experiments, including spectral convergence in the harmonic-degree and fourth-order convergence in time, as well as explicit validation against full GP simulations. The method enables efficient, scalable GP simulations in the small-$\varepsilon$ limit, with potential extensions to more general domains and settings involving magnetic fields, while acknowledging limitations related to vortex collisions and emitted radiation.

Abstract

This paper deals with the numerical simulation of the Gross-Pitaevskii (GP) equation, for which a well-known feature is the appearance of quantized vortices with core size of the order of a small parameter $\varepsilon$. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit $\varepsilon\to0$, through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite-dimensional GP equation for small, but finite, $\varepsilon$. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of $\varepsilon$ typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.

Numerical simulation of the Gross-Pitaevskii equation via vortex tracking

TL;DR

This work develops a reduced-order, vortex-tracking approach to the two-dimensional Gross–Pitaevskii equation in the small- regime, leveraging the point-vortex dynamics governed by the renormalized energy and canonical harmonic maps. The authors construct well-prepared initial data, solve a one-dimensional ODE for vortex trajectories, and reconstruct GP solutions by smoothing the canonical map with a radial profile , achieving substantial computational savings while maintaining asymptotic accuracy as . They provide rigorous error controls for the supercurrents and demonstrate convergence through numerical experiments, including spectral convergence in the harmonic-degree and fourth-order convergence in time, as well as explicit validation against full GP simulations. The method enables efficient, scalable GP simulations in the small- limit, with potential extensions to more general domains and settings involving magnetic fields, while acknowledging limitations related to vortex collisions and emitted radiation.

Abstract

This paper deals with the numerical simulation of the Gross-Pitaevskii (GP) equation, for which a well-known feature is the appearance of quantized vortices with core size of the order of a small parameter . Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit , through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite-dimensional GP equation for small, but finite, . This method allows us to avoid numerical stability issues in solving the GP equation, where small values of typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.
Paper Structure (23 sections, 10 theorems, 101 equations, 13 figures, 2 tables)

This paper contains 23 sections, 10 theorems, 101 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model and motivations
  3. Structure of the paper
  4. Notations
  5. Results on Gross--Pitaevskii vortices and their dynamics
  6. Properties of canonical harmonic maps and the renormalized energy
  7. Asymptotic dynamics of vortices
  8. Well-prepared initial conditions
  9. Numerical simulation of the Hamiltonian dynamics
  10. Numerical simulation of the Gross--Pitaevskii equation: a new method based on vortex tracking
  11. Description of the method and mathematical justification
  12. Vortex positions as initial conditions
  13. Error control on the supercurrents
  14. Numerical study of the approximation error
  15. Numerical convergence of the Hamiltonian dynamics
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\psi_\varepsilon$ solve eq:GPE with well-prepared initial conditions, in the sense of Definition def:WP for some initial vortices with positions $\bm{a}^0 = (a_j^0)_{j=1,\dots,N}$ and degrees $(d_j)_{j=1,\dots,N}$. Then, there exists $\varepsilon_0$, $0~<~\beta$, $\gamma<1$ and $C>0$, depending for any $0\leq t\leq \tau_{\varepsilon,\bm{a}^0}$, where ${\bm a}(t) = (a_j(t))_{j=1,\dots,N}$ solv

Figures (13)

  • Figure 1: Numerical approximation of $f_{\varepsilon,r_0}$.
  • Figure 2: Trajectories of vortices in the singular limit $\varepsilon\to0$, cases 1 to 8, from left to right and up to bottom. Initial conditions are represented by dots and the different degrees used are specified for each trajectory.
  • Figure 3: Diagram summarizing the numerical simulation of the GP equation \ref{['eq:GPE']} via vortex tracking.
  • Figure 4: Case 1: Squared modulus and phase of $\psi_{\varepsilon}^*(t)$ for different times $t$.
  • Figure 5: Case 2: Squared modulus and phase of $\psi_{\varepsilon}^*(t)$ for different times $t$.
  • ...and 8 more figures

Theorems & Definitions (22)

  • Definition 1
  • Theorem 1: jerrardRefinedJacobianEstimates2008
  • Lemma 1: jerrardRefinedJacobianEstimates2008
  • Proposition 1
  • proof
  • Remark 1: Generic smooth domains
  • Remark 2: ODE solvers
  • Proposition 2
  • proof
  • Remark 3: Unknown initial vortex locations
  • ...and 12 more