I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Referenced in 27 articles fortran programs for the timedependent grosspitaevskii equation in a fully anisotropic trap. Twosoliton collision for the grosspitaevskii equation in. Computer physics computer physics communications 00.
Gpelab, a matlab toolbox to solve grosspitaevskii equations ii. An efficient compact finite difference method for the. The grosspitaevskii equation describes the properties of a dilute. The dimensionless grosspitaevskii equation used in gpelab. 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. The model equation that gpelab solves is the grosspitaevskii equation. Grosspitaevskii equation for boseeinstein condensation. A matlab library of numerical functions for gross pitaevskii equation in 1d and 2d. 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. Exact soliton solutions of the 1d generalized gross.
The primary use of this code was for my research on rapidly rotating boseeinstein condensates. In this paper more general scalings shall be considered assuming positivity of the interaction. 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. Numerical solutions to the grosspitaevskii equation for boseeinstein condensates by luigi galati b. The model equation that gpelab solves is the grosspitaevskii equation gpe. This toolbox is available in the download section with a user guide. We present an efficient, unconditionally stable, and accurate numerical method for the solution of the grosspitaevskii equation.
Grosspitaevskii equation gpe determines the quantum wave function of a. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions. The grosspitaevskii equation describes the ground state of a quantum system of identical bosons using the hartreefock approximation and the pseudopotential interaction model. Boseeinstein condensates, nonlinear schrodinger equation, grosspitaevskii equation, stationary states, ground. 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. We study the global behavior of small solutions of the grosspitaevskii equation in three dimensions. 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. This paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and. 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. 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. Numerical solution for the grosspitaevskii equation.
Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to the numerical. 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. In preparation for the numerics we scale the 3d grosspitaevskii equation and obtain a fourparameter model. 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. Fourthorder algorithms for solving the imaginarytime grosspitaevskii equation in a rotating anisotropic trap siu a. This function solves the grosspitaevskii 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 grosspitaevskii equations. We consider the backward euler spectral besp scheme proposed in 10 for computing the stationary states of boseeinstein condensates becs through the grosspitaevskii equation. We study the numerical solution of the timedependent grosspitaevskii equation gpe describing a boseeinstein condensate bec at zero or very low temperature.
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. The problem is boundary and initial conditions that i. Gpelab, a matlab toolbox to solve grosspitaevskii equations i. Siam journal on numerical analysis siam society for.
The result for a large number of particles compares well with the thomasfermi approximation as expected. Numerical solution of the grosspitaevskii equation for. We begin with an introduction on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem. A spectral integral equation solution of the gross. This software is a cudaenabled nonlinear schrodinger gross pitaevskii equation solver. Complex solutions of the time fractional grosspitaevskii. We investigate the 1d generalized grosspitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. Comparing three numerical solvers of the grosspitaevskii equation posted on 03 february 2016. Under certain simplified assumptions small amplitudes, propagation in one direction, etc. Gpelab a free matlab toolbox to compute and simulate bose. Robust and accurate numerical schemes are implemented for computing stationary solutions and the dynamics of becs through the gpe. 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.
The grosspitaevskii equation and boseeinstein condensates. The considered model equation is related to the timedependent grosspitaevskii equation arising in the description of boseeinstein condensates in dilute gases. The model equation that gpelab solves is the gross pitaevskii equation gpe. This function solves the gross pitaevskii equation in a twodimensional space. Fourthorder algorithms for solving the imaginarytime. However the description can only be valid provided the amplitudes can be regarded as large, and this must be the case for all modes.
Solver for the twodimensional grosspitaevskii equation. Grosspitaevskii equation for the density matrix in the. The coefficients of terms of ggpe studied are arbitrary functions of time t. Gsgpes is a matlab gnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. Comparing three numerical solvers of the grosspitaevskii. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in the. We begin with the threedimensional 3d grosspitaevskii equation gpe with an angular momentum rotation term andor longrange ddi, state the two. In this equation only the short range part of the interaction between the. Published 19 january 2004 2004 iop publishing ltd journal of physics a. 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. We obtain conditions for the existence of a global attractor and find bounds for its dimension. Derivation of the time dependent gross pitaevskii equation.
The exact solutions of the ggpe are obtained via expansion method with particular soliton features highlighted. A simple and efficient numerical method for computing the. Gpelab, a matlab toolbox to solve gross pitaevskii equations ii. 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. Pdf gpelab, a matlab toolbox to solve grosspitaevskii. Recently, methods for solving the grosspitaevskii equation gpe arising in boseeinstein condensates becs 1, 20 based on riccati equation were introduced. 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. Some numerical methods have been proposed to obtain approximate solutions for. This software is a cudaenabled nonlinear schrodinger grosspitaevskii equation solver.
Gpelab, an open source matlab toolbox for the numerical. 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. The model equation that gpelab solves is the gross pitaevskii equation. A timesplitting pseudospectral method for the solution of. We show that the fixed point approach introduced in 10 fails to converge. Gsgpes is a matlabgnu octave suite of programs for the computation of the ground state of systems of 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. The gross pitaevskii equation gpe, named after eugene p. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to the numerical. We consider the timedependent grosspitaevskii equation describing the dynamics of rotating boseeinstein condensates and its discretization with. Grosspitaevskii equation for boseeinstein condensation application id. 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. Together with evgeny lifshitz and vladimir berestetskii.
In what follows, students will learn how to use the ndsolve command of mathematica. 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 grosspitaevskii equation laboratory is a flexible matlab toolbox devoted to the numerical computation of stationary and. It has a soliton solution, whose envelope does not change in form over time. 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. Then we present a new numerical method, cfdmaif method, which combines compact finite.