CMMSE: analysis and comparison of some numerical methods to solve a nonlinear fractional Gross–Pitaevskii system

In this work, we introduce and theoretically analyze various computational techniques to approximate the solutions of solve a fractional extension of a double condensate system. More precisely, the continuous model extends the well-known Gross–Pitaevskii equation to the fractional scenario, and considering two interacting condensates. The mathematical system considers two complex-valued regimes with coupling, and a mass and energy functions are associated to this model. Both are constant in time. Here, various discretizations are analyzed to solve this system. Some of them are able to preserve the mass and the energy, some are not. We discuss the existence of solutions, the consistency of the models, the stability and the convergence. Finally, from the computational point of view, some algorithms are simpler to code than others. In fact, those for which the mass and the energy are conserved are more difficult to implement. We discuss here pros and cons.