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.

scholarly journals A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1765
Author(s):  
Adán J. Serna-Reyes ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Initial Conditions ◽  
Type System ◽  
Continuous System ◽  
Mass Conservation ◽  
Continuous Model ◽  
Manuscript Studies ◽  
Multiple Dimensions ◽  
Negative Constant ◽  
Energy Conserving ◽  
Finite Difference Discretization

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

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.

Solvability of a product-type system of difference equations with six parameters

2016 ◽  
Vol 8 (1) ◽  
pp. 29-51 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Closed Form ◽  
Difference Equations ◽  
Type System ◽  
Product Type ◽  
System Of Difference Equations ◽  
Initial Values ◽  
Complex Valued

AbstractClosed form formulas for well-defined complex-valued solutions to a product-type system of difference equations of interest with six parameters are presented. The form of the solutions is described in detail in terms of the parameters and initial values.

scholarly journals Numerical calculating linear vibrations of third order systems involving fractional operators

2012 ◽  
Vol 34 (2) ◽  
pp. 91-99
Author(s):  
Nguyen Van Khang ◽  
Tran Dinh Son ◽  
Bui Thi Thuy
Keyword(s):  
Dynamic Response ◽  
Numerical Method ◽  
Numerical Algorithm ◽  
Fractional Derivatives ◽  
Integration Method ◽  
Fractional Operators ◽  
Dynamic Calculation ◽  
Third Order ◽  
Linear Vibrations ◽  
Definition Of

This paper presents a numerical method for dynamic calculation of third order systems involving fractional operators. Using the Liouville-Rieman's definition of fractional derivatives, a numerical algorithm is developed on base of the well-known Newmark integration method to calculate dynamic response of third order systems. Then, we apply this method to calculate linear vibrations of viscoelastic systems containing fractional derivatives.

scholarly journals Amplitude equation formalism for reaction—subdiffusion systems

2021 ◽  
pp. 1-15
Author(s):  
Dmitry Alexeevich Zenyuk ◽  
Georgii Gennadyevich Malinetskii
Keyword(s):  
Hopf Bifurcation ◽  
Chemical Kinetics ◽  
Fractional Derivatives ◽  
Amplitude Equation ◽  
Two Component System ◽  
Caputo Fractional Derivatives ◽  
Component System ◽  
Two Component ◽  
Normal Diffusion ◽  
Linear Differential

The paper presents derivation of the amplitude equation for the Hopf bifurcation in the two-component system with nonlinear chemical kinetics and subdiffusion. Anomalous diffusion transport is described via Caputo fractional derivatives. The obtained amplitude equation is much more complex compared to the case of normal diffusion because solutions of fractional order linear differential equations have inconvenient behavior.

Formation and interaction characteristics of two-component spatial weak-light soliton in a four-level double-Λ type system

2010 ◽  
Vol 27 (2) ◽  
pp. 208 ◽  
Author(s):  
Yanchao She ◽  
Denglong Wang ◽  
Weixi Zhang ◽  
Zhangming He ◽  
Jianwen Ding
Keyword(s):  
Type System ◽  
Two Component ◽  
Weak Light

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.

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