A two-parameter continuation algorithm for vortex pinning in rotating Bose–Einstein condensates

Jeng, B.-W.; Wang, Y.-S.; Chien, C.-S.

doi:10.1016/j.cpc.2012.10.001

Cited by 21 publications

(16 citation statements)

References 39 publications

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“…Various SCM have been exploited to solve numerical solutions of the GPE [7,14,20,29]. In particular, the Fourier sine functions [7], the Legendre polynomials [20], and the Chebyshev polynomials of the first kind and second kind [14,29] have been used as the basis functions for the trial function space of the SCM to solve the GPE.…”

Section: A Note On Scmsmentioning

confidence: 99%

“…In particular, the Fourier sine functions [7], the Legendre polynomials [20], and the Chebyshev polynomials of the first kind and second kind [14,29] have been used as the basis functions for the trial function space of the SCM to solve the GPE. For the Fourier sine functions, one chooses uniform grids on the domain as the collocation points.…”

Section: A Note On Scmsmentioning

confidence: 99%

“…Further, let T n (x) be the n-th degree Chebyshev polynomial of the first kind, i.e. [14,23]. Thenφ k (±1) = 0 and the trial function space can be expressed as V 0 N = span{φ k (x)φ j (y) : k, j = 0, 1, .…”

Section: A Note On Scmsmentioning

confidence: 99%

“…Accordingly the original tangent vectors were split into two unit tangent vectors which were used as the constraint conditions for solving the bordered linear systems. Further, Jeng et al [14] described a different type of two-parameter continuation algorithm for one single component rotating BEC, where the chemical potential λ and angular velocity ω were treated as the continuation parameters simultaneously, and the unit tangent vector and normalization condition were used as the corresponding constraint conditions, respectively. The numerical results reported therein can effectively imitate experimental results for rotating BECs [30].…”

Section: Introductionmentioning

confidence: 99%

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

Chen

Wang

Jeng

et al. 2014

International Journal of Computer Mathematics

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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.

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Section: A Note On Scmsmentioning

confidence: 99%

Section: A Note On Scmsmentioning

confidence: 99%

Section: A Note On Scmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Chen

Wang

Jeng

et al. 2014

International Journal of Computer Mathematics

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“…Various condensate geometries [19], simplifications and special cases [20] were taken into account. Numerical methods involved finite-difference approach [19,21,22], bi-k-Lagrange elements [23], spectral collocation methods with Chebyshev polynomials of the first and second kind [24] as well as basis set expansion technique [25]. Time-dependent equations were solved with implicit and semi-implicit Crank-Nicolson methods [26,27,18,28,29], Euler scheme [22], third and fourth-order adaptive Runge-Kutta methods [30], splitstep finite difference method [22] and time-splitting sine and Fourier pseudospectral methods [31,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

Voronych

Buraczewski

Matuszewski

et al. 2017

Computer Physics Communications

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A novel, optimized numerical method of modeling of an exciton-polariton superfluid in a semiconductor microcavity was proposed. Exciton-polaritons are spin-carrying quasiparticles formed from photons strongly coupled to excitons. They possess unique properties, interesting from the point of view of fundamental research as well as numerous potential applications. However, their numerical modeling is challenging due to the structure of nonlinear differential equations describing their evolution. In this paper, we propose to solve the equations with a modified Runge-Kutta method of 4th order, further optimized for efficient computations. The algorithms were implemented in form of C++ programs fitted for parallel environments and utilizing vector instructions. The programs form the EPCGP suite which have been used for theoretical investigation of exciton-polaritons.Keywords: exciton-polariton superfluid; Bose-Einstein condensate; microcavity; Gross-Pitaevskii equation; Runge-Kutta method Nature of problem: An exciton-polariton superfluid is a novel, interesting physical system allowing investigation of high temperature Bose-Einstein condensation of exciton-polaritons-quasiparticles carrying spin. They have brought a lot of attention due to their unique properties and potential applications in polariton-based optoelectronic integrated circuits. This is an out-of-equilibrium quantum system confined within a semiconductor microcavity. It is described by a set of nonlinear differential equations similar in spirit to the Gross-Pitaevskii (GP) equation, but their unique properties do not allow standard GP solving frameworks to be utilized. Finding an accurate and efficient numerical algorithm as well as development of optimized numerical software is necessary for effective theoretical investigation of exciton-polaritons. Program summarySolution method: A Runge-Kutta method of 4th order was employed to solve the set of differential equations describing exciton-polariton superfluids. The method was fitted for the exciton-polariton equations and further optimized. The C++ programs utilize OpenMP extensions and vector operations in order to fully utilize the computer hardware.

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Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

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A two-parameter continuation algorithm for vortex pinning in rotating Bose–Einstein condensates

Cited by 21 publications

References 39 publications

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

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

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

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