Theoretical analysis of a conservative finite-difference scheme to solve a Riesz space-fractional Gross–Pitaevskii system

In this work, we propose a fractional extension of the multi-dimensional Gross–Pitaevskii system that describes a two-component Bose–Einstein condensate with an internal atomic Josephson junction. The fractional problem is governed by two parabolic partial differential equations that consider fractional spatial derivatives of the Riesz type along with coupling terms. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of a closed and bounded domain. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant, whence the need to provide energy-conserving schemes to solve the system is pragmatically justified. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved at each temporal step. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservation properties.