GSGPEs-v1.1: A MATLAB code for computing the ground state of systems of Gross–Pitaevskii equations

Abstract

GSGPEs is a MATLAB/GNU Octave suite of programs for the computation of the ground state of systems of Gross–Pitaevskii equations. It can compute the ground state in the defocusing case, for any number of equations with harmonic or quasi-harmonic trapping potentials, in spatial dimension one, two or three. The computation is based on a spectral decomposition of the solution into Hermite functions and direct minimization of the energy functional through a Newton-like method with an approximate line-search strategy.This new version is due to a change in the function eig of Matlab® which requires a new way to compute Gauss–Hermite quadrature nodes and weights. Program summaryProgram Title: GSGPEsProgram Files doi:http://dx.doi.org/10.17632/3rn4z5dzwj.1Licensing provisions: GPLv3Programming language: MATLAB/GNU OctaveJournal reference of previous version: M. Caliari and S. Rainer, Comput. Phys. Commun. 184 (2013) 812–823Does the new version supersede the previous version?: YesReasons for the new version: The computation of eigenpairs (by [V, D] = eig (A)) was used in order to get the Gauss–Hermite quadrature nodes and weights via the Golub–Welsch algorithm. A change in the function eig of Matlab® occurred after version R2011b, which still gives a norm of AV−VD of the order of machine precision, produces inaccurate quadrature weights.Summary of revisions: Gauss–Hermite quadrature nodes and weights are now computed by the fast algorithm by Glaser, Liu and Rokhlin [1], as implemented by Hale and Trefethen [2].Nature of problem: A system of Gross–Pitaevskii Equations (GPEs) is used to mathematically model a Bose–Einstein Condensate (BEC) for a mixture of different interacting atomic species. The equations can be used both to compute the ground state solution (i.e., the stationary order parameter that minimizes the energy functional) and to simulate the dynamics. For particular shapes of the traps, three-dimensional BECs can be also simulated by lower dimensional GPEs.Solution method: The ground state of a system of Gross–Pitaevskii equations is computed through a spectral decomposition into Hermite functions and the direct minimization of the energy functional.