scholarly journals Symmetry operators of the two-component Gross—Pitaevskii equation with a Manakov-type nonlocal nonlinearity

2016 ◽  
Vol 670 ◽  
pp. 012046 ◽  
Author(s):  
A V Shapovalov ◽  
A Yu Trifonov ◽  
A L Lisok
Keyword(s):  
Pitaevskii Equation ◽  
Nonlocal Nonlinearity ◽  
Two Component ◽  
Symmetry Operators

PARAMETRIC EXCITATION OF SOLITONS IN DIPOLAR BOSE–EINSTEIN CONDENSATES

2013 ◽  
Vol 27 (25) ◽  
pp. 1350184 ◽  
Author(s):  
A. BENSEGHIR ◽  
W. A. T. WAN ABDULLAH ◽  
B. A. UMAROV ◽  
B. B. BAIZAKOV
Keyword(s):  
Parametric Excitation ◽  
Einstein Condensate ◽  
Quantum Gases ◽  
Variational Approximation ◽  
Pitaevskii Equation ◽  
Nonlocal Nonlinearity ◽  
Bose Einstein Condensate ◽  
Atomic Interactions ◽  
Theoretical Predictions ◽  
Bose Einstein

In this paper, we study the response of a Bose–Einstein condensate with strong dipole–dipole atomic interactions to periodically varying perturbation. The dynamics is governed by the Gross–Pitaevskii equation with additional nonlinear term, corresponding to a nonlocal dipolar interactions. The mathematical model, based on the variational approximation, has been developed and applied to parametric excitation of the condensate due to periodically varying coefficient of nonlocal nonlinearity. The model predicts the waveform of solitons in dipolar condensates and describes their small amplitude dynamics quite accurately. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation and good agreement between them is found. The results can lead to better understanding of the properties of ultra-cold quantum gases, such as 52 Cr , 164 Dy and 168 Er , where the long-range dipolar atomic interactions dominate the usual contact interactions.

Dynamic vortex evolution of harmonically trapped coupled Bose–Einstein condensate with quintic nonlinearity

2021 ◽  
Author(s):  
Yunsong Guo ◽  
Yubin Jiao ◽  
Xiaoning Liu ◽  
Xiangbo Zhu ◽  
Ying Wang
Keyword(s):  
Periodic Breathing ◽  
Oscillation Period ◽  
Angular Frequency ◽  
Equation Model ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Atomic Systems ◽  
Bose Einstein Condensate ◽  
Two Component ◽  
Bose Einstein

In this study, we investigate the evolution of vortex in harmonically trapped two-component coupled Bose–Einstein condensate with quintic-order nonlinearity. We derive the vortex solution of this two-component system based on the coupled quintic-order Gross–Pitaevskii equation model and the variational method. It is found that the evolution of vortex is a metastable state. The radius of vortex soliton shrinks and expands with time, resulting in periodic breathing oscillation, and the angular frequency of the breathing oscillation is twice the value of the harmonic trapping frequency under infinitesimal nonlinear strength. At the same time, it is also found that the higher-order nonlinear term has a quantitative effect rather than a qualitative impact on the oscillation period. With practical experimental setting, we identify the quasi-stable oscillation of the derived vortex evolution mode and illustrated its features graphically. The theoretical results developed in this work can be used to guide the experimental observation of the vortex phenomenon in ultracold coupled atomic systems with quintic-order nonlinearity.

scholarly journals The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 201
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin ◽  
Andrey Yu. Trifonov
Keyword(s):  
Linear Equation ◽  
Semiclassical Approximation ◽  
Nonlocal Interaction ◽  
Pitaevskii Equation ◽  
Dimensional Manifold ◽  
Complex Germ ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Symmetry Operators ◽  
Interaction Term

We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.

scholarly journals Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 366 ◽  
Author(s):  
Alexander Shapovalov ◽  
Andrey Trifonov
Keyword(s):  
Active Substance ◽  
Semiclassical Approximation ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Kpp Equation ◽  
Activity Effect ◽  
Two Component ◽  
Symmetry Operators ◽  
Model Equations ◽  
Nonlocal Reaction

We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

scholarly journals An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2727
Author(s):  
Jorge E. Macías-Díaz ◽  
Nuria Reguera ◽  
Adán J. Serna-Reyes
Keyword(s):  
Numerical Algorithm ◽  
Fractional Derivatives ◽  
Type System ◽  
Continuous Model ◽  
Pitaevskii Equation ◽  
Mathematical System ◽  
Two Component ◽  
Time Variables ◽  
Complex Valued ◽  
Quadratic Order

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

Vortex states in rotating two-component dipolar Bose–Einstein condensates

2019 ◽  
Vol 33 (10) ◽  
pp. 1950080 ◽  
Author(s):  
Qiang Zhao
Keyword(s):  
Numerical Simulations ◽  
Stationary State ◽  
Significant Role ◽  
Formation Process ◽  
Vortex Structures ◽  
Pitaevskii Equation ◽  
Vortex States ◽  
Dipole Interactions ◽  
Two Component ◽  
Bose Einstein

We consider the stationary state properties of pseudo-spin-1/2 rotating dipolar Bose–Einstein condensates (BECs) by numerical simulations of the Gross–Pitaevskii equation. Different vortex structures in each component are studied, depending on the competition between the dipole–dipole interactions (DDIs) and rotational. We also investigate the differences of vortex number in the two components, showing that anisotropic nature of DDIs plays a significant role in vortices formation process.

Sonic horizon formation for coupled one-dimensional Bose–Einstein condensate in an elongated harmonic potential

2018 ◽  
Vol 32 (29) ◽  
pp. 1850352
Author(s):  
Ying Wang ◽  
Shuyu Zhou
Keyword(s):  
Expansion Method ◽  
Einstein Condensate ◽  
Wave Functions ◽  
Harmonic Potential ◽  
Pitaevskii Equation ◽  
Two Component System ◽  
One Dimensional ◽  
Bose Einstein Condensate ◽  
Two Component ◽  
Bose Einstein

We theoretically studied the sonic horizon formation problem for coupled one-dimensional Bose–Einstein condensate trapped in an external elongated harmonic potential. Based on the coupled (1[Formula: see text]+[Formula: see text]1)-dimensional Gross–Pitaevskii equation and F-expansion method under Thomas–Fermi formulation, we derived analytical wave functions of a two-component system, from which the sonic horizon’s occurrence criteria and location were derived and graphically demonstrated. The theoretically derived results of sonic horizon formation agree pretty well with that from the numerically calculated values.

scholarly journals STABILITY OF BOSE–EINSTEIN CONDENSATES CONFINED IN TRAPS

2000 ◽  
Vol 14 (07) ◽  
pp. 655-719 ◽  
Author(s):  
TAKEYA TSURUMI ◽  
HIROFUMI MORISE ◽  
MIKI WADATI
Keyword(s):  
Time Development ◽  
Theoretical Research ◽  
Einstein Condensation ◽  
Pitaevskii Equation ◽  
Static Properties ◽  
Atomic Vapors ◽  
Dilute Gas ◽  
Two Component ◽  
The Stability ◽  
Bose Einstein

Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.

Spectroscopic Observations of Visual Double Stars

1965 ◽  
Vol 5 ◽  
pp. 109-111
Author(s):  
Frederick R. West
Keyword(s):  
Radial Velocity ◽  
High Dispersion ◽  
Velocity Difference ◽  
Spectrograph Slit ◽  
Spectroscopic Observations ◽  
Double Stars ◽  
Visual Double Stars ◽  
Relative Orbit ◽  
Two Component ◽  
Star Images

There are certain visual double stars which, when close to a node of their relative orbit, should have enough radial velocity difference (10-20 km/s) that the spectra of the two component stars will appear resolved on high-dispersion spectrograms (5 Å/mm or less) obtainable by use of modern coudé and solar spectrographs on bright stars. Both star images are then recorded simultaneously on the spectrograph slit, so that two stellar components will appear on each spectrogram.

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