Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Abstract

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.