A numerical study of the ground state and dynamics of atomic–molecular Bose–Einstein condensates

In this paper, we numerically investigate the ground-state structure and dynamics of atomic–molecular Bose–Einstein condensates at zero temperature, which are modeled by coupled Gross–Pitaevskii equations (GPEs). To get the ground state, we evolve a gradient flow with discrete normalization numerically. To study the dynamics, we employ an efficient numerical method—the time-splitting Fourier pseudospectral method for solving the coupled GPEs. The proposed numerical methods have been numerically tested and employed in studying the mechanism on how an atomic condensate can be converted into an atomic–molecular mixture or a pure molecular condensate from an atomic condensate either in equilibrium or dynamically.     

Numerical exploration of vortex matter in Bose–Einstein condensates

We describe a finite element numerical approach to the full Hartree-Fock-Bogoliubov treatment of a vortex lattice in a rapidly rotating Bose–Einstein condensate. We study the system in the regime of high thermal or significant quantum fluctuations where we are presented with a very large nonlinear unsymmetric eigenvalue problem which is indefinite and which possesses low-lying excitations clustered arbitrarily close to zero, a problem that requires state-of-the-art numerical techniques.     

Distributed Kalman filter based on Metropolis–Hastings sampling strategy

The reasonable extraction and utilization of observation information are considered as the key of design and optimization of filters. By constructing the sampling steps of multi-sensor bootstrapped observations and the validation process of credible observations, a novel distributed Kalman filter in multi-sensor observations based on Metropolis–Hastings (M–H) sampling strategy is proposed in this paper. Firstly, combined with the latest observation information and the accuracy information of sensor which is also used to describe the prior modeling knowledge of observation system, we design the bootstrapped observation sampling for linear observation system. Secondly, aiming to the consistency deviation phenomenon appearing in the bootstrapped observations of single sensor, through constructing the likelihood degree of multi-sensor bootstrapped observations and the accept probability of credible observations, meanwhile, combined with the M–H sampling strategy, we give the validation method of credible observations. Finally, the realization steps of new algorithm are constructed according to the weighted fusion criterion. The advantage of new algorithm is to improve greatly the filtering precision with additional less hardware costs. The theoretical analysis and experimental results show the feasibility and efficiency of the proposed algorithm.     

OCTBEC—A Matlab toolbox for optimal quantum control of Bose–Einstein condensates

OCTBEC is a Matlab toolbox designed for optimal quantum control, within the framework of optimal control theory (OCT), of Bose–Einstein condensates (BEC). The systems we have in mind are ultracold atoms in confined geometries, where the dynamics takes place in one or two spatial dimensions, and the confinement potential can be controlled by some external parameters. Typical experimental realizations are atom chips, where the currents running through the wires produce magnetic fields that allow to trap and manipulate nearby atoms. The toolbox provides a variety of Matlab classes for simulations based on the Gross–Pitaevskii equation, the multi-configurational Hartree method for bosons, and on generic few-mode models, as well as optimization problems. These classes can be easily combined, which has the advantage that one can adapt the simulation programs flexibly for various applications.     

Unified way for computing dynamics of Bose–Einstein condensates and degenerate Fermi gases

In this work we present a very simple and efficient numerical scheme which can be applied to study the dynamics of bosonic systems like, for instance, spinor Bose–Einstein condensates (BEC) with non-local interactions but equally well works for Fermi gases. The method we use is a modification of well known Split Operator Method (SOM). We carefully examine this algorithm in the case of F=1 spinor BEC without and with dipolar interactions and for strongly interacting two-component Fermi gas. Our extension of the SOM method has many advantages: it is fast, stable, and keeps constant all the physical constraints (constants of motion) at high level.     

Multi-parameter continuation and collocation methods for rotating multi-component Bose–Einstein condensates

We describe multi-parameter continuation methods combined with spectral collocation methods for computing numerical solutions of rotating two-component Bose–Einstein condensates (BECs), which are governed by the Gross–Pitaevskii equations (GPEs). Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena. Numerical results on rotating two-component BECs are reported.     

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.     

Efficient continuation methods for spin-1 Bose–Einstein condensates in a magnetic field

We study linear stability analysis for spin-1 Bose–Einstein condensates (BEC). We show that all bounded solutions of this physical system are neutrally stable. In particular, all steady-state solutions of the physical system, and the associated discrete steady-state solutions are neutrally stable. Next, we consider the physical system without the affect of magnetic field. By exploiting the physical properties of both ferromagnetic and antiferromagnetic cases, we develop efficient multi-level pseudo-arclength continuation algorithms combined with a spectral collocation method for these two cases, respectively. When the magnetic field is imposed on the physical system, an additional multi-level continuation algorithm is described for the ferromagnetic case. Extensive numerical results for spin-1 BEC in a magnetic field, and in optical lattices are reported.     

Excited Bose–Einstein condensates: Quadrupole oscillations and dark solitons

We study the dynamics of atomic Bose–Einstein condensates (BECs), when the quadrupole mode is excited. Within the Thomas–Fermi approximation, we derive an exact first-order system of differential equations that describes the parameters of the BEC wave function. Using perturbation theory arguments, we derive explicit analytical expressions for the phase, density and width of the condensate. Furthermore, it is found that the observed oscillatory dynamics of the BEC density can even reach a quasi-resonance state when the trap strength varies according to a time-periodic driving term. Finally, the dynamics of a dark soliton on top of a breathing BEC are also briefly discussed.     

Projection gradient method for energy functional minimization with a constraint and its application to computing the ground state of spin–orbit-coupled Bose–Einstein condensates

In this study, we propose a projection gradient method for energy functional minimization with a constraint, which we use to compute the ground state of spin–orbit-coupled Bose–Einstein condensates at extremely low temperatures. The method has the advantage that it maintains the constraint when evolving a gradient flow to find the energy functional minimization under a constraint. The original gradient projection method for energy functional minimization under a constraint only considers an energy functional with real functions as variables. Thus, we extend it to consider complex functions as independent variables. We apply the newly proposed method to study the ground state solution of spin–orbit-coupled pseudo-spin 1/2 Bose–Einstein condensates. Detailed numerical results demonstrate the effectiveness of our method. Using this method, we found various types of ground state structures of spin–orbit coupled Bose–Einstein condensates.     

A modified conditional Metropolis–Hastings sampler

A modified conditional Metropolis–Hastings sampler for general state spaces is introduced. Under specified conditions, this modification can lead to substantial gains in statistical efficiency while maintaining the overall quality of convergence. Results are illustrated in two settings: a toy bivariate Normal model and a Bayesian version of the random effects model.     

A splitting compact finite difference method for computing the dynamics of dipolar Bose–Einstein condensate

We numerically study the nonlocal Gross–Pitaevskii equation (NGPE) which describes the dynamics of Bose–Einstein condensates (BEC) with dipole–dipole interaction at extremely low temperature. In preparation for the numerics, first we reformulate the dimensionless NGPE into a Schrödinger–Poisson system. Then, we discretize the three-dimensional Schrödinger–Poisson system in space by a sixth-order compact finite difference method and in time by a splitting technique. By means of three-dimensional discrete fast Sine transform, we develop a fast solver for the resulting discretized system. Finally, we present numerical examples in three dimensions to demonstrate the power of the numerical methods and to discuss some physics of dipolar BEC. The merits of the proposed method for the NGPE are that it is fast and unconditionally stable. Moreover, the method is of spectral-like accuracy in space, and conserves the particle number and the energy of the system in the discretized level.     

Intelligent computing for numerical treatment of nonlinear prey–predator models

In this study, a new computing paradigm is presented for evaluation of dynamics of nonlinear prey–predator mathematical model by exploiting the strengths of integrated intelligent mechanism through artificial neural networks, genetic algorithms and interior-point algorithm. In the scheme, artificial neural network based differential equation models of the system are constructed and optimization of the networks is performed with effective global search ability of genetic algorithm and its hybridization with interior-point algorithm for rapid local search. The proposed technique is applied to variants of nonlinear prey–predator models by taking different rating factors and comparison with Adams numerical solver certify the correctness for each scenario. The statistical studies have been conducted to authenticate the accuracy and convergence of the design methodology in terms of mean absolute error, root mean squared error and Nash-Sutcliffe efficiency performance indices.     

The numerical simulation of periodic solutions for a predator–prey system

In this article, the existence and global stability of periodic solutions for a semi-ratio-dependent predator–prey system with Holling IV functional response and time delays are investigated. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results.     

2d numerical simulations of an electron–phonon hydrodynamical model based on the maximum entropy principle

2d numerical solutions of a new macroscopic model describing the electron transport in semiconductors coupled with the heating of the crystal lattice are presented. The model has been obtained with the use of the maximum entropy principle. Numerical simulations of a nanoscale MOSFET are presented and the influence of self heating on the electrical characteristics is analyzed.     

A nonstandard numerical scheme of predictor–corrector type for epidemic models

In this paper we construct and develop a competitive nonstandard finite difference numerical scheme of predictor–corrector type for the classical SIR epidemic model. This proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model. The nonstandard finite difference scheme with Conservation Law (NSFDCL) developed here satisfies some important properties associated with the considered SIR epidemic model, such as positivity, boundedness, monotonicity, stability and conservation of frequency of the oscillations. Numerical comparisons between the NSFDCL numerical scheme developed here and Runge–Kutta type schemes show its effectiveness.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

Formulation and study of thermal–mechanical coupling for saturated porous media

A formulation for the thermo-poro-elasto-plastic coupling analysis is presented, in which the energy balance equation is re-derived based on the concept of free enthalpy. The corresponding finite element procedures are developed and implemented into the commercial software ADINA. A complete solution strategy is described. The poro-mechanical and thermal coupling is conducted based on the simultaneous solution scheme for the isothermal poro-elasto-plastic interaction and the conventional thermal analysis. The complete solution procedures can be employed to solve both transient static and dynamic problems. The one-dimension column consolidation scenario is effectively re-examined with the proposed solution procedures, with heat and pore pressure dissipated on the top of the column. Accurate and reliable results with temperature and pore pressure distributions at different time steps are provided.     

An atomistic geometrical model of the B-DNA configuration for DNA–radiation interaction simulations

In this paper, an atomistic geometrical model for the B-DNA configuration is explained. This model accounts for five organization levels of the DNA, up to the 30 nm chromatin fiber. However, fragments of this fiber can be used to construct the whole genome. The algorithm developed in this work is capable to determine which is the closest atom with respect to an arbitrary point in space. It can be used in any application in which a DNA geometrical model is needed, for instance, in investigations related to the effects of ionizing radiations on the human genetic material. Successful consistency checks were carried out to test the proposed model.