Adaptation and assessement of projected Nesterov accelerated gradient flow to compute stationary states of nonlinear Schrödinger equations

Abstract

The aim of the paper is to derive minimization algorithms based on the Nesterov accelerated gradient flow [Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion paper, (2007). Available at http://www.ecore.be/DPs/dp_1191313936.pdf; Y. Nesterov, A method of solving a convex programming problem with convergence rate $\mathcal{O}(1/k^2)$. In Doklady Akademii Nauk, Vol. 269, Russian Academy of Sciences, 1983, pp. 543–547; Y. Nesterov, Introductory Lectures on Convex Optimization: A basic course, Kluwer Academic Publishers, Massachusetts, 2004.] to compute the ground state of nonlinear Schrödinger equations, which can potentially include a fractional laplacian term. A comparison is developed with standard gradient flow formulations showing that the Nesterov accelerated gradient flow has some interesting properties but at the same time finds also some limitations due to the nature of the problem. A few simulations are finally reported to understand the behaviour of the algorithms and open the path to further complicate questions that require more advanced studies concerning the application of the Nesterov accelerated gradient flow to nonlinear Schrödinger equations.

KEYWORDS:

• Bose-Einstein condensation
• nonlinear space fractional Schrödinger equation
• stationary states
• gradient flow
• Nesterov accelerated gradient flow
• pseudo-spectral method

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 81V80
• 65N25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

X. ANTOINE and C. BENTAYAA sincerely thank the funding support of the University of Lorraine under the program LUE-UL.