Tackling the Gross-Pitaevskii energy functional with the Sobolev gradient - Analytical and numerical results

TL;DR

This paper proves the global existence and uniqueness of solutions converging to the minimizer of the Gross-Pitaevskii energy functional and demonstrates the effectiveness of Sobolev gradient methods through numerical experiments.

Contribution

It provides a rigorous proof of convergence for Sobolev gradient methods applied to the Gross-Pitaevskii functional and illustrates their practical benefits in numerical simulations.

Findings

Proved global existence and uniqueness of the minimizer trajectory.

Demonstrated high numerical stability of Sobolev gradient method.

Showed rapid convergence towards the energy functional's minimizer.

Abstract

In the first part of this contribution we prove the global existence and uniqueness of a trajectory that globally converges to the minimizer of the Gross-Pitaevskii energy functional for a large class of external potentials. Using the method of Sobolev gradients we can provide an explicit construction of this minimizing sequence. In the second part we numerically apply these results to a specific realization of the external potential and illustrate the main benefits of the method of Sobolev gradients, which are high numerical stability and rapid convergence towards the minimizer.