A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

Abstract

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

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

“…A Matlab implementation of the numerical model is provided in the Appendix of this work, and we used to produce approximations to the solutions of our continuous model. The results show that the energy and mass are approximately constant, in agreement with the theoretical results derived in this work, and improving computationally efforts already reported [20,26].…”

“…Moreover, the uniqueness in the present case is derived at the same time as the existence. In terms of conservation of energy, all the numerical models are capable of preserving this quantity except [26]. All the numerical models have a consistency of the second order in both space and time, and they are all stable and quadratically convergent in both space and time, for sufficiently small values of the computer parameters.…”

“…It is worth pointing out that there are already some reports by these same authors on the design of numerical methods to solve double Gross-Pitaevskii-type systems with fractional diffusion, most notably the papers by Serna-Reyes et al [26] and Serna-Reyes and Macías-Díaz [20]. In both works, the same system of fractional partial differential equations is investigated using a finite-difference methodology, the fractional derivatives are of the Riesz type, and they are discretized using fractional-order centered differences.…”

“…In both works, the same system of fractional partial differential equations is investigated using a finite-difference methodology, the fractional derivatives are of the Riesz type, and they are discretized using fractional-order centered differences. To start with, the proofs for the existence of solutions of the numerical model are more complicated in the case of the published papers [20,26]. Indeed, the arguments in those cases require using some fixed-point theorem in view of the nonlinear nature of those works.…”

“…Finally, in terms of implementation, the present numerical methodology is easy to implement computationally, in view that the code only requires solving a couple of linear systems at each time steps. On the other hand, the finite-difference schemes reported in [20,26] are harder to code. This is again due to the nonlinear nature of the numerical models.…”