Computing the least action ground state of the nonlinear Schrödinger equation by a normalized gradient flow

• Published in: Journal of computational physics Vol. 471; p. 111675
• Main Author: Wang, Chushan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.12.2022
• Subjects: Least action ground state; Least energy ground state; Nonlinear Schrödinger equation; Normalized gradient flow; Orbital stability; Orbital stability; Least energy ground state; Least action ground state; Normalized gradient flow; Nonlinear Schrödinger equation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111675

In this paper, we generalize the normalized gradient flow method which was first applied to computing the least energy ground state to compute the least action ground state. A continuous normalized gradient flow (CNGF) will be presented and the action diminishing property will be proved to provide a mathematical justification of the gradient flow with discrete normalization (GFDN). Then we use backward-forward Euler method to further discretize the GFDN in time which leads to the GFDN-BF scheme. It is shown that the GFDN-BF scheme preserves the positivity and diminishes the action unconditionally. We compare it with other three schemes which are modified from corresponding ones designed for the least energy ground state and the numerical results show that the GFDN-BF scheme performs much better than the others in accuracy, efficiency and robustness for large time steps. Extensive numerical results of least action ground states for several types of potentials are provided. We also use our numerical results to verify some existing results and lead to some conjectures. •Propose a gradient flow with discrete normalization to compute the least action ground state.•Present a continuous normalized gradient flow to give mathematical justification.•Find the best scheme is the backward-forward Euler discretization of the discrete normalized gradient flow.•Report extensive numerical results and make a conjecture about the orbital stability.