We study the global behavior of small solutions of the grosspitaevskii equation in three dimensions. We study the numerical solution of the timedependent gross pitaevskii equation gpe describing a boseeinstein condensate bec at zero or very low temperature. We prove that in a certain asymptotic regime, solutions of the gross pitaevskii equation converge to solutions of the incompressible euler equation, and solutions to the parabolic ginzburglandau equations converge to solutions of a limiting equation which we identify. The grosspitaevskii equation for in appropriate coordinates has the form. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. I spend my weekend to get a solution with mathematica without success. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of gross pitaevskii equations that arise in the physics of boseeinstein condensates. Quantum grosspitaevskii equation jutho haegeman,1 damian draxler,2 vid stojevic,3 j. Learning matlab society for industrial and applied. Then we present a new numerical method, cfdmaif method, which combines compact finite difference method cfdm. The prompt is still there, but it is now just one familiar face in a crowd. We present an efficient, unconditionally stable, and accurate numerical method for the solution of the gross pitaevskii equation. Numerical solution of the grosspitaevskii equation for bose. Usually it does not have analytic solution and numerical methods, such as the splitstep cranknicolson 4 and fourier spectral 5 methods, are used for its solution.
Osborne,4 and frank verstraete1,2 1department of physics and astronomy, university of ghent, krijgslaan 281 s9, b9000 ghent, belgium 2faculty of physics, university of vienna, boltzmanngasse 5, a1090 wien, austria 3maxplanckinstitut fur quantenoptik, hanskopfermannstr. Together with evgeny lifshitz and vladimir berestetskii. The grosspitaevskii equation is a partial differential equation in space and time variables. Galati, luigi, numerical solutions to the gross pitaevskii equation for boseeinstein condensates 20. 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. By means of a new variational method we report a direct solution for quintic and cubicquintic nonlinear 1d grosspitaeskii equations gpe in a harmonic confined potential. 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 dynamics of gross. 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 dynamics of gross.
Gpelab is a matlab toolbox developed to help physicists for computing ground states or dynamics of quantum systems modeled by grosspitaevskii equations. A timesplitting pseudospectral method for the solution of the grosspitaevskii equations using spherical harmonics with generalisedlaguerre basis functions hayder salman 20 journal of computational physics. Solve the 2d grosspitaevskii equation for boseeinstein condensate in a static external potential. Pitaevskii equation on unbounded domains a linear system. Decay for travelling waves in the grosspitaevskii equation philippe gravejat 1 abstract we study the limit at in. Gpelab, a matlab toolbox to solve grosspitaevskii equations. We solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by symplectic shooting method ssm. Currently, there are standard software packages like gpelab in matlab or. This software is a cudaenabled nonlinear schrodinger grosspitaevskii equation solver. Gpelab grosspitaevskii equation laboratory is a flexible matlab toolbox devoted to the numerical computation of stationary and.
International audiencegpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of gross pitaevskii equations that arise in the physics of boseeinstein condensates. Marco caliaris software profs area scienze ed ingegneria. Gsgpe, ground state for systems of coupled grosspitaevskii equations, download code, references cort09. Solving the gpe lets us describe a dilute becs ground state as well as its linear and nonlinear excitations and transport properties at ultralow temperatures. A numerical study of adaptive space and time discretisations. Blowup solutions for two coupled grosspitaevskii equations. The instability occurs as a result of a linear instability of a vortexfree. This software is a cudaenabled nonlinear schrodinger gross pitaevskii equation solver. Gpelab 1 g ross p itaevskii e quation lab oratory is a flexible matlab toolbox devoted to the numerical computation of stationary and dynamical solutions of 1d2d3d grosspitaevskii equations gpes,, arising in the modeling of boseeinstein condensates becs. 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. Home browse by title periodicals journal of computational physics vol.
Solving the gpe lets us describe a dilute becs ground state as well as its linear and nonlinear excitations and. 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. A spectral integral equation solution of the gross. Gpelab, an open source matlab toolbox for the numerical.
I am trying to solve the gross pitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Probing quasiintegrability of the grosspitaevskii equation. The grosspitaevskii equation gpe, that describes the wave function of a number of coherent bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions. Upon starting matlab 7 in 2008, one gets four tabbed windows, six main menus, and a dozen or so clickable buttons. Octbeca matlab toolbox for optimal quantum control of bose. Padua2dm a matlaboctave code for interpolation and cubature at the padua points, download code, references cdmsv09. The result for a large number of particles compares well with the thomasfermi approximation as expected. The primary use of this code was for my research on rapidly rotating boseeinstein condensates. Numerical solution for the grosspitaevskii equation. Gpelab, a matlab toolbox to solve grosspitaevskii equations i. Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bound state eigenvalues as well as the corresponding wavefunctions are. 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. Vortex solutions of the discrete grosspitaevskii equation.
Galati, luigi, numerical solutions to the grosspitaevskii equation for boseeinstein condensates 20. The condensates of alkali vapours are pure and dilute, so that the grosspitaevskii gp. 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. You wil find many working examples when you search for matlab runge kutta. Robust and accurate numerical schemes are implemented for computing stationary solutions and the dynamics of becs through the gpe. Gpelab a free matlab toolbox to compute and simulate.
We study the numerical solution of the timedependent grosspitaevskii equation gpe describing a boseeinstein condensate bec at zero or very low temperature. This toolbox allows the user to define a large range of physical problems 1d2d3d equations, general nonlinearities, rotation term, multicomponents problems. Pitaevskii equation on unbounded domains sarra, scott a. The model equation that gpelab solves is the grosspitaevskii equation gpe. Gsgpes is a matlab gnu octave suite of programs for the computation of the ground state of systems of gross pitaevskii equations. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the thomasfermi radius. The model equation that gpelab solves is the grosspitaevskii equation. Learning matlab society for industrial and applied mathematics. Robust and accurate numerical schemes are implemented for computing stationary. International audiencegpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in. We investigate the 1d generalized grosspitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. Oct 19, 2019 gsgpes is a matlabgnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. Grosspitaevskii pitaevskii gp equation macroscopic quantum wavefunction a quantum state with large number of bosons occupying the same oneparticle states can be described by a macroscopic wavefunction 0 r t, 0 r t n r t, remark.
Gsgpes is a matlab gnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. Download wolfram player under certain simplified assumptions small amplitudes, propagation in one direction, etc. Gpue is a grosspitaevskii equation solver that is accelerated on gpu hardware. This function solves the grosspitaevskii equation in a twodimensional space. In preparation for the numerics we scale the 3d grosspitaevskii equation and obtain a fourparameter model. I am also interested in solving the grosspitaevskiiequation. Pdf gpelab, a matlab toolbox to solve grosspitaevskii. The model equation that gpelab solves is the gross pitaevskii equation gpe.
We present an efficient, unconditionally stable, and accurate numerical method for the solution of the grosspitaevskii equation. This function solves the gross pitaevskii equation in a twodimensional space. Gpelab is a matlab toolbox developed to help physicists for computing ground states or dynamics of quantum systems modeled by gross pitaevskii equations. Gpelab, a matlab toolbox to solve grosspitaevskii equations ii. Solver for the twodimensional gross pitaevskii equation. A system of grosspitaevskii equations gpes is used to. The square of the wave function, times the above mentioned factor, is defined as the hartree potential. It can compute the ground state in the defocusing case, for any. The gross pitaevskii equation gpe, that describes the wave function of a number of coherent bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. Grosspitaevskii equation gpe determines the quantum wave function of a gas of bosons in the limit in which thermal and quantum fluctuations are negligibly small. The computation is based on a spectral decomposition of the solution into hermite. Matlab began life as a friendly interface to numerical libraries for linear algebra.
Decay for travelling waves in the grosspitaevskii equation. 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. Quantum grosspitaevskii equation for a complex manifold, the tdvp can be understood as a replacement of schrodingers equation by. Onedimensional cubicquintic grosspitaevskii equation. 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. Vortex solutions of the discrete grosspitaevskii equation starting from the anticontinuum limit vortex solutions of the discrete grosspitaevskii equation starting from the anticontinuum limit cuevas, j james, g kevrekidis, p.
This toolbox is available in the download section with a user guide. A hermite pseudospectral method for solving systems of grosspitaevskii equations weishaupl, rada m. The coefficients of terms of ggpe studied are arbitrary functions of time t. Siam journal on numerical analysis siam society for. The model equation that gpelab solves is the gross pitaevskii equation. In preparation for the numerics we scale the 3d gross pitaevskii equation and obtain a fourparameter model. As a basic principle, benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations. Gsgpes is a matlabgnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. We begin with an introduction on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem.
Grosspitaevskii equation for boseeinstein condensation. Update the question so its ontopic for mathematica stack exchange. In the last decade the experimental realization of boseeinstein condensation in trapped alkalimetal gases at ultralow temperatures has stimulated an intense interest in the production of vortices and vortex arrays and theoretical investigations of their structure, energy, dynamics and stability. Sep 19, 2006 we solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by symplectic shooting method ssm. The grosspitaevskii equation gpe, named after eugene p. Every variable in matlab was a matrix, which made it easy to learn how to solve certain core problems and interact with the results. An efficient compact finite difference method for the. The toolbox provides a variety of matlab classes for simulations based on the grosspitaevskii equation, the. Rungekutta 4th order matlab answers matlab central. 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 equations. Pade approximations of solitary wave solutions of the gross. Gpelab a free matlab toolbox to compute and simulate bose. Grosspitaevskii equation for boseeinstein condensation application id. We explore the influence of the quintic nonlinear term on the ideal 1d pure cigarlike shape model for attractive and repulsive atomatom interaction in boseeinstein.
For a dissipative variant of the twodimensional grosspitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. We work in the setting of the whole plane and possibly the whole threedimensional space in the gross. A spectral integral equation solution of the grosspitaevskii. Numerical solution of the grosspitaevskii equation for. Solver for the twodimensional grosspitaevskii equation. Twosoliton collision for the grosspitaevskii equation in. This thesis open access is brought to you for free and open access by the graduate studies, jack. The exact solutions of the ggpe are obtained via expansion method. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to the numerical. I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Upon starting matlab version 4 in 1992, one got a simple prompt demanding that the user start defining variables and running functions on them. International audiencethis 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. Siam journal on numerical analysis society for industrial. Exact soliton solutions of the 1d generalized gross.