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Authors

G. Sadaka, P. Jolivet, E. G. Charalampidis and I. Danaila

Abstract

We present and distribute a parallel finite-element toolbox written in the free software FreeFem++ for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFem++ with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFem++ with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox’s results with known theoretical and numerical findings that have been reported in the literature.

Supplemental material: 3D one-component BEC continuation

3D one-component BEC with single-charged vortex-line configuration	Real part $\omega_r$ and imaginary part $\omega_i$ of eigenvalues
3D one-component BEC with vortex-ring with two (oppositely charged) vortex-line handles configuration	Real part $\omega_r$ and imaginary part $\omega_i$ of eigenvalues

Supplemental material: 2D two-component BEC continuation

The dark-bright soliton state

Snapshots of the solution of the 1st component	Snapshots of the solution of the 2nd component	Real and imaginary part of eigenvalues

The ground state and soliton necklace state

Snapshots of the solution of the 1st component	Snapshots of the solution of the 2nd component	Real and imaginary part of eigenvalues

The ground state and multipole state

Snapshots of the solution of the 1st component	Snapshots of the solution of the 2nd component	Real and imaginary part of eigenvalues

Supplemental material: 3D two-component BEC continuation

The dark-bright soliton stripe (with the ground state in $\phi_1$ and the planar dark soliton in $\phi_2$)

Snapshots of the solution of the 1st component	Snapshots of the solution of the 2nd component	Real and imaginary part of eigenvalues

The vortex-ring-bright state (with the ground state in $\phi_1$ and the vortex-ring state in $\phi_2$)

Snapshots of the solution of the 1st component	Snapshots of the solution of the 2nd component	Real and imaginary part of eigenvalues