The dimensionless grosspitaevskii equation used in gpelab. Solver for the twodimensional grosspitaevskii equation. Comparing three numerical solvers of the grosspitaevskii. Boseeinstein condensates, nonlinear schrodinger equation, grosspitaevskii equation, stationary states, ground. Under certain simplified assumptions small amplitudes, propagation in one direction, etc. We begin with an introduction on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem. Gpue is a gpubased gross pitaevskii equation solver via the splitoperator method for superfluid simulations of both linear and nonlinear schrodinger equations, emphasizing superfluid vortex dynamics in two and three dimensions. The considered model equation is related to the timedependent grosspitaevskii equation arising in the description of boseeinstein condensates in dilute gases. Fourthorder algorithms for solving the imaginarytime. The exact solutions of the ggpe are obtained via expansion method with particular soliton features highlighted. Pdf gpelab, a matlab toolbox to solve grosspitaevskii. Gpelab, a matlab toolbox to solve grosspitaevskii equations ii. However the description can only be valid provided the amplitudes can be regarded as large, and this must be the case for all modes.
Computer physics computer physics communications 00. We begin with the threedimensional 3d grosspitaevskii equation gpe with an angular momentum rotation term andor longrange ddi, state the two. Published 19 january 2004 2004 iop publishing ltd journal of physics a. Due to the complexity and timescales needed to simulate such system, it was essential to write some accelerated code to understand the behaviour of such systems. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in the. Twosoliton collision for the grosspitaevskii equation in. It may be exploited to simulate the evolution of boseeinstein condensate in a static external potential, or to calculate the groundstate using the imaginary time evolution. Grosspitaevskii equation for the density matrix in the. This function solves the grosspitaevskii equation in a twodimensional space. This paper presents gpelab gross pitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. We show that the fixed point approach introduced in 10 fails to converge. Grosspitaevskii equation for boseeinstein condensation.
I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Recently, methods for solving the grosspitaevskii equation gpe arising in boseeinstein condensates becs 1, 20 based on riccati equation were introduced. A timesplitting pseudospectral method for the solution of. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in the physics of boseeinstein condensates. Complex solutions of the time fractional grosspitaevskii. The aim of this second paper, which follows \citegroundgpelab, is to first present the various pseudospectral schemesavailable in gpelab for computing the deterministic and stochastic nonlinear. The gross pitaevskii equation gpe, named after eugene p. Gpelab a free matlab toolbox to compute and simulate bose. The result for a large number of particles compares well with the thomasfermi approximation as expected. Gpelab, an open source matlab toolbox for the numerical.
The primary use of this code was for my research on rapidly rotating boseeinstein condensates. Gsgpes is a matlabgnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. We consider the timedependent grosspitaevskii equation describing the dynamics of rotating boseeinstein condensates and its discretization with. Numerical solutions to the grosspitaevskii equation for boseeinstein condensates by luigi galati b. This software is a cudaenabled nonlinear schrodinger grosspitaevskii equation solver. We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating boseeinstein condensates becs in a rotational frame with or without longrange dipoledipole interaction ddi. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. In this equation only the short range part of the interaction between the. Fourthorder algorithms for solving the imaginarytime grosspitaevskii equation in a rotating anisotropic trap siu a. Gross and lev petrovich pitaevskii describes the ground state of a quantum system of identical bosons using the hartreefock approximation and the pseudopotential interaction model in the hartreefock approximation the total wavefunction of the system of bosons is taken as a product of singleparticle functions. In what follows, students will learn how to use the ndsolve command of mathematica. The model equation that gpelab solves is the grosspitaevskii equation gpe. We investigate the 1d generalized grosspitaevskii equation ggpe with quadratic potential and parameterized nonlinearity.
It can compute the ground state in the defocusing case, for any number of equations with harmonic or quasiharmonic trapping potentials, in spatial dimension one, two or three. Previous simulations of the onedimensional grosspitaevskii equation gpe with repulsive nonlinearity and a harmonicoscillator trapping potential hint towards the emergence of quasiintegrable dynamicsin the sense of quasiperiodic evolution of a moving dark soliton without any signs of ergodicityalthough this model does not belong to the list of integrable. International audiencethis paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. This equation is a general version of the dissipative grosspitaevskii equation including terms with firstorder derivatives in the spatial coordinates which allow for rotational contributions.
We present an efficient, unconditionally stable, and accurate numerical method for the solution of the grosspitaevskii equation. We obtain conditions for the existence of a global attractor and find bounds for its dimension. A matlab library of numerical functions for gross pitaevskii equation in 1d and 2d. Siam journal on numerical analysis siam society for. Robust and accurate numerical schemes are implemented for computing stationary solutions and the dynamics of becs through the gpe. Gpelab, a matlab toolbox to solve grosspitaevskii equations i. Derivation of the time dependent gross pitaevskii equation. Then we present a new numerical method, cfdmaif method, which combines compact finite. Gpelab is a matlab toolbox developed to help physicists for computing ground states or dynamics of quantum systems modeled by grosspitaevskii equations. This paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. An efficient compact finite difference method for the.
The model equation that gpelab solves is the grosspitaevskii equation. Grosspitaevskii equation for boseeinstein condensation application id. In this paper more general scalings shall be considered assuming positivity of the interaction. A spectral integral equation solution of the gross. Referenced in 27 articles fortran programs for the timedependent grosspitaevskii equation in a fully anisotropic trap.
Recently this method was used to derive the hartree equation for singular interactions 4 and the gross pitaevskii equation without positivity condition on the interaction 10 where one had to restrict the scaling behaviour of the interaction. A simple and efficient numerical method for computing the. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions. Grosspitaevskii equation gpe determines the quantum wave function of a. Pitaevskii equation on unbounded domains, numerical methods for partial differential equations on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The coefficients of terms of ggpe studied are arbitrary functions of time t. The grosspitaevskii equation describes the ground state of a quantum system of identical bosons using the hartreefock approximation and the pseudopotential interaction model. We consider the generalized purestate density matrix, which depends on different time moments, and obtain the evolution equation for this density matrix for the case where the density matrix corresponds to solutions of the grosspitaevskii equation. Numerical solution for the grosspitaevskii equation. Together with evgeny lifshitz and vladimir berestetskii. Exact soliton solutions of the 1d generalized gross. This toolbox is available in the download section with a user guide. This software is a cudaenabled nonlinear schrodinger gross pitaevskii equation solver. Solver for the twodimensional gross pitaevskii equation.
Gpue is a fast, robust, and accessible software suite to simulate physics for fundamental research in the area of. We consider the backward euler spectral besp scheme proposed in 10 for computing the stationary states of boseeinstein condensates becs through the grosspitaevskii equation. Numerical solution of the grosspitaevskii equation for. This function solves the gross pitaevskii equation in a twodimensional space. The problem is boundary and initial conditions that i. The model equation that gpelab solves is the gross pitaevskii equation. The model equation that gpelab solves is the gross pitaevskii equation gpe. In preparation for the numerics we scale the 3d grosspitaevskii equation and obtain a fourparameter model. We study the global behavior of small solutions of the grosspitaevskii equation in three dimensions. It has a soliton solution, whose envelope does not change in form over time. Comparing three numerical solvers of the grosspitaevskii equation posted on 03 february 2016. Ed, university of florida a thesis submitted to the graduate faculty of georgia southern university in partial ful llment of the requirement for the degree master of science statesboro. This paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and.
The grosspitaevskii equation and boseeinstein condensates. Some numerical methods have been proposed to obtain approximate solutions for. We solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by symplectic shooting method ssm. Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to the numerical. The grosspitaevskii equation describes the properties of a dilute. Gpelab grosspitaevskii equation laboratory is a flexible matlab toolbox devoted to the numerical computation of stationary and.
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